ABSTRACT HOMOMORPHISMS OF ALGEBRAIC GROUPS: RIGIDITY AND GROUP ACTIONS By Joshua Ruiter A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics—Doctor of Philosophy 2022 PUBLIC ABSTRACT ABSTRACT HOMOMORPHISMS OF ALGEBRAIC GROUPS: RIGIDITY AND GROUP ACTIONS By Joshua Ruiter By way of analogy, an algebraic group is like a houseboat. A houseboat is both house and boat, so construction involves solving engineering problems common to houses (e.g. plumbing) and problems common to boats (e.g. floating). Additional complications emerge from the interplay of the two structures – for instance, the presence of water may require more safeguards in the electrical system. An algebraic group is a mathematical object with two structures: it is both an algebraic variety and a group. To study them, we use tools from from algebraic geometry and from group theory, as well as tools related to the interplay between the two structures. Mathe- maticians use group homomorphisms to describe relationships between different groups. To study algebraic groups, we usually use algebraic group homomorphisms, group homomor- phisms that are compatible with the algebraic structure. In the 1970’s, Armand Borel and Jacques Tits discovered that sometimes the “algebraic” assumption is superfluous. They proved that, in certain circumstances, an ordinary group homomorphism between two alge- braic groups has to be almost algebraic. After establishing their theorem, Borel and Tits conjectured that this was just one exam- ple of a more general phenomenon. Since then, mathematicians have made partial progress in confirming this, and this thesis extends that work by verifying the conjecture for a family of algebraic groups called special unitary groups. We hope that the methods developed here can later prove the conjecture for the larger family of quasi-split algebraic groups. ABSTRACT ABSTRACT HOMOMORPHISMS OF ALGEBRAIC GROUPS: RIGIDITY AND GROUP ACTIONS By Joshua Ruiter We investigate two related problems involving abstract homomorphisms between the groups of rational points of algebraic groups. First, we show that under appropriate as- sumptions, abstract representations of quasi-split special unitary groups associated with quadratic extensions of the field of definition have standard descriptions, i.e. can be factored as a group homomorphism induced by a morphism of algebras, followed by a homomorphism arising from a morphism of algebraic groups. This establishes a new case of a longstanding conjecture of Borel and Tits. In the second part, we apply existing results on standard de- scriptions for abstract representations of Chevalley groups to study some rigidity properties of actions of elementary subgroups on algebraic varieties. The thesis is organized as follows. To provide context for the study of abstract homo- morphisms, in §1 we give a historical overview of key developments going back to Cartan’s work on homomorphisms of Lie groups. In §2, we prove our rigidity result for special unitary groups, using a strategy inspired by [Rap11] which depends crucially on the construction of certain algebraic rings associated to abstract representations. In §3, we apply existing rigidity statements for representations of elementary subgroups of Chevalley groups to study rigidity properties of these groups acting on affine algebraic varieties and projective surfaces. We discuss some open questions and plans for future work in §4. In the appendices, we collect some relevant background material on algebraic rings, and also provide details on the computations of commutator relations needed for the constructions in §2. ACKNOWLEDGMENTS I am thankful to my advisor, Igor Rapinchuk, for advice in choosing a problem and guidance while working on it, as well as very clear exposition in his courses. I am also thankful to Rajesh Kulkarni for his excellent teaching in many courses and seminars, especially a summer course on algebraic groups. Thanks to Anastasia Stavrova for several helpful discussions about root systems and Steinberg groups. Finally, thanks to my parents, siblings, and friends for their support throughout my years at Michigan State University. iii TABLE OF CONTENTS LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v KEY TO SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Chapter 1 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Chapter 2 Abstract homomorphisms of special unitary groups . . . . . . . 9 2.1 Special unitary group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Elementary subgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Steinberg group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.4 Algebraic ring associated to an abstract representation . . . . . . . . . . . . 29 2.5 Rationality of σ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Chapter 3 Group actions on varieties . . . . . . . . . . . . . . . . . . . . . . 45 3.1 Elementary groups acting on affine varieties . . . . . . . . . . . . . . . . . . 47 3.2 Elementary groups acting on projective surfaces . . . . . . . . . . . . . . . . 57 Chapter 4 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.1 Algebraic rings associated to abstract representations . . . . . . . . . . . . . 63 4.2 Steinberg symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Appendix A Algebraic rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Appendix B Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Appendix C Logical dependency chart . . . . . . . . . . . . . . . . . . . . . . . . 104 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 iv LIST OF FIGURES Figure 2.1: The nontrivial automorphism of the A2n−1 Dynkin diagram . . . . . . . 14 Figure 2.2: Orbits in A2n−1 corresponding to nodes in Cn . . . . . . . . . . . . . . . 14 v KEY TO SYMBOLS A Algebraic ring associated to an abstract representation Aα Algebraic ring associated to a particular root α An Root system of type An An Affine space of dimension n AffAlgGpk Category of affine algebraic groups (representable functors Algk → Gp) Algk Category of finitely-generated commutative k-algebras αi Character which picks off the ith diagonal entry of the diagonal α, β (Relative) roots, elements of Φk (α, β) Positive integer linear combinations of roots α and β Aut(X) Group of biregular automorphisms, identified with a subgroup of Bir(X) Bα Algebraic variety from Lemma 2.4.2 Bir(X) Group of birational automorphisms of a variety X C Complex numbers cα (u, v) Unitary Steinberg symbol cij 1 if i < j, −1 if i > j Cn Root system of type Cn conjϕ Conjugation by ϕ √ d Primitive element of the quadratic extension L/k ∆ Finite-index subgroup of Γ (Dn ) True if dimK Derg (R, K) ≤ n for all g Derg (R, K) Space of g-derivations of R into K (−)δ Identity if δ = 1, conjugation if δ = −1 eij (x) Matrix with x in ij-th entry, ones on the diagonal, zeros elsewhere Eij (x) Matrix with x in ij-th entry, zeros elsewhere f Ring homomorphism k → A with Zariski-dense image vi fα Ring homomorphism k → Aα with Zariski-dense image F Group homomorphism G(k) → G(A) induced by f Fe Group homomorphism G(k) e → G(A) e induced by f G Special unitary group SU2n (L, h) Ge Steinberg group of G G(k) Group of k-rational points of G G0 Split subgroup of a group G G0 (k) Group of k-rational points of G0 GA Base change of G to A G(R) Group of points of G = SU2n (L, h) over a k-algebra R G(R) e Steinberg group of G(R) G(A, J) Congruence subgroup of G(A) for an ideal J ⊂ A Gal(L/k) Galois group of L/k, order 2 Γ, G+ , G(R)+ Classical elementary subgroup, generated by k-points of unipotent radicals of k-defined parabolic subgroups Γ.f Orbit of f for an action of the group Γ Ga Additive group, Ga (k) = (k, +) Gm Multiplicative group, Gm (k) = (k × , ×) GLn General linear group GL(V ) General linear group of a vector space V Gp Category of groups h Skew-hermitian form on V with maximal Witt index H H = ρ(G(k)) H Quotient group H/Z(H) hα (v) Weyl-group element hα (v) e Lift of hα (v) to Steinberg group inv Inversion map from a group to itself, G 7→ G, g 7→ g −1 vii J Jacobson radical k Field k× Multiplicative group of a field k, k × = k \ {0} k[x] Ring of polynomials in the variable x with coefficients in k K Field, usually assumed char K = 0 and algebraically closed K[X] Coordinate ring of a variety X √ L Quadratic extension of k, L = k( d) LieGp Category of real Lie groups Mn (R) n × n matrices with entries in a ring R αβ Nij Homogeneous polynomial map arising in Steinberg commutator formula Nice pair 2 ∈ R× if Φ contains a copy of B2 , and 2, 3 ∈ R× if Φ is type G2 ν Inverse map to π in Lemma 2.4.2 Φ Absolute root system (with respect to a fixed maximal torus) Φk Relative root system (with respect to a fixed maximal k-split torus) π Compatibility isomorphism of Lemma 2.4.2 πR Canonical homomorphism G(R) e → G(R) ψα Functions arising in Proposition 2.4.6 ρ Abstract representation G(k) → GLm (K) ρ0 Restriction of ρ to split subgroup G0 (k) r Group action r : Γ → Aut(X) r∗ Associated action on coordinate ring, r∗ : Γ → Aut(K[X]) R Arbitrary k-algebra RL Tensor product/extension of scalars R ⊗k L Ru (H) Unipotent radical of H RA/k Weil restriction/restriction of scalars functor rA/k Canonical isomorphism associated with RA/k Ru Unipotent radical viii S Maximal k-split torus (contained in T ) SLn Special linear group Sp2n Symplectic group SU2n (L, h) Isometry group of hermitian form h σ Map constructed in Theorem 2.5.6 σ Map constructed in Proposition 2.5.2 σ e Map constructed in Proposition 2.4.8 Sym(V ) Symmetric algebra on a vector space V (T ) Kazhdan’s property (T) T Maximal torus (containing S) τ Unique nontrivial Galois automorphism in Gal(L/k) TopGp Category of topological groups Tr Field trace map L → k, or the extension RL → R Uα Root subgroup associated to α, Uα (R) = Xα (Vα (R)) V L2n , viewed as an L-vector space Vα Vector space, dimension equal to dimension of root space for α wα β Reflection of β across the hyperplane perpendicular to α wα (v) Weyl-group element weα (v) Lift of wα (v) to Steinberg group Xα Root subgroup embedding Vα (R) → G(R) Xeα (v) Generator of Steinberg group X ∗ (S) Character group Hom(S, Gm ) [x, y] Commutator xyx−1 y −1 (Z) True if Z(H) ∩ U = {e} Z(H) Center of H Convention: All rings are assumed to be unital, associative, and commutative. ix Chapter 1 Rigidity In this chapter, we provide historical background on rigidity for algebraic groups, starting with Cartan’s rigidity result for Lie groups and leading into the Borel-Tits conjecture. We discuss partial results towards the Borel-Tits conjecture, and outline how the current work fits into and extends known results. One of the earliest forms of rigidity is Cartan’s theorem regarding continuous group homomorphisms between Lie groups. In general, a continuous map f : M → N between smooth manifolds cannot be expected to be smooth. However, Cartan’s theorem says that if M, N are Lie groups and f is a group homomorphism, then continuity implies smoothness1 . In other words, group homomorphisms between Lie groups are much more rigid that they first appear. Following Cartan, there was a considerable amount of work on rigidity properties of classical groups, by people including Schreier and van der Waerden [SVdW28], Dieudonné [Die80], and O’Meara [HO89]. However, all of these results were obtained in a somewhat ad hoc manner. The first organized approach was taken by Steinberg in [Ste16], who systemat- ically studied all simply-connected Chevalley groups over fields, culminating in the rigidity result [Ste16, Theorem 30], which gives a structured factorization description for any abstract 1 This goes back to [Car30] and [vN29], though Cartan’s statement is not in these terms. Cartan proved the closed-subgroup theorem, of which this is a corollary. See [Lee13, Theorem 20.12] for the closed-subgroup theorem, and [Lee13, Exercise 20-11(b)] for the corollary as stated here. 1 automorphism of the group of points of a Chevalley group over a perfect field. Later work of Borel and Tits in [BT73] generalized Steinberg’s results and focused the subject around a main conjecture. Before discussing this conjecture, we motivate the transition from Cartan’s theorem to analogous statements about algebraic groups. To translate Cartan’s theorem into a conjecture for algebraic groups, we first rephrase it in more categorical terms: it is equivalent to the statement that the forgetful functor LieGp → TopGp is full. We replace Lie groups with the category AffAlgGpk of affine algebraic groups over a field k (viewed as representable functors Algk → Gp, where Algk is the category of finitely generated commutative k-algebras), and replace topological groups by the category Gp of abstract groups (groups with no additional structure). We replace the forgetful functor LieGp → TopGp with the functor P : AffAlgGpk → Gp, G 7→ G(k) We think of this as a parallel to the forgetful functor in the sense that G(k) “forgets” the algebraic group structure of G. Then Cartan’s theorem suggests the question: is P full? We introduce some additional terminology to rephrase this more concretely. Definition 1.0.1. Let G, G0 be algebraic k-groups. An abstract group homomorphism ρ : G(k) → G0 (k) is algebraic if there is a morphism of algebraic groups σ : G → G0 such that ρ coincides with the map induced by σ on k-points. In other words, ρ is algebraic if it is in the image of the functor P : AffAlgGpk → Gp. In the analogy between algebraic groups and Lie groups, algebraic group homomorphisms correspond to smooth group homomorphisms and abstract homomorphisms correspond to continuous homomorphisms. Our earlier question of whether AffAlgGpk → Gp is full can 2 now be rephrased as: is every abstract group homomorphism ρ : G(k) → G0 (k) algebraic? Unfortunately, the answer is no. We give some counterexamples. Example 1.0.2. Let k = C and G = G0 = Ga be the additive group and let ρ : C → C be complex conjugation. All algebraic group homomorphisms Ga → Ga are given by (linear) polynomials, which ρ is not, so ρ is not algebraic. A similar counterexample is obtained by replacing Ga by Gm . More generally, let G = G0 be an algebraic subgroup of GLn and let ρ : G(C) → G(C) be complex conjugation on each matrix entry; this is not algebraic. Example 1.0.3. Let k be a perfect field of characteristic p > 0, let G = G0 = Ga and let ρ : k → k, x 7→ x1/p be the inverse of the Frobenius map. Then ρ : Ga (k) → Ga (k) is not algebraic. Example 1.0.4. Let k be a field, and let G = G0 = GLn over k. Let φ : k → k be a field automorphism, which induces a map ρ : GLn (k) → GLn (k) applying φ to each matrix entry. If φ is not algebraic (as an abstract homomorphism Ga → Ga ), then ρ is also not algebraic. As these examples demonstrate, field automorphisms are a major source of non-algebraic homomorphisms between linear algebraic groups. However, the general philosophy of rigidity is that maps arising from field automorphisms (and more generally, morphisms of algebras) are essentially the only obstruction to an abstract homomorphism being algebraic. To explain the source of this philosophy, we formulate one of the main results of the landmark 1973 paper of Borel and Tits [BT73]. Let k, k 0 be infinite fields and let G, G0 be algebraic groups over k, k 0 respectively. Suppose that G is absolutely almost simple k- isotropic, and G0 is absolutely simple adjoint. Let G+ denote the elementary subgroup of G(k), the group generated by the k-points of unipotent radicals of k-defined parabolic subgroups, and let ρ : G+ → G0 (k 0 ) be an abstract homomorphism with Zariski-dense 3 image. Then there exists a field embedding f : k → k 0 and a morphism of algebraic groups σ : G → G0 such that ρ = σ ◦ F |G+ , where F : G(k) → G(k 0 ) is the group homomorphism induced by f . We depict this in the following commutative diagram. ρ G+ G0 (k 0 ) F σ G(k 0 ) A factorization of ρ as above is called a standard description. Borel and Tits also ex- tended of this result in [BT73, Theorem 8.16], where, roughly speaking, the hypothesis on G0 is replaced by assuming that G0 is reductive. Borel and Tits also gave the following example, which illustrates two aspects of the subject: (1) more abstract homomorphisms may have standard descriptions if we allow morphisms of algebras rather than just field embeddings, and (2) the image of a reductive group under an abstract homomorphism need not be reductive. Example 1.0.5. Fix an infinite field k and let G ⊂ GLn be a simple linear algebraic k- group, with Lie algebra g ⊂ Mn . Let δ : k → k be a nontrivial derivation, which induces a map ∆ : G(k) → G(k) by applying δ on each matrix entry2 . Let G0 = g n G, using the adjoint representation of G on g. Define   ρ : G(k) → G0 (k) g 7→ g −1 · ∆(g), g More conceptually, ρ can be described in the following way which explicitly shows how ρ factors as predicted by the result of Borel and Tits. Let A = k[ε] with ε2 = 0, and note that G(A) ∼ = G0 (k) via the algebraic isomorphism   σ : G(A) → G0 (k) X + Y ε 7→ X −1 · Y, X 2 For concreteness, one can take G = SL , g = sl , k = C(x), and δ = d to be the differentiation n n dx operator. 4 Define f : k → A by f (x) = x + δ(x)ε, which induces F : G(k) → G(A) by applying f to each matrix entry. It is straightforward to verify ρ = σ ◦ F , i.e. ρ essentially arises from f . ρ G(k) G0 (k) F σ G(A) To summarize, ρ is a non-algebraic morphism between groups of k-points which essentially arises from a morphism of k-algebras f : k → k[ε]. Also, ρ has Zariski-dense image in g n G. In particular, even if G is semisimple, the Zariski closure of the image ρ(G(k)) = G0 (k) is not reductive (it has nontrivial unipotent radical g). So in general, the image of a reductive group under an abstract morphism need not be reductive. Furthermore, the example demonstrates that in order to obtain a standard description for an abstract homomorphism, it may not be possible to factor using a map induced by a field embedding; it may be necessary to use a map induced by a morphism of k-algebras as F is in the example. This led them to formulate the following conjecture [BT73, 8.19]. Conjecture 1.0.6 (BT). Let G and G0 be algebraic groups defined over infinite fields k and k 0 , respectively. If ρ : G(k) → G0 (k 0 ) is any abstract homomorphism such that ρ(G+ ) is Zariski-dense in G0 (k 0 ), then there exists a commutative finite-dimensional k 0 -algebra A and a ring homomorphism f : k → A such that ρ = σ ◦ rA/k0 ◦ F where F : G(k) → GA (A) is induced by f , rA/k0 : GA (A) → RA/k0 (GA )(k 0 ) is the canonical isomorphism, and σ is a rational k 0 -morphism of RA/k0 (GA ) to G0 . In the conjecture, GA is the group obtained by base change from k to A, and RA/k0 is the functor of Weil restriction of scalars. Generalizing our previous usage, if an abstract 5 homomorphism ρ : G(k) → G0 (k 0 ) admits a factorization as in (BT), we will say that ρ has a standard description. We think of (BT) as a generalized, algebraic version of Cartan’s theorem. Notably, it explains our previous counterexamples. Example 1.0.7. We return to the setting of Example 1.0.2. Let k = k 0 = C and G = G0 ⊂ GLn , and let ρ : G(C) → G(C) be complex conjugation on each entry. Let A = C, let f : C → C be complex conjugation, which induces F = ρ : G(C) → G(C), and let σ = Id. This gives a (somewhat vacuous) standard description of ρ. ρ G(C) G0 (C) F σ=Id G(C) Around the time Conjecture 1.0.6 was formulated, Tits sketched an argument in [Tit71, §4] for the case k = k 0 = R. Later, Weisfeiler [Wei81] proved a case in which G is split by a quadratic extension and ρ is an abstract isomorphism. Seitz [Sei97] obtained a result when k is a perfect field of positive characteristic, and L. Lifschitz and A. Rapinchuk [LR01] gave (essentially) a proof of (BT) in the case where G is an absolutely simple, simply-connected Chevalley group, char k = 0, and G0 has commutative unipotent radical. Note that using the results of [CGP15, Chapter 9] (particularly Proposition 9.9.1), it is possible to give counterexamples to (BT) over any field k of characteristic 2 such that [k : k 2 ] = 2, using perfect and k-simple groups, so we do not expect (BT) to hold over fields of characteristic 2. Most recently, I. Rapinchuk obtained significant results towards (BT) in [Rap11], [Rap13], and [Rap19], essentially resolving the conjecture for all split groups (avoiding characteristic 2 and 3), including results for groups over more general commutative rings. In [Rap11], he introduced a method for studying abstract representations of the elementary subgroups of simply-connected Chevalley groups over commutative rings based on the construction 6 and analysis of certain algebraic rings. These techniques led to a general result on abstract representations that, in particular, yielded (BT) in the case where k is a field of characteristic 6= 2 or 3, k 0 = K is an algebraically closed field of characteristic zero, and G is a split simply- connected k-group. Subsequently, this approach was extended in [Rap13] to confirm (BT) for abstract representations of groups of the form SLn,D , where D is a finite-dimensional central division algebra over a field of characteristic zero. We refer the reader to [Rap15] for a more extensive overview of work on (BT) and its connections to various classical forms of rigidity. Except for results concerning the groups SLn,D , essentially all existing progress on (BT) has concerned split groups. Our main contribution is to extend the methods of [Rap11] to prove (BT) for a class of quasi-split special unitary groups (see §2). The primary obstacle in extending results to non-split groups is the fact that root spaces and root subgroups are no longer necessarily 1-dimensional. In the split case, analysis of abstract representations is aided by a construction involving SLn going back to Kassabov and Sapir [KS09], where they put an algebraic ring structure on the closure of the image (under an abstract representation) of a 1-dimensional root subgroup. The generalization of this construction to more general Chevalley groups is the heart of the strategy of [Rap11] to prove (BT) for those groups. We have managed to extend this construction to a quasi-split group with 2-dimensional (relative) root subgroups. We hope that eventually the method can be further extended to more quasi-split groups. Our second contribution to the study of rigidity of a more geometric nature. In §3, we apply rigidity statements in the vein of (BT) from [Rap19] to obtain rigidity statements for elementary subgroups of Chevalley groups acting on varieties. We think of these results as a kind of algebraic analog of the Zimmer program, where we replace diffeomorphism groups 7 of smooth manifolds by biregular or birational automorphism groups of varieties. 8 Chapter 2 Abstract homomorphisms of special unitary groups In this chapter, we prove that the conjecture of Borel and Tits holds for abstract represen- tations of certain even-dimensional quasi-split special unitary groups, modulo an additional technical hypothesis on a certain unipotent radical. The precise statement is as follows. √ Theorem 2.0.1. Let L = k( d) be a quadratic extension of a field k of characteristic zero, and for n ≥ 2, set G = SU2n (L, h) to be the special unitary group of a (skew-)hermitian form h : L2n × L2n → L of maximal Witt index. Let K be an algebraically closed field of characteristic zero and consider an abstract representation ρ : G(k) → GLm (K). Set H = ρ(G(k)) to be the Zariski closure of the image of ρ. Then if the unipotent radical U = Ru (H) of H is commutative, there exists a commutative finite-dimensional K-algebra A, a ring homomorphism f : k → A with Zariski-dense image, and a morphism of algebraic K-groups σ : G(A) → H such that ρ = σ ◦ F , where F : G(k) → G(A) is the group homomorphism induced by f . ρ G(k) GLm (K) F σ G(A) 9 In the statement of the theorem, we are using the functor of restriction of scalars to view G(A) as an algebraic K-group. Namely, denoting by GA the base change of G from k to A, restriction of scalars gives a natural isomorphism rA/K : GA (A) → RA/K (GA )(K). Since A is a finite-dimensional K-algebra, RA/K (GA )(K) is an affine algebraic K-group, and the isomorphism rA/K allows us to endow G(A) = GA (A) with the structure of an algebraic K-group. Note that the group H appearing in the theorem is connected by Lemma 2.5.1. Also note that there is no meaningful difference between assuming h is hermitian or skew- hermitian, see Remark 2.1.2. The proof of Theorem 2.0.1 spans this chapter, and proceeds as follows. First, since G = SU2n (L, h) is a simply-connected k-group with relative root system Φk of type Cn , it follows that G contains a k-split simply connected k-group G0 = Sp2n of type Cn (see [BT65, Théorème 7.2] or [CGP15, Theorem C.2.30]). We describe one particular such split subgroup in Definition 2.2.5. Given an abstract representation ρ : G(k) → GLm (K), we consider the restriction of ρ to G0 (k), and use the construction in [Rap11] to associate to ρ|G (k) an 0 algebraic ring A, as well as a ring homomorphism f : k → A with Zariski-dense image. Since k and K are both fields of characteristic zero, A is a finite-dimensional K-algebra by [Rap11, Lemma 2.13(ii), Proposition 2.14]. See Appendix A for results we use concerning algebraic rings. In the methodology of [Rap11], the algebraic ring A plays a central role in proving that ρ|G (k) has a standard description. We show here that A also suffices for the analysis of 0 ρ. More precisely, following the general strategy of [Rap11] and [Rap13], we first show that ρ lifts to a representation σe : G(A) e → GLm (K), where G(A) e is the generalized Steinberg group introduced by Stavrova [Sta20] (which builds on an earlier construction of Deodhar [Deo78]). Then, using the fact that the kernel of the canonical map G(A)e → G(A) is central 10 (which extends a result of Stavrova to the present situation), together with our assumption that the unipotent radical Ru (H) is commutative, we establish the existence of the required algebraic representation σ : G(A) → GLm (K). The structure of this chapter is as follows. We begin by describing the special unitary groups SU2n (L, h) (§2.1), their elementary subgroups (§2.2), and associated Steinberg groups (§2.3). Then in §2.4 we describe an algebraic ring A associated with an abstract representa- tion of SU2n (L, h), describe how it is a commutative finite-dimensional K-algebra as claimed by the theorem, and show that ρ lifts to a representation σ e of G(A). e Finally, in §2.5, we descend σe to obtain the morphism σ and finish the proof of the theorem. The main results of this chapter also appear in more condensed form in the paper [RR22]. 2.1 Special unitary group We define the special unitary group SU2n (L, h). Let k be a field of characteristic zero and √ let L/k be a quadratic extension, so L = k( d) for some d ∈ k. We set τ : L → L to be the nontrivial element of Gal(L/k) and note that for any commutative k-algebra R, the action of τ naturally extends to RL := R ⊗k L via the second factor. More precisely, given a simple tensor a ⊗ b ∈ R ⊗k L with a ∈ R and b ∈ L, τ acts by τ (a ⊗ b) = a ⊗ τ (b) We write τ (x) = x for x ∈ RL . Next, fix an integer n ≥ 2 and let V = L2n be a 2n- dimensional L-vector space equipped with a (skew-)hermitian form h : V × V → L, and assume that h has maximal Witt index. Because of the Witt index assumption, with respect to a suitable basis of V the matrix of h is 11   0 −1 1 0    H=  ..  .     0 −1 1 0 Definition 2.1.1. With k, L, h as above, let G = SU2n (L, h) be the isometry group of h. Explicitly in terms of matrices, for a commutative k-algebra R, we have G(R) = {X ∈ SL2n (RL ) | X ∗ HX = H}, where for X = (aij ), we let X ∗ = (aji ) denote the conjugate transpose matrix. Note in particular that the group of k-points G(k) is a group of matrices with entries in L, not just in k. For the rest of this chapter, G denotes the special unitary group SU2n (L, h). The choice of basis (or equivalently the choice of matrix representation for h) only affects G up to inner automorphism. Remark 2.1.2. The special unitary groups considered here belong to a larger class of unitary groups described in [Bor91, V.23.9] (also see [Mil17, Theorem 24.44, Remark 24.46] and [PR94, Ch. 2, §2.3.3]). In a more general setting considered by Borel, L can be any division √ algebra over k( d), τ is an involution of the second kind on L, and the form h may have arbitrary Witt index. As noted in [Bor91, V.23.8], “there is no essential difference” between hermitian and skew-hermitian when dealing with an involution of the second kind, in the sense that if the form h is τ -hermitian then a scalar multiple of h is τ -skew-hermitian. The group G also belongs to the class of unitary groups studied by Hahn and O’Meara in [HO89, Ch 5]. 12 Remark 2.1.3. G is a k-form of SL2n . More precisely, G becomes isomorphic to SL2n over L, as we now describe. Let R be an L-algebra. Then we have the following sequence of group homomorphisms, all of which are functorial in R. SU2n (R) ,→ SL2n (RL ) → SL2n (R2 ) → SL2n (R)2 → SL2n (R) The first map is inclusion, and the second is induced by the following isomorphism of L- algebras with involution1 . RL ∼ = R2 √  √ √  a ⊗ 1 + b ⊗ d 7→ a + b d, a − b d a, b ∈ R The third map takes matrices with ordered pair entries and turns it into an ordered pair of matrices, and the final map is projection onto the first copy of SL2n (R). The entire composition gives an isomorphism SU2n (R) ∼ = SL2n (R) which is functorial in R. Let G be any quasi-split k-group and let L be a splitting field for G, then extend if necessary so that L/k is Galois. Let S ⊂ T ⊂ G with S a maximal k-split torus and T a maximal torus (which splits over L). Let Φ = Φ(G, T ) be the absolute root system and Φk = Φ(G, S) be the relative root system, viewing Φ as a subset of the character group X ∗ (T ) and Φk as a subset of the character group X ∗ (S). The natural restriction map X ∗ (T ) → X ∗ (S) takes Φ onto Φk , and the Galois group Gal(L/k) acts on Φ by inducing automorphisms of the associated Dynkin diagram, and restriction of characters from T to S gives a bijection ∼ = {Gal(L/k)-orbits in Φ} Φk 1 The involution on R2 is (x, y) 7→ (y, x). In fact, this isomorphism the unique isomorphism R ∼ 2 L = R of L-algebras with involution, up to multiplying by the scalar −1. 13 That is, two absolute roots in Φ restrict to the same relative root if and only if they lie in the same Gal(L/k)-orbit. In the particular case of G = SU2n (L, h), Φ is type A2n−1 and Φk is type Cn . The Galois group is Gal(L/k) ∼ = Z/2Z hτ i. The generator τ acts as on the An−1 Dynkin diagram by reflection across the central node. This reflection is depicted in Figure 1 below. ··· ··· Figure 2.1: The nontrivial automorphism of the A2n−1 Dynkin diagram There are n orbits; one singleton orbit and (n − 1) orbits each containing two nodes. As depicted below in Figure 2, the (n − 1) orbits with two nodes correspond to the (n − 1) short roots in a base for Cn , and the lone singleton orbit corresponds to the long root in a base for Cn . A2n−1 orbits Cn ··· ∼ ··· < ··· Figure 2.2: Orbits in A2n−1 corresponding to nodes in Cn See §2.2 of [Deo78] and Table II of [Tit66] for more discussion on this correspondence for general quasi-split groups. 14 2.2 Elementary subgroup We describe the elementary subgroup of SU2n (L, h), first by recalling some aspects of the theory of elementary subgroups of isotropic reductive group schemes, developed by Petrov and Stavrova [PS08]. Suppose G is a reductive group scheme over a ring R that is isotropic of rank ≥ 1 (i.e. every semisimple normal R-subgroup of G contains a 1-dimensional split R-torus). Then G contains a pair of opposite parabolic R-subgroups P and P − that inter- sect properly every semisimple normal R-subgroup of G (such subgroups are called strictly parabolic). In [PS08], the corresponding elementary subgroup EP (R) is then defined as the subgroup of G(R) generated by UP (R) and UP − (R), where UP and UP − are the unipotent radicals of P and P − , respectively (note that when R = k is a field, then EP (k) coincides with the group G+ appearing in the Borel-Tits Conjecture 1.0.6). The main result of [PS08] (see also [Sta14, Theorem 2.4]) is that if for any maximal ideal m ⊂ R, the group GRm is isotropic of rank ≥ 2, then EP (R) does not depend on the choice of a strictly parabolic subgroup P . This assumption is automatically satisfied in all situations considered here, so to simplify notations, we will denote the elementary subgroup simply by E(R). The elementary subgroup is functorial in R (i.e. a ring homomorphism R1 → R2 gives rise to a group homomorphism E(R1 ) → E(R2 )) and it is compatible with finite products (i.e. E(R1 × · · · × Rn ) = E(R1 ) × · · · × E(Rn )). Furthermore, since G is a quasi-split simply- connected k-group, this observation and [Sta20, Lemma 5.2] imply that G(R) = E(R) for any ring R that is a finite product of local k-algebras. In particular G(k) = E(k), and G(A) = E(A) where A is the ring constructed later in Proposition 2.4.6 (note that while A may contain nontrivial idempotents, it is a product of local rings which do not). In analogy with elementary subgroups of Chevalley groups, Petrov and Stavrova provide 15 a description of E(R) in terms of generators that satisfy certain generalized Steinberg com- mutator relations. The following statement collects the relevant parts of [PS08, Theorem 2] and [Sta20, Lemma 2.14] in the case of G = SU2n (L, h) that will be needed for our analysis. Theorem 2.2.1. Let G = SU2n (L, h), fix a maximal k-split torus S ⊂ G, and denote by Φk the corresponding relative root system (of type Cn ), viewed as a subset of the character group X ∗ (S). Then for every α ∈ Φk , there exists a vector k-group scheme Vα and a closed embedding of schemes Xα : Vα → G such that for any k-algebra R, we have the following: (1) For any v, w ∈ Vα (R), Xα (v) · Xα (w) = Xα (v + w) In particular, Xα (0) = 1. (2) For any s ∈ S(R) and v ∈ Vα (R),   s · Xα (v) · s−1 = Xα α(s)v (3) (Steinberg commutator formula) For any α, β ∈ Φk such that α 6= ±β, and for all u ∈ Vα (R), v ∈ Vβ (R), h i   αβ Y Xα (u), Xβ (v) = Xiα+jβ Nij (u, v) i,j≥1 iα+jβ∈Φk αβ αβ for some polynomial maps Nij : Vα (R) × Vβ (R) → Viα+jβ (R). The map Nij is homogeneous of degree i in the first variable and homogeneous of degree j in the second variable. (4) The elementary subgroup E(R) is generated by the elements Xα (v) for all α ∈ Φk and all v ∈ Vα (R). 16 Remark 2.2.2. We make some remarks on the theorem. 1. The more general formulas in [PS08, Theorem 2] and [Sta20, Lemma 2.14] include complicated product terms in (1) and (2) above, but these terms are trivial in our case because Φk is reduced. 2. In view of the structure of the root system Cn , the product on the right hand side of (3) contains at most two terms with the possible values of i and j lying in {1, 2}. Moreover, when there are two terms, these commute with each other. In fact, all nontrivial commutator relations in Theorem 2.2.1 (3) arise from a copy of C2 or A2 sitting inside Cn . That is, if α, β ∈ Φk and α + β ∈ Φk so that the product on the  right side is nontrivial, then iα + jβ ∈ Φk : i, j ∈ Z≥0 sits inside a copy of C2 or A2 . See Lemma B.1 for a more precise statement. 3. The dimension of Vα is the dimension of the relative root space associated to α. Thus there are two possibilities, distinguished by the two root lengths in Cn : if α is a long root, then Vα = Ga , so that Vα (R) = R for any k-algebra R. If α is a short root, then Vα ∼ = (Ga )2 , and we have Vα (R) = R ⊗k L = RL (to make the identification (Ga )2 (R) = R2 ∼= RL , we use the fact that k 2 ∼ = L as a k-vector space). For the calculations that we will carry out in subsequent sections, it will be useful to make the statement of Theorem 2.2.1 more explicit, as follows. The full diagonal subgroup T ⊂ G is a non-split maximal torus, which contains the maximal k-split subtorus S consisting of elements of T fixed by conjugation. 17     s1  s−1           1    T (R) =   ..  : si ∈ RL  × .         sn     −1   sn       s1  s−1           1     S(R) = X ∈ T (R) : X = X =   ...  : si ∈ R ×          sn     −1   sn   Clearly T splits over L (this also follows from Remark 2.1.3). Let αi : T → Gm be the character which picks off the ith diagonal entry. We abuse notation slightly and denote the restriction αi |S by αi as well. The character group X ∗ (S) is free abelian with basis {α1 , α3 , . . . , α2n−1 }. A computation shows that the relative root spaces of G with respect to S are  Φk = {±2αi : i odd} ∪ ±αi ± αj : i 6= j, both odd 1 ≤ i, j, ≤ 2n Roots of the form ±2αi are long, and roots of the form ±αi ±αj with i 6= j are short. The root spaces for long roots are 1-dimensional, and root spaces for short roots are 2-dimensional. Furthermore, the morphisms Xα of Theorem 2.2.1 look as follows. For a ring R, we denote by Eij (x) ∈ M2n (R) the matrix with x in the ij-th entry and 0 in all other entries. Recall that for a k-algebra R, the conjugation map τ (x) = x extends to RL . We denote matrix transpose by X t . Then the root group morphisms for the long roots are X2αi (R) : R → G(R) x 7→ 1 + Ei,i+1 (x) X−2αi (R) : R → G(R) x 7→ X2αi (x)t = 1 + Ei+1,i (x), 18 and for the short roots, the morphisms are Xαi −αj (R) : RL → G(R) x 7→ 1 + Eij (x) − Ej+1,i+1 (x) X−αi +αj (R) : RL → G(R) x 7→ Xαi −αj (x)t = 1 + Eji (x) − Ei+1,j+1 (x), Xαi +αj (R) : RL → G(R) x 7→ 1 + Ei0 ,j 0 +1 (x) + Ej 0 ,i0 +1 (x), X−αi −αj (R) : RL → G(R) x 7→ Xαi +αj (x)t = 1 + Ej 0 +1,i0 (x) + Ei0 +1,j 0 (x), where for a pair i, j, we set i0 = min(i, j) and j 0 = max(i, j). Note that the definition of X−αi +αj is redundant (and consistent) with the definition of Xαi −αj . One can check by direct calculation that the maps Xα defined here have the properties asserted by Theorem 2.2.1. Immediately from the definitions, they have the additional property that negating a root α corresponds to taking the matrix transpose of Xα (x). X−α (x) = Xα (x)t ∀α ∈ Φk Example 2.2.3. We write out the maps Xα in the case n = 2.     1 v 1  1  v 1  X2α1 (v) =   X−2α1 (v) =    1   1  1 1     1 1  1   1  X2α3 (v) =   X−2α3 (v) =    1 v  1  1 v 1     1 v 1  1   1 −v  Xα1 −α3 (v) =   X−α1 +α3 (v) =    1   v 1  −v 1 1     1 v 1  1   1 v  Xα1 +α3 (v) =  v 1   X−α1 −α3 (v) =    1  1 v 1 19 αβ By direct calculation, we can obtain explicit formulas for the polynomial maps Nij appearing in Theorem 2.2.1. To formulate the result, given a k-algebra R, we let Tr : RL → R a 7→ a + ā denote the extension of the usual trace map TrL/k : L → k to RL . Also, for δ = ±1 and v ∈ RL , we define    v  δ=1 vδ = (2.1)   v  δ = −1 √ √ The notation is chosen so that ( d)δ = δ d. The following lemma makes use of this αβ notation to describe the maps Nij arising in all nontrivial Steinberg commutator relations for SU2n (L, h). Lemma 2.2.4. Let α, β ∈ Φk be relative roots such that α + β ∈ Φk , and let u ∈ Vα (R), v ∈ Vβ (R). (1) Suppose α, β are both short and α + β is short. Then relabelling α, β if necessary we have α = αi − αj , β = αj − α` for distinct indices i, j, `, and αβ βα N11 (u, v) = uv N11 (v, u) = −uv (2) Suppose α, β are both short and α + β is long. Then relabelling α, β if necessary we have α = ε(αi − αj ), β = ω(αi + αj ) for some ε = ±1, ω = ±1, with i < j, and αβ βα N11 (u, v) = ω Tr(u−εω v) N11 (v, u) = −ω Tr(u−εω v) (3) Suppose α is short and β long. Then we have α = εαi + ωαj and β = −ε2αi for some 20 ε = ±1, ω = ±1 and i 6= j, and αβ βα N11 (u, v) = ωu−cij v N11 (v, u) = −ωu−cij v αβ βα N21 (u, v) = −εωvuu N12 (v, u) = εωvuu where    1  ij αβ In particular, whenever it is defined, the map N11 is surjective. Proof. See Appendix B, Lemma B.4. A key step in the proof of Theorem 2.0.1 involves restricting a given representation ρ : G(k) → GLm (K) to a split subgroup G0 (k) = Sp2n (k), so we now describe one such split subgroup explicitly. Definition 2.2.5. Let H be the matrix from Definition 2.1.1. Let G0 (k) = G(k) ∩ GL2n (k) = {X ∈ SL2n (k) | X t HX = H} = Sp2n (k) √ Recall that if β ∈ Φk is a short root, then Vβ (k) = L = k ⊕ k d. Our calculation then shows that G0 (k) is the subgroup of G(k) = E(k) generated by the images of the maps Xα (k) : Vα (k) → G(k) for long roots α and by the images of the maps Xβ (k) : Vβ (k) → G(k) restricted to the first component for long roots β. Definition 2.2.6. For later use, following Steinberg (see [Ste16, Ch. 3]) we define the elements wα (v) = Xα (v) · X−α (−v −1 ) · Xα (v) hα (v) = wα (v) · wα (1)−1 21 for α ∈ Φk and v ∈ Vα (k)× . Note that wα (1), hα (1) ∈ G0 (k). We also have the following analog of Steinberg’s relation (R7), wα (1) · Xβ (v) · wα (1)−1 = Xwα β (ϕv), (2.2) where ϕ(v) = ±v±1 (so that ϕ2 = Id) and wα β denotes the action of the corresponding Weyl group element wα on the root β. The element wα (v) normalizes the split torus S(k), and wα (v)2 centralizes S(k), so wα (v) corresponds to an element of the Weyl group N (S)/Z(S) = N (S)/T of order 2. In particular, wα (1) corresponds to the element of the Weyl group of Φk which is reflection in the hyperplane orthogonal to α. In Appendix B, we do explicit computations to verify a case of equation (2.2) on the level of the Steinberg group (see §2.3 for the definition of the Steinberg group). Example 2.2.7. We write out the matrices wα (v) and hα (v) in the case n = 2.     v v −v −1   v −1  w2α1 (v) =   h2α1 (v) =    1   1  1 1 −v −1    −1  v v   v  w−2α1 (v) =   h−2α 1 (v) =    1   1  1 1     1 1  1   1  w2α3 (v) =   h2α3 (v) =    v   v  −v −1 v −1     1 1  1   1  w−2α3 (v) =  −1  h−2α 3 (v) =  −1   −v   v  v v 22     v v  v −1   v −1  wα1 −α3 (v) =  hα1 −α3 (v) =  −v −1 v −1       −v v  −1 −v −1    v  −v   v  w−α1 +α3 (v) =   h−α1 +α3 (v) =    v   v  v −1 v −1     v v  −v −1   v −1  wα1 +α3 (v) =   hα1 +α3 (v) =    v   v  −v −1 v −1 −v −1    −1  v  v   v  w−α1 −α3 (v) =  h−α1 −α3 (v) =  −v −1 v −1       v v Lemma 2.2.8. Let R be a k-algebra, let α ∈ Φk , and let u, v ∈ Vα (R)× . Then hα (v) is a diagonal matrix with diagonal entries from 1, v ±1 , v ±1 , wα (v) is a monomial matrix with  nonzero entries from ±v ±1 , ±v ±1 , and  hα (v)−1 = hα (v −1 ) hα (u) · hα (v) · hα (uv)−1 = 1 Proof. Obvious for n = 2 by inspection of the tables in Example 2.2.7. The general n ≥ 2 case reduces to the n = 2 case because hα (v), wα (v) are contained in the subgroup generated by Xα (v), X−α (v), which is contained in a copy of SU4 (L, h) sitting inside SU2n (L, h), corresponding to a copy of C2 sitting inside Cn . The relations follow immediately from the description of hα (v) as a diagonal matrix. The elements wα (v) correspond to similar elements described by Deodhar in [Deo78, Lemma 1.3] and also more explicitly in §2.322 . In particular, one can check that wα (v) = 2 There is potential for confusion between the notation here and in Deodhar – in his notation, w (u) is α defined for nontrivial u ∈ Uα (k), but our wα (v) is defined for v ∈ Vα (k)× . Since Xα : Vα (k) → Uα (k) is an isomorphism, this is merely a notational difference. 23 w−α −v −1 , or more explicitly,        Xα (v) · X−α −v −1 · Xα (v) = X−α −v −1 · Xα (v) · X−α −v −1 It follows from this and the uniqueness aspect of [Deo78, Lemma 1.3] that the element wα (v) we have defined agrees with Deodhar’s definition of wα (v). 2.3 Steinberg group Next we recall Stavrova’s generalization of the classical Steinberg group from [Sta20, Defi- nition 3.1], noting that this is in turn inspired by a construction of Deodhar [Deo78, §1.9]. We establish a key centrality property for a unitary version of the group K2 from algebraic K-theory (Proposition 2.3.2) which extends a centrality result of Stavrova [Sta20, Theorem 1.3]. Although Stavrova works in the general context of reductive group schemes, for the sake of concreteness, we will restrict ourselves to the case G = SU2n (L, h). As in the classical setting, for a k-algebra R, the generalized Steinberg group G(R) e is the abstract group generated by symbols X eα (v), for α ∈ Φk and v ∈ Vα (R), subject to the relations (1) and (3) of Theorem 2.2.1 but replacing Xα with X eα . More precisely, it is defined as follows. Definition 2.3.1. Let R be a k-algebra. The Steinberg group G(R) e is the group generated by symbols X eα (v), for all α ∈ Φk and all v ∈ Vα (R), subject to the relations (R1) For α ∈ Φk and v, w ∈ Vα (R), eα (v) · X X eα (w) = X eα (v + w); That is, Xeα is a group homomorphism Vα (R) → G(R). e 24 (R2) For α, β ∈ Φk such that α 6= ±β, and for all u ∈ Vα (R), v ∈ Vβ (R), h i   eiα+jβ N αβ (u, v) ; Y X eα (u), X eβ (v) = X ij i,j≥1 iα+jβ∈Φk αβ where Nij are the same maps as in Theorem 2.2.1(3). A morphism of k-algebras φ : R → R0 induces an additive group homomorphism φ : Vα (R) → Vα (R) which then induces a group homomorphism e 0)  G(R) e → G(R eα (v), 7→ X X eα φ(v) . This process of inducing maps on the Steinberg group is functorial in R. Furthermore, for every k-algebra R, we have a natural surjective homomorphism πR : G(R) e → E(R), eα (v) 7→ Xα (v). X The kernel of πR is a unitary analog of the group K2 (R) from classical algebraic K-theory. The main result of this subsection is the following statement, which partially extends [Sta20, Theorem 1.3]. Proposition 2.3.2. Suppose R is a k-algebra that is a finite product of local k-algebras. Then ker πR is a central subgroup of G(R).e Before proving this, we prove some lemmas. Lemma 2.3.3. Any generator X eα (v) of G(R) e can be written as a product of generators Xeγ (ui ) with γi 6= α and some ui ∈ Vγ (R). Furthermore, X eα (v) is contained in the com- i i mutator subgroup [G(R), e G(R)]. e In particular, G(R) e and E(R) are perfect groups, i.e. they coincide with their commutator subgroups3 . 3 See [LS11, Theorem 1] for a more general result on elementary subgroups of reductive groups being perfect. 25 Proof. First, suppose α is a long root. Then we can write α = γ1 + γ2 for appropriate short roots γ1 and γ2 , and it follows from (R2) that eα N γ1 γ2 (u1 , u2 ) h i   X eγ (u1 ), X 1 eγ (u2 ) = X 2 11 γ γ Since N111 2 is surjective (Lemma 2.2.4), we can find u1 ∈ Vγ1 (R) and u2 ∈ Vγ2 (R) so that γ γ N111 2 (u1 , u2 ) = v, which yields our claim in this case. Next, suppose α is short. Then we can write α = γ1 + γ2 for an appropriate long root γ1 and short root γ2 . Hence, by (R2) we have eα N γ1 γ2 (u1 , u2 ) · X eγ +2γ N γ1 γ2 (u1 , u2 ) h i     Xeγ (u1 ), X 1 eγ (u2 ) = X 2 11 1 2 12 γ γ Again by Lemma 2.2.4 we can choose u1 , u2 so that N111 2 (u1 , u2 ) = v. Multiplying both −1 eγ +2γ N γ1 γ2 (u1 , u2 )  sides by X 1 2 12 then yields our first claim. Furthermore, since γ1 + 2γ2 −1 γ1 γ2  is a long root, the preceding case shows that Xγ1 +2γ2 N12 (u1 , u2 ) e is contained in the commutator subgroup, and hence so is X eα (v). Thus, since all generators of G(R) e are contained in [G(R), e G(R)], e it follows that G(R) e is perfect. Since the natural map π : G(R) e → E(R) is surjective, we conclude that E(R) is perfect as well. Lemma 2.3.4. The Steinberg group commutes with finite products. That is, for any k- e 1 × · · · × Rn ) = G(R algebras R1 , . . . , Rn , we have G(R e 1 ) × · · · × G(Re n ). Proof. By induction, it suffices to show that G(A e × B) ∼ = G(A) e × G(B) e for any k-algebras A and B. We do this by mimicking an argument of Stein [Ste73, Lemma 2.12]; note that our result is not a special case as Stein considers only split groups. First, we note that the projections of A × B onto its components induce a surjective group homomorphism   e × B) → G(A) p : G(A e × G(B), e Xeα (v) = Xeα (v1 , v2 ) 7→ X eα (v1 ), X eα (v2 ) 26 Next we define a map in the reverse direction by s : G(A) e × G(B) e → G(A e × B)   Xeα (v1 ), 1 7→ X eα (v1 , 0)   1, Xα (v2 ) 7→ X e eα (0, v2 ) (This defines s on a set of generators for G(A) e × G(B), e and we then extend it to the whole group by multiplicativity.) It follows immediately from the definitions that p ◦ s and s ◦ p are the respective identity maps on generating sets. Thus, it remains to show that s is a homomorphism by verifying that s takes all the defining relations in G(A)× e G(B) e to relations e × B). We need to check that s preserves three kinds of relations: in G(A   (i) The defining relations of G(A)e applied to the generators X eα (v1 ), 1 .   (ii) The defining relations of G(B) applied to the generators 1, Xα (v2 ) . e e h   i (iii) Xeα (a), 1 , 1, X eβ (b) = 1 for all α, β ∈ Φk and all a ∈ Vα (A), b ∈ Vβ (B). It is clear that s preserves (i) and (ii), so it remains to show that s preserves (iii). First consider the case α 6= −β. Let a ∈ Vα (A) and b ∈ Vβ . Then h   i h i   αβ Y s Xeα (a), 1 , 1, X eβ (b) = X eα (a, 0), Xeβ (0, b) = Xiα+jβ Nij (a, 0), (0, b) i,j≥1 iα+jβ∈Φk αβ Since Nij is homogeneous of degree i in the first argument and of degree j in the second αβ  argument, it is at least linear in each argument. So each term of Nij (a, 0), (0, b) includes αβ  a factor of (a, 0) · (0, b) = (0, 0) = 0 ∈ A × B. Hence Nij (a, 0), (0, b) = 0 and the product is trivial, so s preserves (iii) when α 6= −β. 27 Now consider relation (iii) with α = −β. For any b ∈ B, by Lemma 2.3.3, the element Xe−α (0, b) ∈ G(0 e × B) ⊂ G(A e × B) can be written as a product Y X e−α (0, b) = Xeγ (0, ui ) i i where γi 6= −α for all i, with each ui ∈ B. Thus, we have " # h   i h i Y s Xeα (a), 1 , 1, X e−α (b) = X eα (a, 0), Xe−α (0, b) = X eα (a, 0), X eγ (0, ui ) i i Since γi 6= −α, by the previous case, X eα (a, 0) commutes with each factor X eγ (0, ui ). Con- i sequently, it commutes with the whole product, hence the last commutator vanishes. This shows that s preserves (iii) when α = −β, completing the proof. With the previous two lemmas in hand, we can proceed to the proof of Proposition 2.3.2. Proof of Proposition 2.3.2. Suppose R = R1 × · · · × Rn , where R1 , . . . , Rn are local k- algebras. Since the elementary subgroup and the Steinberg group both commute with finite products (Lemma 2.3.4), it follows that n Y πR = πR i , i=1 and hence n n e i) ∼ Y Y ker πR = ker πRi ⊂ G(R = G(R). e i=1 i=1 Applying [Sta20, Theorem 1.3] to each local factor Ri , we see that ker πRi is central in G(R e i ). Consequently, ker πR is central in G(R), e as claimed.  Our main application of Proposition 2.3.2 will be to the case where R = A is the algebra obtained in Proposition 2.4.6. This application is used in Proposition 2.5.2. 28 2.4 Algebraic ring associated to an abstract represen- tation With our descriptions of the elementary and Steinberg groups of SU2n (L, h), we now embark on the proof of Theorem 2.0.1, starting with the construction of A and f as in the statement of the theorem. In this section, we describe an algebraic ring A associated to an abstract represtation ρ : G(k) → GLm (K), then show how ρ can be lifted to a representation σ e of the Steinberg group G(A). e These techniques go back to a construction of Kassabov and Sapir [KS09], and more importantly extend the methods introduced in [Rap11] and [Rap13] to verify the Borel-Tits conjecure for split groups. Let ρ : G(k) → GLm (K) be an abstract representation (with K an algebraically closed field of characteristic zero). We set H = ρ(G(k)) to be the Zariski closure of the image of ρ. Furthermore, we let ρ0 : G0 (k) → GLm (K) denote the restriction ρ|G (k) (see Definition 0 2.2.5). By [Rap11, Theorem 3.1], we can associate to ρ0 an algebraic ring A, together with a ring homomorphism f : k → A with Zariski-dense image, as follows. Recall that if α ∈ Φk √ is a long root, then Vα (k) = k, whereas if α is short, then Vα (k) = L = k ⊕ k d. Definition 2.4.1. For α ∈ Φk , define Aα = ρ(Xα (k)) and fα : k → A α , u 7→ ρ0 (Xα (u)). Note that for α short, the root subgroup Xα (Vα (k)) has dimension 2 (over k), and in this case, Aα is the closure of the image of the 1-dimensional subgroup Xα (k) ⊂ Xα (Vα (k)), arising from the natural embedding k ,→ L = Vα (k). As shown in [Rap11, Theorem 3.1], each Aα has the structure of an algebraic ring (see Appendix A for some background on algebraic rings) — we recall that the addition operation 29 is obtained simply by restricting matrix multiplication in H to Aα , whereas the multiplication operation is defined using the Steinberg commutator relations. Moreover, for any α, β ∈ Φk , there exists an isomorphism παβ : Aα → Aβ of algebraic rings such that παβ ◦ fα = fβ ([Rap11, Lemma 3.3]). We denote this common algebraic ring by A, and, for each α ∈ Φk , we fix an isomorphism of algebraic rings πα : A → Aα such that παβ ◦ πα = πβ . So we have a ring homomorphism f : k → A with Zariski-dense image such that πα ◦ f = fα for all α ∈ Φk . The following commutative diagram depicts the situation. k fα f fβ A πα πβ ∼ = ∼ ∼ = = Aα παβ Aβ Also from [Rap11, Theorem 3.1], we have (injective) regular maps ψα1 : A → H satisfying (ψα1 ◦ f )(u) = (ρ0 ◦ Xα )(u) for all u ∈ k. In other words, the following diagram commutes. Xα k G0 (k) f ρ0 ψα1 A H The algebraic ring A plays a pivotal role in the proof of the main results of [Rap11], i.e. obtaining a standard description of ρ0 . It turns out that A also suffices for the analysis of the representation ρ. The precise statement that is needed in our context will be given in Proposition 2.4.6 below. First, we make the following construction. Given a short root α ∈ Φk , let us define a subset Bα ⊂ GLm (K).   √  Bα = ρ Xα k d We also define the following map:   √  gα : k → Bα , u 7→ ρ Xα u d . 30 Our next step is to give an isomorphism analogous to παβ to identify Aβ and Bα for some relative roots α, β. Rapinchuk constructs παβ in [Rap11, Lemma 3.3] as follows: for a particular pair of roots α, β ∈ B2 with α short, β long, and β − α ∈ Φk , παβ is given by    1 παβ (x) = x, fβ−α (2.3) 2 After translating through an identification B2 ∼ = C2 , for such a pair α, β as above, they span a copy of C2 sitting inside Φk consisting of the roots {±α, ±β, ±(β − α), ±(2α − β)}, where ±α and ±(β − α) are short and the others are long. All of the Steinberg commutator relations in the following lemma involve only this copy of C2 inside Φk . Lemma 2.4.2. Take the short root α = α1 − α3 and the long root β = −2α3 . Then there is an isomorphism of algebraic varieties π : Bα → Aβ such that π ◦ gα = fβ . Proof. We perform a similar computation to that in [Rap11, Lemma 3.3]. Define a regular map π : Bα → H by    −1 π(x) = x, gβ−α . 2d (Compare with equation (2.3).) Note that 2d is invertible as char k = 0. This commutator occurs inside GLm (K) and multiplication is regular, so π is regular. By Lemma 2.2.4 (2), α,β−α we have N11 (u, v) = − Tr(uv). Now let s ∈ k. Then "  √  √ !#   √    − d −1 π ◦ gα (s) = ρ ◦ Xα s d , ρ ◦ Xβ−α = ρ Xα s d , Xβ−α √ 2d 2 d √ −1       α,β−α −s  = ρ ◦ Xβ N11 s d, √ = ρ ◦ Xβ − Tr = ρ Xβ (s) = fβ (s) 2 d 2 This shows that π ◦ gα = fβ , and that π maps gα (k) into fβ (k). Since π is regular, it follows that π(Bα ) ⊂ Aβ . It remains to show that π is invertible. First, using Lemma 2.2.4 (3), we 31 obtain β,α−β N11 (v, u) = −uv β,α−β N12 (v, u) = vuu Let h = h2α−β (1/2) = h2α1 (1/2) be the element introduced in Definition 2.2.6 and define −1 · ρ(h)−1 .    ν : Aβ → Bα , ν(y) = ρ(h) · y, gα−β (−1) · y, gα−β (1) It is clear that ν is a regular map; we claim it is an inverse for π. Let t ∈ k. Using the commutator relation in Theorem 2.2.1(3), we have h  √ i  β,α−β  √   β,α−β  √  Xβ (t), Xα−β − d = Xα N11 t, − d · X2α−β N12 t, − d √    = Xα t d · X2α−β − td . Thus h  √ i h √ i−1 Xβ (t), Xα−β − d · Xβ (t), Xα−β d √  √ −1  √   −1   :   :     = Xα t d · X 2α−β   − td · X 2α−β    − td · Xα − t d = Xα 2t d .   We also have the relation h · Xα (2v) · h−1 = Xα (v) for all v ∈ L. Putting everything together, we obtain  h  √ i h √ i−1  ν ◦ fβ (t) = ρ h · Xβ (t), Xα−β − d · Xβ (t), Xα−β d ·h −1   √     √  = ρ h · Xα 2t d · h −1 = ρ Xα t d = gα (t). Thus, ν ◦ π and π ◦ ν are the respective identity maps on dense subsets of Aβ and Bα . Since they are regular, it follows that they are the respective identities on the whole space, so ν is the inverse of π as claimed. 32 Remark 2.4.3. Although we fixed the roots α and β in the preceding argument for the sake of concreteness, essentially the same calculations can be carried for any short root α and long root β such that α − β is a root (with appropriate modifications to the definitions of π and ν, depending on the signs arising in computing N11 and N12 ). We carry this out in Lemma B.9. Remark 2.4.4. In Appendix A, we describe an algebraic ring structure on Bα1 −α3 and show that π is not just an isomorphism of varieties, but an isomorphism of algebraic rings. Next, we make the following observation, which will streamline the proof of Proposition 2.4.6 by allowing us to consider just a single root of each length. In the statement, we refer to the group schemes Vα introduced in Theorem 2.2.1. Given a k-algebra homomorphism f : k → A, we denote by Vα (f ) : Vα (k) → Vα (A) the associated group homomorphism. Note that if α is a long root, then Vα (f ) can be identified with f , and if α is short, then Vα (f ) is √ √ the map a + b d 7→ f (a) + f (b) d — in particular, if α and β have the same length, then the homomorphisms Vα (f ) and Vβ (f ) coincide. Lemma 2.4.5. Let α, β ∈ Φk be roots of the same length. Then (1) There exists w ∈ E(k) such that for all v ∈ Vα (k), we have w · Xα (v) · w−1 = Xβ (ϕv), where ϕv = ±v±1 . (2) Let f : k → A be a k-algebra homomorphism. Suppose there exists a regular map ψα : Vα (A) → H such that ψα ◦ Vα (f ) = ρ ◦ Xα . Let w, ϕ be as in part (1), and define ψβ : Vβ (A) → H by ψβ (v) = ρ(w) · ψα (ϕv) · ρ(w)−1 Then ψβ is regular and satisfes ψβ ◦ Vβ (f ) = ρ ◦ Xβ . 33 Proof. (1) It is well-known that the Weyl group W acts transitively on roots of the same length, so there exists we ∈ W such that wα e = β. Write w e as a product of simple reflections, we = wγ1 · · · wγn for roots γi ∈ Φk . Using the relation (2.2), we have wγi (1) · Xα (v) · wγi (1)−1 = Xwγ α (ϕi v) i where ϕi v = ±v±1 . Let w = wγ1 (1) · · · wγn (1) and ϕ = ϕ1 · · · ϕn . Then repeatedly applying the above relation yields w · Xα (v) · w−1 = Xwγ ···wγn α (ϕ1 · · · ϕn v) = Xwαe (ϕv) = Xβ (ϕv) 1 Since each ϕi is a composition of negation and conjugation, ϕv = ±v±1 . (2) This is proved by a direct calculation using the result of part (1), the fact that ϕ2 = Id, and the assumption that ψα ◦ Vα (f ) = ρ ◦ Xα . Namely, let v ∈ Vβ (A). Then   ψβ ◦ Vβ (f )(v) = ρ(w) · ψα ϕ ◦ Vβ (f )(v) · ρ(w)−1   = ρ(w) · ψα ◦ Vα (f )(ϕv) · ρ(w)−1   = ρ(w) · ρ ◦ Xα (ϕv) · ρ(w)−1   = ρ w · Xα (ϕv) · w−1   = ρ Xβ (ϕ2 v) = ρ ◦ Xβ (v) We now come to one of the main statements of this section. Proposition 2.4.6. Let G = SU2n (L, h) and let ρ : G(k) → GLm (K) be an abstract representation, with K an algebraically closed field of characteristic zero. Set H = ρ(G(k)). 34 There exists a finite-dimensional K-algebra A, a ring homomorphism f : k → A with Zariski- dense image, and for each α ∈ Φk , a regular map ψα : Vα (A) → H such that ψα ◦ Vα (f ) = ρ ◦ Xα . Xα Vα (k) G(k) Vα (f ) ρ ψα Vα (A) H Proof. We take A and f : k → A to be the algebraic ring and ring homomorphism constructed from the restriction ρ|G (k) using [Rap11, Theorem 3.1], as described at the beginning of 0 this section. Recall that for each α ∈ Φk , there is a regular map ψα1 : Aα → H, and an isomorphism πα : A → Aα such that πα ◦ f = fα and ψα1 ◦ Vα (f )|k = ρ ◦ Xα |k . By Lemma 2.4.5, it suffices to construct ψα : Vα (A) → H satisfying ψα ◦ Vα (f ) = ρ ◦ Xα for a single root of each length. If α is a long root, then, since the corresponding root space is 1-dimensional, we can simply set ψα = ψα1 . Now let us consider the short root α = α1 − α3 . Then, taking β = −2α3 , Lemma 2.4.2 yields an isomorphism of algebraic varieties π : Bα → Aβ such that π −1 ◦ fβ = gα . Define ψα2 : A → H, ψα2 = ιB ◦ π −1 ◦ πβ , where πβ : A → Aβ is the previously fixed isomorphism satisfying fβ = πβ ◦f , and ιB : Bα ,→ √ H is the natural inclusion. Using the identification Vα (A) = AL ' A2 , v = v1 + v2 d 7→ (v1 , v2 ), we define ψα : Vα (A) → H, v = (v1 , v2 ) 7→ ψα1 (v1 ) · ψα2 (v2 ). Note that for any u ∈ k, we have (ψα2 ◦ f ) (u) = (ιB ◦ π −1 ◦ πβ ◦ f ) (u) = (ιB ◦ π −1 ◦ fβ )(u)  √  = (ιB ◦ gα )(u) = (ρ ◦ Xα ) u d 35 √ Hence, for v = (v1 , v2 ) ∈ Vα (k) = L = k ⊕ k d, we have         ψα ◦ Vα (f ) (v) = ψα f (v1 ), f (v2 ) = ψα1 ◦ f (v1 ) · ψα2 ◦ f (v2 ) =     √  = ρ ◦ Xα (v1 ) · ρ ◦ Xα v1 d =  √  = ρ ◦ Xα v1 + v2 d = (ρ ◦ Xα )(v) Thus, ψα ◦ Vα (f ) = ρ ◦ Xα , as needed. Remark 2.4.7. The algebraic ring A ∼ = Aβ ∼ = Bα above is a connected algebraic ring over the algebraically closed field K of characteristic zero, so it is a finite-dimensional K-algebra (see [Gre64, Proposition 5.1] or [Rap11, Lemma 2.13, Proposition 2.14], or Appendix A). By [Rap11, Lemma 2.9], A is artinian so it is a finite product of local rings, and each of those rings is a k-algebra. In particular, A is connected and satisfies the hypotheses of Proposition 2.3.2. This is a key input for Proposition 2.5.2. Note that since f : k → A has Zariski-dense image in A, for any root α the map Vα (f ) : Vα (k) → Vα (A) has Zariski-dense image in Vα (A). If α is long this is vacuous, and if α is short this just says that applying f to both components of k 2 has dense image in A2 . To conclude this section, we show the representation ρ can be lifted to a representation σ e of the Steinberg group G(A). e The precise statement is as follows. Proposition 2.4.8. Let ρ : G(k) → GLm (K) be an abstract representation, with K an algebraically closed field of characteristic zero. Set H = ρ(G(k)), and let A, f, ψα be as in Proposition 2.4.6. Let Fe : G(k) e → G(A) e be the homomorphism induced by f and πk : G(k) e → G(k) be the natural map. Then there exists a group homomorphism σ e : G(A) e → H such e ◦ Fe = ρ ◦ πk and σ that σ e◦X eα = ψα for all α ∈ Φk ; i.e. σ e makes the following diagrams commute. 36 Xeα πk Vα (A) G(A) e G(k) e G(k) Fe ρ ψα σ e σ H G(A) e e H Proof. In order for the relation σ e◦X eα = ψα to hold, we must define σ e on the generators of G(A) e by   σ e X eα (v) := ψα (v). To show that σ e is well defined, we need to verify that the relations (R1) and (R2) hold, replacing X eα with ψα . For this, we imitate the proof of [Rap11, Proposition 4.2], starting with (R1). To make the notation less burdensome, let us set f˜ = Vα (f ). Let a, b ∈ f˜(Vα (k)), and choose v, w ∈ Vα (k) such that f˜(v) = a and f˜(w) = b. Then     ˜ ˜ ψα (a) · ψα (b) = ψα ◦ f (v) · ψα ◦ f (w) = (ρ ◦ Xα )(v) · (ρ ◦ Xα )(w)     = ρ Xα (v) · Xα (w) = (ρ ◦ Xα )(v + w) = ψα ◦ f˜ (v + w)   = ψα f˜(v) + f˜(w) = ψα (a + b). Thus, we have two regular maps Vα (A) × Vα (A) → H given by (a, b) 7→ ψα (a) · ψα (b) and (a, b) 7→ ψα (a + b) that agree on the Zariski-dense subset f˜(Vα (k)) ⊂ Vα (A). So, they must coincide on all of Vα (A). This verifies (R1). By a similar calculation, for any α, β ∈ Φk with α 6= ±β, we have two regular maps Vα (A) × Vβ (A) → H given by h i   αβ Y (a, b) 7→ ψα (a), ψβ (b) and (a, b) 7→ ψiα+jβ Nij (a, b) i,j>0 that agree on the Zariski-dense subset f˜(Vα (k)), and hence on all of Vα (A). Thus, (R2) holds as well. This shows that σ e◦X e is well defined, and, by construction, satisfies σ eα = ψα 37 for all α ∈ Φk . Finally, for any α ∈ Φk and v ∈ Vα (k), we have          σ ◦ Fe) X (e eα (v) = σ e X eα f˜(v) = ψα f˜(v) = (ρ ◦ Xα )(v) = (ρ ◦ πk ) X eα (v) , from which it follows that σ e ◦ Fe = ρ ◦ πk . 2.5 Rationality of σ We retain the notations of the previous section. Namely, we let G = SU2n (L, h), consider an abstract representation ρ : G(k) → GLm (K), with K an algebraically closed field of characteristic zero, and set H = ρ(G(k)). By Proposition 2.4.6, one can associate to ρ an algebraic ring A together with a ring homomorphism f : k → A with Zariski-dense image. Then in Proposition 2.4.8, we constructed a group homomorphism σ e : G(A) e → H that lifts ρ to a representation of the Steinberg group G(A). e More precisely, the diagrams formed by the solid arrows below commute. Fe G(k) e G(A) e πk πA σ F e (2.4) G(k) G(A) σ ρ H (Here F and Fe denote the group homomorphisms induced by f .) To complete the proof of Theorem 2.0.1, it remains to show the existence, under the assumptions of the theorem, of a morphism of algebraic groups σ : G(A) → H as indicated in the diagram. Note that before this section, we have not made use of the assumption that Ru H is commutative; it appears in Lemma 2.5.5. Before addressing the algebraicity of σ, we show that H is connected and perfect. In the following lemma, H ◦ denotes the connected component of the identity of H. 38 Lemma 2.5.1. The homomorphism σ e : G(A) e → H is surjective and the algebraic group H   is connected and perfect. That is, H = H ◦ = σ e G(A) e = [H ◦ , H ◦ ] = [H, H]. Proof. Let H ⊂ H be the (abstract) subgroup generated by the elements ψα (v) = σ e◦Xeα (v) for all α ∈ Φk and v ∈ Vα (A), where ψα : Vα (A) → H are the regular maps introduced in Proposition 2.4.6. H = hψα (v) : v ∈ Vα (A), α ∈ Φk i   Since, by definition, the Xα (v) generate G(A), their images generate σ e e e G(A) , so H = e σ e(G(A)). e Now, A is connected by Remark 2.4.7, so Vα (A) is connected, and hence ψα (Vα (A)) is connected. Thus, it follows from [Bor91, Proposition 2.2] that H is Zariski-closed and connected, so H ⊂ H ◦ . On the other hand, H contains ρ(E(k)), which is Zariski-dense in H. So, H is Zariski-dense in H, and since H is closed, we see that H = H. This shows that H = H = H ◦ . Furthermore, by Lemma 2.3.3, G(A) e is equal to its commutator subgroup, so the same is true for H = σ e(G(A)). e In the remainder of the this section, we will complete the proof of Theorem 2.0.1 using a strategy inspired by that of [Rap11, §5.6]. Namely, let Z(H) be the center of H, set H = H/Z(H), and denote by ν : H → H the corresponding quotient map. We first show that σe gives rise to a group homomorphism σ : G(A) → H satisfying σ ◦ πA = ν ◦ σ e, and verify that σ is in fact a morphism of algebraic groups. Then, using the assumption that the unipotent radical U = Ru (H) is commutative (together with our standing hypothesis that char K = 0), we lift σ to the required morphism of algebraic groups σ : G(A) → H. Proposition 2.5.2. There exists a group homomorphism σ : G(A) → H such that σ ◦ πA = ν◦σ e, where πA : G(A) e → G(A) is the canonical map. 39 πA G(A) e G(A) σ e σ H ν H Proof. First, it follows from Remark 2.4.7 that G(A) = E(A). Moreover, since πA is sur- jective, we have E(A) ∼ = G(A)/ e ker πA . Now, according to Proposition 2.3.2, ker πA is central in G(A), e and by Lemma 2.5.1, σ e : G(A) e → H is surjective. Consequently, we have e(ker πA ) ⊂ Z(H), and hence σ σ e induces a map σ : G(A) → H on the quotients satisfying σ ◦ πA = ν ◦ σ e. The following commutative diagram with exact rows depicts the situation. πA 1 ker πA G(A) e G(A) 1 σ e σ e σ ν 1 Z(H) H H 1 Next we show that σ is algebraic. Proposition 2.5.3. The group homomorphism σ : G(A) → H from Proposition 2.5.2 is a morphism of algebraic groups. Proof. Using [Bor91, Proposition 2.2] or [Sta20, Lemma 2.14(iv)], we can write G(A) = E(A) as a product m e Uαeα Y Y G(A) = = Uαii α∈Φk i=1 for some sequence of roots {α1 , . . . , αm } ⊂ Φk , where Uαi = Xαi (Vαi (A)) is the root sub- Y m group associated with α ∈ Φk and each ei = ±1. Let X = Aαi be a product of copies of i=1 A indexed by the αi , and define a regular map s : X → G(A) by s(a1 , . . . , am ) = Xα1 (a1 )e1 · · · Xαm (am )em . 40 The maps Xαi are all regular, so s is regular. Let us also define a regular map t0 : X → H by t0 (a1 , . . . , am ) = ψα1 (a1 )e1 · · · ψαm (am )em , where the ψαi are the morphisms from Proposition 2.4.6. Since each ψαi is regular, t0 is regular. Set t = ν ◦ t0 . One easily checks that σ ◦ s = t. In particular, for any x1 , x2 ∈ X, the condition s(x1 ) = s(x2 ) implies t(x1 ) = t(x2 ). So, by [Rap13, Lemma 3.10], σ is a rational map. Hence, there is an open subset of G(A) on which σ is regular. Then, applying [Rap13, Lemma 3.12], we conclude that σ is a morphism of algebraic groups. To conclude the argument, we will show that σ can be lifted to a morphism σ : G(A) → H. For this, we first discuss several preliminary statements, which are analogues in the present setting of results established in [Rap11, §§5,6]. Let A be the algebraic ring associated with the representation ρ, and let J ⊂ A be its Jacobson radical. As previously noted, A is a finite-dimensional K-algebra; in partic- ular, A is artinian, and hence J d = {0} for some d ≥ 1 (see [AM16, Proposition 8.4]). Moreover, by the Wedderburn-Malcev theorem (see [Pie82, Theorem 11.6]), there exists a semisimple subalgebra A ⊂ A such that A = A ⊕ J as K-vector spaces and A ∼ = A/J as K-algebras. We note that [Rap11, Proposition 2.20] implies that A ' K ×· · ·×K (r copies). Since G = SU2n (L, h) is K-isomorphic to SL2n , it follows that G(A) is a connected, simply connected, semisimple algebraic group. For the next statement, we consider the canonical homomorphism A → A/J and set G(A, J) = ker(G(A) → G(A/J)) to be the corresponding congruence subgroup. We then have the following. Lemma 2.5.4. Retain the notation above. 41 (1) The congruence subgroup G(A, J) is nilpotent. (2) We have a Levi decomposition G(A) = G(A, J) n G(A). Proof. (1) Fixing an embedding G(A) ,→ GL2n (AL ), it is straightforward to show that G(A, J a ), G(A, J b ) ⊂ G(A, J a+b )   for any a, b ∈ Z≥1 . Since J is a nilpotent ideal, our claim follows. (2) Using the fact that G(A) is perfect (see Lemma 2.3.3), this statement is proved by the same argument as [Rap11, Proposition 6.5]. Next, we note the following analogue of [Rap11, Proposition 5.5] — this result is proved exactly as in that context, employing Lemma 2.5.1 in place of the corresponding statements in loc. cit.. Following the discussion there, we say that H satisfies condition (Z) if Z(H) ∩ U = {e}. As previously noted, the following lemma is is the first and only place where the assumption on the commutativity of the unipotent radical U = Ru (H) is used. Lemma 2.5.5. Let ρ, H be as above, and let U be the unipotent radical of H. (1) If U = Ru (H) is commutative and char K = 0, then H satisfies (Z). (2) If H satisfies (Z), then Z(H) is finite. If additionally char K = 0, then Z(H) is contained in any Levi subgroup of H. Proof. (1) Let H = U n S be a Levi decomposition of H. Since char K = 0, by [Bor91, Remark II.7.3] U ∼ = (K m , +) where m = dim U , and the action of S on U gives a ra- tional representation of S on K m . By Weyl’s Theorem, this representation is completely reducible (see [Hum72, Theorem 6.3] and [Hum75, Theorem 13.2], and note that these rely on 42 char K = 0). Since H = [H, H] (Lemma 2.5.1) the representation cannot contain the trivial representation, so U has no nonzero elements fixed by the S-action, so Z(H) ∩ U = {e}. (2) Consider the quotient H/U , which is a reductive algebraic group that coincides with its commutator. So by [Bor91, Corollary 14.2], Z(H/U ) is finite. Since Z(H) ∩ U = {e}, the restriction of the quotient map H → H/U is injective, so Z(H) is finite. Now suppose char K = 0. Let S be a Levi subgroup of H, so H = U n S. Since Z(H) is a finite abelian group, it is reductive, so some conjugate of it is contained in S (see [Mos56]). Since Z(H) is central, it is itself contained in S. The next statement completes the proof of Theorem 2.0.1. Theorem 2.5.6. Assume that U = Ru (H) is commutative and char K = 0. Then there exists a morphism of algebraic groups σ : G(A) → H making the diagram below (same as 2.4) commute. Fe G(k) e G(A) e πk πA σ F e (2.5) G(k) G(A) σ ρ H Proof. (cf. [Rap11, Proposition 6.6]) Let G(A) = G(A, J) n G(A) be the Levi decomposition from Lemma 2.5.4, and set U = σ(G(A, J)), S = σ(G(A)), and S = (ν −1 (S))◦ , where ν : H → H is the quotient map. Then H = U n S and H = U n S are also Levi decompositions. By Lemma 2.5.5, we have Z(H) ⊂ S. Consequently, S = S/Z(H) and the restriction ν|U : U → U is an isomorphism. 43 Now, since the quotient map ν : H → H is a central isogeny, and, as we observed above, G(A) is simply connected, it follows from [BT72, Proposition 2.24(i)] that there exists a morphism of algebraic groups σS : G(A) → S such that ν|S ◦ σS = σ|G(A) . S⊂H σS ν|S G(A) S⊂H σ   Define σU = ν|−1 U ◦ σ| G(A,J) . Then σ = (σU , σS ) : G(A) → H is a morphism of algebraic groups such that ν ◦ σ = σ. It remains to show that σ makes the diagram (2.4) commute. Define χ : G(A) e → H, g 7→ σe(g)−1 · (σ ◦ πA )(g). e and σ ◦ πA are group homomorphisms, so is χ. Also, by Proposition 2.5.2, we have Since σ σ ◦ πA = ν ◦ σ e, which yields ν ◦ σ ◦ πA = ν ◦ σ e. Thus, the image of χ is contained in ker ν = Z(H). Since G(A) e coincides with its commutator subgroup, if the image is central in H then we conclude that χ is trivial, and hence σ ◦ πA = σ e. Finally, the equality σ ◦ F = ρ follows from the commutativity of the rest of the diagram and the surjectivity of πk . 44 Chapter 3 Group actions on varieties In this chapter, we discuss applications of rigidity statements for Chevalley groups to obtain rigidity statements for actions of elementary groups on algebraic varieties. We think of this as an algebraic analog of the Zimmer program (see [Fis11]). Throughout the chapter, K will denote an algebraically closed field of characteristic zero, and all algebraic groups and varieties will be assumed to be K-defined. As in the preceeding chapter, we take a classical approach of identifying algebraic groups and varieties with their K-points. An overline such as H denotes the Zariski closure of H. For a quasi-projective variety X, we denote by Bir (X) the group of birational morphisms X 99K X, and Aut (X) the subgroup of biregular maps. For a K-algebra A, Aut (A) is its group of automorphisms. If X is affine, the groups Aut(K[X]) and Aut(X) are naturally (anti-)isomorphic (see §3.1). Definition 3.0.1. Let Γ be an abstract group and X an affine variety. An abstract action of Γ on X is given by a group homomorphism ρ : G → Aut(X). Equivalently, there is a map Γ × X → X such that for every γ ∈ Γ, the map X → X, x 7→ ρ(γ)(x) is biregular. Definition 3.0.2. Let G be an algebraic group and X an affine variety. An abstract action of G on X is algebraic if the map G × X → X is a morphism of varieties (G × X is the product variety)1 . An action is almost algebraic if G contains a finite-index subgroup ∆ 1 It is tempting to characterize an algebraic action as one in which the associated group homomorphism ρ : G → Aut(X) is algebraic, but this does not work because the automorphism group Aut(X) is “too big” to be an algebraic group. Instead, Aut(X) has the structure of an ind-group, see [Sha81]. 45 such that ρ|∆ gives an algebraic action of ∆ on X. In this chapter, we study a different but related form of ridigity to that in Conjecture 1.0.6. We prove rigidity statements of the form: under some hypotheses, an abstract action of Γ on X agrees with an algebraic action of some algebraic group G on X, at least on a finite-index subgroup ∆ ⊂ Γ and after passing through some abstract homomorphism F : Γ → G, which may or may not be an embedding. We obtain two main results of this type, both of which rely critically on [Rap19, Theorem 1.1], a rigidity result for Chevalley groups over commutative rings in the spirit of the Borel-Tits conjecture. We set some notation and terminology. Let Φ be a reduced irreducible root system of rank ≥ 2 and let R be a commutative ring. The pair (Φ, R) is nice if 2 ∈ R× whenever Φ contains a subsystem of type B2 and 2, 3 ∈ R× if Φ is of type G2 . Let G be the corresponding universal Chevalley-Demazure group scheme over Z, and let Γ = G(R)+ be the elementary subgroup of G, the group generated by R-points of unipotent radicals of R-defined parabolic subgroups. Let g : R → K be a ring homomorphism. A g-derivation is map δ : R → K such that for any a, b ∈ R we have δ(a + b) = δ(a) + δ(b) δ(ab) = δ(a)g(b) + g(a)δ(b) The set of all g-derivations is denoted Derg (R, K), and has a natural K-vector space struc- ture. We say that R has property (Dn ) if dimK Derg (R, K) ≤ n for every ring homorphism g : R → K. We focus on the case n = 1. Example 3.0.3. A ring R of S-integers in a number field has (D1 ). See [Rap19, Lemma 6.1] and [Rap13, Lemma 4.7] for more general statements. We recall and rephrase the notion of a standard description from §1. Let ρ : Γ → GLm (K) be an abstract representation, where Γ = G(R)+ as above. We say ρ has a 46 standard description if there exists a finite-dimensional commutative K-algebra A, a ring homomorphism f : R → A with Zariski-dense image, a morphism of algebraic groups σ : G(A) → GLm (K), and a finite-index subgroup ∆ ⊂ Γ such that ρ|∆ = (σ ◦ F )|∆ , where F : G(R) → G(A) is induced by f . Using this terminology, [Rap19, Theorem 1.1] says that if G is the universal Chevalley group of a reduced irreducible root system Φ of rank ≥ 2, R is a commutative ring with (D1 ), (Φ, R) is a nice pair, and Γ = G(R)+ then any abstract homomorphism ρ : Γ → GLm (K) has a standard description. 3.1 Elementary groups acting on affine varieties In this section we give two closely related rigidity results for elementary subgroups of Cheval- ley groups acting on affine varieties. We begin by recalling the correspondence between actions on a variety and on its coordinate ring. For an affine variety X defined over K, there is a natural anti-automorphism ηX : Aut(X) ∼ = Aut(K[X]) which takes φ ∈ Aut(X) to φ∗ ∈ Aut(K[X]), where φ∗ (f ) = f ◦ φ. Let Γ be a group acting on X via r : Γ → Aut(X). Note that r induces an action r∗ : Γ → Aut(K[X]) of Γ on Aut(K[X]) characterized by      r∗ (γ)f x = f r γ −1 x for all x ∈ X and f ∈ K[X]. That is, r∗ = ηX ◦ r ◦ inv, where inv is the inversion map γ 7→ γ −1 . We will refer to r∗ as the associated action on K[X], or the coaction induced by r. If we suppress r from the notation (as in e.g. Springer [Spr09, 2.3.5]) and denote r(γ)x = γx and r∗ (γ)x = γ ∗ x, this can be rewritten as (γ ∗ f )(x) = f (γ −1 x). For a comorphism ψ ∈ Aut(K[X]), denote the associated morphism by ψ ∨ = ηX −1 (ψ). Clearly (φ∗ )∨ = φ and (ψ ∨ )∗ = ψ. Given an action ρ : Γ → Aut(K[X]), we define ρ∨ = ηX −1 ◦ ρ ◦ inv ∨ to obtain an action ρ∨ : Γ → Aut(X), given explicitly by ρ∨ (γ) = ρ γ −1 . It is clear 47 that (r∗ )∨ = r and (ρ∨ )∗ = ρ. For a group Γ acting on a K-algebra A, we will say the action is locally finite- dimensional if the orbit of any single element spans a finite-dimensional K-subspace of A. Recall the following classical result (see [Spr09, Proposition 2.3.6] or [Bor91, Proposition I.1.9]). Proposition 3.1.1. If G is an affine algebraic group acting algebraically on an affine variety X, the associated action on K[X] is locally finite-dimensional. We think of the main rigidity statements from this section as a partial converses to this statement. To formulate them, we set some notation. Let K be an algebraically closed field of characteristic zero. Let Φ be a reduced, irreducible root system of rank ≥ 2 and G be the corresponding universal Chevalley-Demazure group scheme over Z. Let R be a commutative ring with (D1 ), such that (Φ, R) is a nice pair. Let Γ = G(R)+ be the elementary subgroup of G. Theorem 3.1.2. Let K, Φ, G, R, Γ be as above. Let X be an affine K-variety, and let Γ act abstractly on X via r : Γ → Aut(X), such that the associated action on K[X] is locally finite- dimensional. Then there exists an algebraic group G, an algebraic action re : G → Aut(X), and a finite-index subgroup ∆ ⊂ Γ such that r|∆ = (e r ◦ F )|∆ , where F : G(R) → G(A) is an abstract group homomorphism arising from a ring homomorphism f : R → A with Zariski-dense image. Note that in the proof we will define G = F (Γ), so the composition re ◦ F is well-defined. Conceptually, the theorem describes how the abstract action of Γ is rigidified by agreeing with an algebraic action of G. More precisely, the equality r|∆ = (e r ◦ F )|∆ says that the 48 abstract action of Γ on X agrees with the algebraic action of G on X, after passing through F and restricting to the finite-index subgroup ∆. We explain how the theorem is a partial converse to Proposition 3.1.1. Suppose an abstract group Γ acts abstractly on an affine K-variety X via r, an algebraic group G acts algebraically on X via re, and there is a group homomorphism F : Γ → G such that r|∆ = (e r ◦F )|∆ for some finite-index subgroup ∆ ⊂ Γ. By Proposition 3.1.1, the coaction of G on K[X] is locally finite-dimensional; we claim it follows that that the coaction of Γ on K[X] is also locally finite-dimensional. Indeed, fix a set {γ1 , . . . , γn } of left coset representatives for ∆ in Γ. Then for any f ∈ K[X] the orbit Γ.f can be written [n n [ Γ.f = {r(γ)f : γ ∈ γi ∆} = {r(γi δ)f : δ ∈ ∆} i=1 i=1 [n [n = r(γi ) {r(δ)f : δ ∈ ∆} = r(γi ) {e r (F (δ)) f : δ ∈ ∆} i=1 i=1 Observe that {e r (F (δ)) f : δ ∈ Γ} ⊂ {e r(g)f : g ∈ G} and the right hand set is finite-dimensional. Thus the orbit Γ.f is a finite union of translations (via K-linear automorphisms r(γi )) of these finite-dimensional subsets, so it is finite-dimensional. Because of Example 3.0.3, Theorem 3.1.2 applies when R is a ring of S-integers in a number field. However, we can also obtain this case more directly with stronger conclusions, as detailed in the next theorem. Theorem 3.1.3. Under the same hypotheses as Theorem 3.1.2, assume in addition that R is a ring of S-integers in a number field. Then there exists an algebraic group G containing Γ, an algebraic action re : G → Aut(X), and a finite-index subgroup ∆ ⊂ Γ such that re|∆ = r|∆ . The remainder of this section is occupied with proving Theorems 3.1.2 and 3.1.3. We begin with some remarks and lemmas. 49 Remark 3.1.4. Let A be a finitely generated K-algebra with a locally finite-dimensional action of a group Γ. Then A contains a finite-dimensional Γ-invariant subspace V which generates A as a K-algebra. Concretely, take a finite generating set {f1 , . . . , fn } of A and let V be the sum of the K-spans of the orbits of the fi . By assumption each orbit is finite- dimensional, so V has the desired properties. In particular, this implies that if X is an affine variety and Γ acts on X via r such that the coaction r∗ on K[X] is locally finite-dimensional, then the image r∗ (Γ) ⊂ Aut(K[X]) is contained in GL(V ). As previously noted, Aut(X) and Aut(K[X]) are not algebraic groups, but we avoid complications of ind-groups by ensuring that everything happens inside the algebraic group GL(V ). Remark 3.1.5. Let Γ be a group and V a finite-dimensional K-vector space. In this situation, an action of Γ on V via ρ : Γ → GL(V ) is called a representation, and the coaction is called the corepresentation. Every K-linear automorphism of V extends uniquely to a K-algebra automorphism of the symmetric algebra Sym(V ); this gives an embedding GL(V ) ,→ Aut(Sym(V )). Hence an abstract representation ρ : Γ → GL(V ) uniquely extends to an abstract representation ρ : Γ → Aut(Sym(V )) making the following diagram commute. Aut(Sym(V )) ρ Γ ρ GL(V ) Next we prove a lemma which establishes that the corepresentation of an algebraic repre- sentation is algebraic. Lemma 3.1.6. Let G be an algebraic group, V a K-vector space of dimension d < ∞, and σ : G → GL(V ) an algebraic representation. Extend σ to a representation σ : G → Aut(B), where B = Sym(V ) = K[Y ] and Y = AdK . Then the associated representation σ ∨ : G → Aut(Y ) is algebraic. 50 Proof. Remark 3.1.5 ensures that such an extension of σ exists. Fix an isomorphism of varieties φ : Y = AdK → V , and consider the following diagram. σ∨ G×Y Y inv×φ φ σ G×V V We have abused notation slightly by having σ, σ ∨ refer to maps G → GL(V ), G → Aut(Y ) and also to the associated maps G × V → V, G × Y → Y . For (g, y) ∈ G × Y we have ∗ σ(g −1 )(φ(y)) = σ ∨ (g) (φ(y)) = φ σ ∨ (g)(y)  σ so the diagram commutes. Since σ is algebraic, G × V − → V is regular, so we have written σ ∨ as the composition φ−1 ◦ σ ◦ (inv × φ) of three regular maps, so it is regular, i.e. G acts algebraically on Y . Lemma 3.1.7. Let Γ be an abstract group acting on affine varieties X, Y , and let θ : X → Y be regular. Then θ is Γ-equivariant if and only if the comorphism θ∗ is Γ-equivariant. More precisely, let Γ act on X via rX and on Y via rY , and fix γ ∈ Γ. Then one of the following diagrams commutes if and only if the other also commutes. θ θ∗ X Y K[X] K[Y ]     rX γ −1 rY γ −1 ∗ (γ) rX ∗ (γ) rY θ θ∗ X Y K[X] K[Y ] In the diagram on the right, rX ∗ , r ∗ are the respective coactions of r , r . Y X Y Proof. Straightforward calculation utilizing separatedness of X and Y . Remark 3.1.8. Suppose G is an algebraic group with an abstract subgroup H and H 0 is a finite-index abstract subgroup of H. Then H (Zariski closure) is an algebraic subgroup of 51 G, and H 0 is a finite-index2 algebraic subgroup of H. Furthermore, if G is connected and 0 H = G, then H = G (use [Bor91, Proposition I.1.2]). We are now ready to prove Theorem 3.1.2. We start with an overview. First, we use [Rap19, Theorem 1.1] to obtain A, f, F, σ, ∆ as in the theorem, and define G = F (Γ). Then we construct an algebraic action re = σ ∗ : G → Aut(Y ), where Y = AdK . Then we describe a closed Γ-equivariant embedding θ : X ,→ Y , and use it to show that r|∆ = (e r ◦ F )|∆ . Finally, we show that re : G → Aut(Y ) leaves X invariant, so that we can think of re as an algebraic action re : G → Aut(X). Proof. Let V ⊂ K[X] be a finite-dimensional Γ-invariant K-subspace (Remark 3.1.4), let d = dimK V , and let ρ be the restriction of the corepresentation r∗ to V . ρ : Γ → GL(V ) ρ(γ) = r∗ (γ) By [Rap19, Theorem 1.1], ρ has a standard description. That is, there exists a finite- dimensional K-algebra A, a ring homomorphism f : R → A with Zariski-dense image, an algebraic representation σ : G(A) → GL(V ), and a finite-index subgroup ∆ ⊂ Γ such that ρ|∆ = (σ ◦ F )|∆ , where F : G(R) → G(A) is the (abstract) homomorphism induced by f . Define G = F (Γ) (Zariski closure) and ρe = σ|G . The diagrams below depict the situation. The left diagram is commutative, and the outer triangle of the right diagram commutes. F σ F σ G(R) G(A) GL(V ) G(R) G(A) GL(V ) ρe ρ=r∗ F Γ G Γ ρ|∆ ρ◦F )|∆ ρ|∆ =(σ◦F )|∆ =(e ∆ ∆ 2 H is the union (not necessarily disjoint) of closures of cosets of H 0 in H, so H : H 0 ≤ H : H 0 . h i h i 52 Let Y = AdK and B = Sym(V ) = K[Y ]. Next we construct an algebraic action re : G → Aut(Y ). Using Remark 3.1.5, the abstract action ρ : Γ → GL(V ) and algebraic action ρe : G → GL(V ) extend to actions on B. ρ : Γ → Aut(B) ρe : G → Aut(B) These actions on the coordinate ring B = K[Y ] induce associated actions on Y . ρ∨ : Γ → Aut(Y ) ρ)∨ : G → Aut(Y ) (e We set re = (e ρ)∨ : G → Aut(Y ). Because ρe = σ is algebraic, re is also algebraic by Lemma 3.1.6. Next we describe a Γ-equivariant closed embedding θ : X ,→ Y . The inclusion V ,→ B induces a K-algebra homomorphism θ∗ : B → K[X] as follows. Fix a K-basis {t1 , . . . , td } of V , and set θ∗ to be the map θ∗ : B → K[X] ti 7→ ti Since B is generated as a K-algebra by the ti , this defines a K-algebra homomorphism. We claim that θ∗ is Γ-equivariant; that is, the follow diagram commutes for all γ ∈ Γ. θ∗ B K[X] ρ(γ) r∗ (γ) θ∗ B K[X] It is clear that this commutes starting with a generator ti ∈ B, since by definition ρ(γ) just acts as r∗ (γ) on V , and then since all the maps in the diagram are K-algebra homomorphisms, they agree on all of B. As θ∗ is a surjective K-algebra homomorphism K[Y ] → K[X], it is the comorphism of a closed embedding θ : X ,→ Y . Since θ∗ is Γ-equivariant, so is θ by 53 Lemma 3.1.7. Note that Γ acts on θ(X) ⊂ Y via ρ∨ = (r∗ )∨ = r, so after identifying X with its image θ(X) this is just the original action of Γ on X via r. In particular, ρ∨ = r as maps Γ → Aut(X)3 . Next, we show that r|∆ = (e r ◦ F )|∆ . We have ρ|∆ = (e ρ ◦ F )|∆ , so taking coactions we obtain r|∆ = ρ∨ |∆ = (e ρ ◦ F )∨ |∆ . Working through the definitions and a short calculation shows that for all γ ∈ Γ, y ∈ Y, and α ∈ K[Y ],       α ρ ◦ F )∨ (γ) (y) = α (e (e r ◦ F ) (γ) (y) Since Y is separated, it follows that (e ρ ◦ F )∨ |∆ = (e r ◦ F )|∆ , so we obtain r|∆ = (e r ◦ F )|∆ as desired. We are not quite done, because re maps G to Aut(Y ), not to Aut(X) as claimed. To finish the proof, we need to show that G acting on Y via re leaves X invariant. Consider the restriction of the action of G on Y using re to X. G×X →Y (g, x) 7→ re(g)(x) We need to verify that re(g)(x) ∈ X. We know that if g ∈ F (∆) so that g = F (δ) for some δ ∈ ∆, then re(g) = r(δ), and when ∆ acts on X via r, the image lands back in X. For a fixed x ∈ X, since G acts algebraically, we get a regular (hence continuous) map `x : G → Y g 7→ re(g)(x) Note that G is connected by [Bor91, Proposition I.2.2], and since [F (Γ) : F (∆)] < [Γ : ∆] < ∞, it follows from Remark 3.1.8 that F (∆) is dense in G. Consider the image of G under   `x . Since `x is continuous, `x (G) = `x F (∆) = `x (F (∆)). Then because ∆ acting on X 3 A potential point of confusion here is that ρ∨ can be either a map Γ → Aut(X) or a map Γ → Aut(Y ). These are really the same map; the alternative interpretations just come from the fact that Γ leaves X (identified with θ(X)) invariant inside Y . 54 via re ◦ F = r leaves X invariant, `x (G) = `x (F (∆)) ⊂ X = X. Thus G acting on Y via re leaves X invariant, so we can view re as a map G → Aut(X). Using a very similar argument, we now prove Theorem 3.1.3. The only significant differ- ence is that we substitute the use of [Rap19, Theorem 1.1] by the refined statement [Rap13, Proposition 5.1] which gives more detailed information about the algebraic ring A and ring homomorphism f obtained in the standard description of ρ, coming from the fact that R is a ring of S-integers. Proof. Let V ⊂ K[X] be a finite-dimensional Γ-invariant subspace, let d = dimK V , and let ρ be the restriction of the corepresentation r∗ to V . Fix a K-basis {t1 , . . . , td } of V , and with it fix an isomorphism GL(V ) ∼ = GLd (K) in the usual way, so we think of ρ as a map Γ → GLd (K). By [Rap13, Proposition 5.1], ρ has a special standard description, where A is a product of copies of K, and the ring homomorphism f : R → A has the form f = (f (1) , . . . , f (m) ) with each f (i) : R → K is the restriction of a (distinct) embedding L ,→ K, and we have ρ|∆ = σ|∆ where σ : G(A) → GLd (K) is a morphism of algebraic groups and ∆ is a finite-index subgroup of Γ. In particular, each f (i) is injective, so the induced map F : G(R) → G(A) is injective, allowing us to identify Γ with its image in G(A). Let G = Γ (technically F (Γ)), which is an algebraic subgroup of G(A) (Remark 3.1.8). From this point, the same arguments as in the proof of Theorem 3.1.2 suffice to complete the proof. We let Y = AdK and B = Sym(V ) = K[Y ], and let re : G → Aut(Y ) be the coaction of σ|G , which is algebraic because σ is algebraic (Lemma 3.1.6). The obvious map θ∗ : B = K[Y ] → K[X] is a Γ-equivariant surjection of K-algebras, so it is the comorphism of a closed Γ-equivariant embedding of affine varieties θ : X ,→ Y (Lemma 3.1.7). Then 55 one can show that r|∆ = re|∆ , and that re leaves (the image under θ of) X invariant in the same way as in the proof of Theorem 3.1.2, so re can be viewed as an algebraic action re : G → Aut(X). The following diagram depicts the situation of Theorem 3.1.3. G re re|Γ Γ Aut(X) r ∆ r|∆ r|∆ =e The upper and lower triangles obviously commute (simply function restriction), as does the triangle involving re|Γ and re|∆ . The content of Theorem 3.1.3 is that the outer triangle commutes, though we note that the theorem makes no claim regarding equality of re|Γ and r as maps on Γ, only that they agree on the subgroup ∆. Example 3.1.9. We give a somewhat trivial example in which the locally finite-dimensional hypothesis of Theorem 3.1.2 and Theorem 3.1.3 holds. Let X = A1K be the affine line, whose coordinate ring is the polynomial ring in one variable K[x]. Any K-algebra automorphism of K[x] is a linear transformation, i.e. of the form x 7→ ax + b for some nonzero a ∈ K × and some b ∈ K. Hence for any abstract group Γ acting on K[x], the Γ-orbits are finite- dimensional. Then we can apply Theorem 3.1.3 with Γ = SLn (Z) (n ≥ 3) and conclude that any action of SLn (Z) on A1K coincides on a finite-index subgroup ∆ with an algebraic action of an algebraic group G containing Γ. Remark 3.1.10. It was pointed out to us by Friedrich Knop that in general, an action of any abstract group Γ on an affine variety X which has locally finite-dimensional coaction is close to being algebraic in the following sense. As in the arguments above, let V ⊂ K[X] be 56 a finite-dimensional Γ-invariant subspace which generates K[X] as an algebra, and consider the corresponding closed embedding θ : X ,→ V ∗ . Then G = {g ∈ GL(V ∗ ) : g(X) = X} is a Zariski-closed subgroup of GL(V ∗ ) acting algebraically on V ∗ , leaving X invariant, and in particular the action on X agrees with the original action of Γ on X. 3.2 Elementary groups acting on projective surfaces Let Γ be an abstract group and X be a projective variety with Bir(X) the group of birational automorphisms. In this section we study abstract homomorphisms α : Γ → Bir(X). Because an element of Bir(X) generally has an indeterminacy locus, such a map does not give a true action of Γ on X; nevertheless, in such a situation we say that Γ acts birationally on X. Let d ≥ 2 be an integer which is not a perfect square. In [CdC19, Theorem 9.1], Cantat  √  and de Cornulier give a rigidity result for SL2 Z[ d] acting by birational maps on projec- tive surfaces, showing that such an action is in some sense close to being an algebraic action by biregular maps. More precisely, they prove that if  √  ˆ Γ is a finite-index subgroup of SL2 Z[ d] , and ˆ X is an irreducible projective surface over an algebraically closed field K, and ˆ α : Γ → Bir(X) is an (abstract) group homomorphism with infinite image, then char K = 0 and there exists a finite-index subgroup ∆ ⊂ Γ and a birational map ϕ : Y 99K X such that ˆ Y is one of P2 , the ruled surface Fm , or C × P1 for some curve C, and ˆ conjϕ maps α(Γ) into Aut(Y ), where conjϕ is the group isomorphism conjϕ : Bir(X) → Bir(Y ) ψ 7→ ϕ−1 ◦ ψ ◦ ϕ 57 ˆ There is a (unique) algebraic homomorphism β : G → Aut(Y ) such that β ◦ j|∆ =  √  conjϕ ◦α|∆ , where G = SL2 (K) × SL2 (K) and j : SL2 Z[ d] ,→ G is the embedding √ arising from the two distinct embeddings Q( d) ,→ K. That is, the following diagram commutes. j ∆ G α β (3.1) α(Γ) Aut(Y ) conjϕ We focus on the rigidity implications of the second and third points. The second says that, after replacing X by a birational equivalent Y and conjugating by an appropriate ϕ, the (abstract) birational action of Γ is actually an (abstract) biregular action. The third says that this action by biregular maps is essentially algebraic, after passing to a finite index subgroup ∆. Cantant and de Cornulier’s proof has two main ingredients: [CdC19, Theorem 2] which gives a criterion for the image of conjϕ to land in Aut(Y ), and [CdC19, Lemma 9.2] which is a ridigity statement based on Margulis’ superrigidity. More precisely, the criterion in Theorem 2 is the so-called property (FW); for our purposes, it is sufficient that Kazhdan’s property (T) implies property (FW) ([CdC19, Remark 3.2]). In this section, we prove a result parallel to the above rigidity result. Our approach combines the (FW) criterion with a more algebraic rigidity statement, namely [Rap19, Theorem 1.1]. √ We also replace the ring Z[ d] with the class of rings R with property (D1 ) (note that √  √  Z[ d] is such a ring by Example 3.0.3), and replace the group SL2 Z[ d] with the elemen- tary subgroup Γ = G(R)+ of a universal Chevalley group with root system Φ of rank ≥ 2 (SL2 (R) is the elementary subgroup of the Chevalley group SL2 with root system Φ = A1 of rank 1). Ours is not a generalization, since we assume char K = 0, and more importantly we require rankΦ ≥ 2 which excludes SL2 . Our new result is as follows. 58 Theorem 3.2.1. Let K, Φ, R, G, Γ be as in Theorem 3.1.2, and assume R is finitely generated as a ring. Let X be an irreducible projective surface over K, and let α : Γ → Bir(X) be an abstract homomorphism with infinite image. Then there exists a birational map ϕ : Y 99K X, a finite-dimensional K-algebra A, a ring homomorphism f : R → A with Zariski-dense image, a morphism of algebraic groups σ : G(A) → GLm (K), and a finite-index subgroup ∆ ⊂ Γ such that ˆ Y is one of P2 , the ruled surface Fm , or C × P1 for some curve C, and ˆ conjϕ maps α(Γ) into Aut(Y ), and ˆ (conjϕ ◦α)|∆ = (σ ◦ F )|∆ , where F : G(R) → G(A) is induced by f : R → A. That is, the following diagram commutes (compare with diagram (3.1).). F ∆ G(A)+ α σ α(Γ) Aut(Y ) GLm (K) conjϕ Conceptually, Γ acts abstractly on X by birational maps, but if we replace X with a suitable birational equivalent Y , then the abstract action of Γ on Y coincides with an algebraic action of an algebraic group G, after restricting to a finite-index subgroup ∆ and passing through an abstract homomorphism F . Proof. Since R is finitely generated, Γ has Kazhdan’s property (T) (by [EJZK17, Theorem 1.1]), so Γ0 = α(Γ) also has (T), so Γ0 has (FW). Thus we can apply [CdC19, Theorem 2] to Γ0 ⊂ Bir(X). This gives a birational map ϕ : Y 99K X where Y is one of P2 , Fm , C × P1 , and conjϕ ◦α(Γ) ⊂ Aut(Y ). Because the possibilities for Y are sufficiently limited, Aut(Y ) is a linear algebraic K-group (see the discussion following [CdC19, Theorem 9.1]). Now we 59 have an abstract group action r := conjϕ ◦α : Γ → Aut(Y ) γ 7→ r(γ) = ϕ−1 ◦ α(γ) ◦ ϕ Since Aut(Y ) is linear, composing with an inclusion map we get a linear (abstract) repre- sentation r : Γ → GLm (K). Then using [Rap19, Theorem 1.1], r has a standard description, so there is a finite dimensional K-algebra A, a ring homomorphism f : R → A with Zariski- dense image, a morphism of algebraic groups σ : G(A) → GLm (K), and a finite-index subgroup ∆ ⊂ Γ such that (conjϕ ◦α)|∆ = (σ ◦ F )|∆ , where F : Γ → G(A) is the map induced by f . We depict the resulting equality of the previous result some commutative diagrams. Philo- sophically, we would like to factor the abstract homomorphism α as a composition σ ◦ F . α Γ Bir(X) F ? G(A) However, we cannot quite do this. Instead, we pass to a birational equivalent Y via conju- gation by ϕ, and restrict to the finite-index subgroup ∆, extending the above picture. α conjϕ ∆ Γ Bir(X) Aut(Y ) GLm (K) F ? F |∆ G(A) σ We can also depict this with the diagrams below. All arrows are group homomorphisms, though only σ is algebraic. F σ ∆ G(A)+ GLm (K) r Γ Aut(Y ) α conjϕ Bir(X) Bir(Y ) 60 F σ ∆ Γ G(A) GLm (K) α conjϕ Γ Bir(X) Bir(Y ) Aut(Y ) To emphasize, the composition (σ ◦ F )|∆ lands in the subgroup Aut(Y ) ⊂ GLm (K), and the composition r|∆ = (conjϕ ◦α)|∆ also lands the subgroup Aut(Y ) ⊂ Bir(Y ), so the equality r|∆ = (σ ◦ F )|∆ is an equality of maps ∆ → Aut(Y ). 61 Chapter 4 Future directions In this chapter, we discuss future directions for extending the methods of §2 and their potential as related to the Borel-Tits conjecture. In particular, we hope that eventually our techniques can resolve the Borel-Tits conjecture for all quasi-split groups of isotropic rank ≥ 2. In §4.1, we discuss the history and generalizations of a foundational aspect of the methods of §2, namely the construction of the algebraic ring A associated to an abstract representation and its role in the analysis of the abstract representation. The main difficulty, we which have resolved for the special unitary groups considered in §2, is how to handle multi-dimensional root subgroups. In §4.2, we revisit the ideas of §2.3, especially those related to centrality of the kernel ker πA and generalizations of Proposition 2.3.2. We describe an approach to establishing rationality of σ via different methods than that in §2.5. In particular, we bypass the technical hypothesis on Ru (H) required for Lemma 2.5.5, but replace it by the conjectural hypothesis that G(A) e is generated by Steinberg symbols, so we also discuss reasons to expect that G(A) e is generated by symbols. 62 4.1 Algebraic rings associated to abstract representa- tions In this section, we discuss future research directions for algebraic rings associated to abstract representations. In §2.4, we described an algebraic ring associated to an abstract represe- tation ρ : SU2n (L, h)(k) → GLm (K), essentially by imitating the methods of [Rap11] for associating an algebraic ring to an abstract representation of a Chevalley group. We pro- vide more historical background on this construction and discuss aspects of this construction which may generalize to other groups. The earliest form of this construction appeared in a 2009 note of Kassabov and Sapir [KS09]. Let R be an associative ring with unity, and ELn (R) the classical elementary sub- group of GLn (R) with n ≥ 3. Let ρ : ELn (R) → GLm (K) be an abstract representation, where K is an algebraically closed field. Let xij (r) ∈ ELn (R) be the elementary matrix with r in the (i, j) entry, 1’s on the diagonal, and zeros elsewhere. Then consider the root subgroup U13 = x13 (R), and let V = ρ(U13 ) be the Zariski closure of ρ(U13 ). It is clear that V is an abelian group under matrix multiplication. Less obviously, one can put an algebraic ring structure on V , with the multiplication defined by h i −1 u1 × u2 := w23 u1 w23 , w12 u2 w12−1 where w12 , w23 are the following matrices representing elements of the Weyl group of GLn (R) (if n ≥ 3 extend w12 , w23 with 1’s on the diagonal as needed).     0 −1 0 1 0 0 w12 = 1 0 0 w23 = 0 0 1 0 0 1 0 −1 0 63 Verifying that the multiplication as defined above gives a map V × V → V utilizes the Steinberg commutator relation [x12 (r), x23 (s)] = x13 (rs) and additional relations describ- ing how one can conjugate by wij to move between root subgroups, such as the relation −1 = x (r). w12 x13 (r)w12 23 We summarize this construction more conceptually. Given an abstract representation ρ of ELn (R), consider the 1-parameter subgroup ρ(U13 ) inside GLm (K). After taking the Zariski closure to obtain V , we put an algebraic ring structure on V in which addition is matrix multiplication, and multiplication is obtained by the action of the Weyl group and the Steinberg commutator relations. Later, Igor Rapinchuk generalized this construction by replacing ELn (R) with the ele- mentary subgroup of any simply-connected Chevalley group. Let Φ be a reduced, irreducible root system and G be the associated simply-connected Chevalley group, and let G+ ⊂ G(R) be the elementary subgroup. Assume that (Φ, R) is a nice pair, and let ρ : G+ → GLm (K) be an abstract representation. Then one can associate an algebraic ring to ρ in much the same way as the construction of Kassabov and Sapir: choose a root α1 , take the root sub- group Uα ⊂ G+ , and let A = ρ(Uα ) be the Zariski closure of ρ(Uα ). As before, addition in A is given by matrix multiplication in GLm (K). Then define multiplication in A using Steinberg commutator relations and Weyl group conjugations in G+ . If Φ is a type A root system, the same relations used by Kassabov and Sapir suffice. If Φ is type B or C, the assumption that (Φ, R) is a nice pair allows for use of 2 as a denominator in the relevant calculations2 , and similarly if Φ is type G2 both 2 and 3 appear as denominators. After analyzing abstract representations of split groups, Rapinchuk further extended this method 1 The construction appears to depend on an arbitrary choice of root α, but the symmetry of Φ implies that the computations should work equally well for any choice of root. 2 See Equation 2.3 and the map π in Lemma 2.4.2 for an example of 2 appearing as a denominator. 64 to analyze groups of the form SLn,D where D is a division algebra ([Rap13]), essentially reusing the construction for type A root systems. Our main contribution in extending this construction is to consider abstract represen- tations of a group with 2-dimensional root subgroups. In particular, we have analyzed G = SU2n (L, h) which has a mix of 1-dimensional and 2-dimensional root subgroups, and described how to associate an algebraic ring to an abstract representation of G(k). Our construction is based on the following general idea: according to the structure theory of reductive groups, a group G contains a split subgroup G0 whose root system has the same type as the relative root system of G ([CGP15, Theorem C.2.30]). For our arguments, we exhibited such a subgroup G0 (k) explicitly (2.2.5). Restricting an abstract representation ρ : G(k) → GLm (K) to G0 (k), we then associate an algebraic ring A to ρ|G (k) using the 0 construction summarized above. Through explicit computations, we are then able to in- corporate the additional components in the 2-dimensional root subgroups into the picture, which ultimately shows that A is sufficient for the analysis of the representation ρ on the whole group G(k). This strategy naturally leads to the following question, whose resolution will significantly expand the scope of the techniques developed in this thesis: given a k-isotropic group G of k-rank ≥ 2 and an abstract representation ρ : G(k) → GLm (K), can the additional components in the root groups of dimension ≥ 2 be incorporated in such a way that the algebraic ring A constructed from the restriction ρ|G (k) to a split subgroup suffice for the 0 analysis of ρ? 65 4.2 Steinberg symbols Let G = SU2n (L, h) as in §2, and let R be a k-algebra. We discuss the Steinberg group G(R) e in more detail, particularly the canonical map πR : G(R) e → G(R) and its kernel. We showed in Proposition 2.3.2 that under suitable hypotheses on R, the kernel is central in G(R). e Although the argument depended on certain specific features of the Steinberg commutator relations of SU2n (L, h), we are confident that this argument can be carried out in more generality, or at least in many other specific cases. In particular, we conjecture that if G is an isotropic quasi-split reductive group with associated Steinberg group G e (defined by Stavrova [Sta20]), and R is a product of local rings, then ker πR is central in G(R). e We anticipate this being relevant for future analysis of rigidity for other quasi-split groups. Knowing that ker πR is central was enough for §2, but we would like to understand ker πR in terms of generators. For split groups over fields, the answer is given by Matsumoto’s theorem, which says that the kernel, known as K2 (R) in this context, is generated by symbols with only a short list of relations needed to give a presentation. This generating set is critical for the methods of [Rap11], [Rap19] in verifying the Borel-Tits conjecture for split groups. The work of Deodhar [Deo78] extends the ideas of Matsumoto’s theorem to quasi-split groups over fields; that is, Deodhar shows that the general quasi-split version of ker πR is generated by symbols for a field R. We believe methods of Deodhar may extend to show that ker πR is generated by symbols when G is quasi-split and R is a local ring (or product of local rings). In particular, if Deodhar’s main result on generation of ker πR could be extended from fields to products of local rings, then we could obtain Theorem 2.0.1 in a more direct way, bypassing the hypothesis on commutativity of the unipotent radical. We describe this more direct method in the remainder of this section. 66 We begin by explicitly describing Steinberg symbols for SU2n (L, h). Let R be a k-algebra. For α ∈ Φk , we have a vector k-group scheme Vα , so Vα (R) has the structure of an additive group. Furthermore, we know the dimension is one for long roots and two for short roots. √ Recall that L = k( d) and write Vα (R) as  R α short Vα (R) = R ⊕ R√d α long This allows us to view Vα (R) as not just an additive group, but as ring. This agrees with √ our previous identification of Vα (k) ∼ = k⊕k d∼ = L in the case where R = k and α is long. Now that we have a ring structure on Vα (R), we can consider its group of units Vα (R)× , and define   wα (v) = Xα (v) · X−α −v −1 · Xα (v) hα (v) = wα (v) · wα (1)−1 for v ∈ Vα (R)× . Note that Lemma 2.2.8 still applies to these more generally defined wα (v) and hα (v) by the same arguments. We also generalize these elements to the Steinberg group G(R) e by setting   eα (v) = Xα (v) · X−α −v w e e −1 ·Xeα (v) hα (v) = w e eα (1)−1 eα (v) · w  Let πR : G(R) e → G(R) be the canonical map, and let α ∈ Φk . It is clear that πR w eα (v) =   wα (v) and πR e hα (v) = hα (v). It is immediate from the definitions that eα (v) · w w eα (−v) = 1 weα (v)−1 = w eα (−v) hα (1) = 1 e Applying πR , versions of the above equations also hold in G(R). Finally, we define hα (u) · e cα (u, v) = e hα (uv)−1 hα (v) · e 67 for u, v ∈ Vα (R)× . The elements cα (u, v) are called Steinberg symbols. As an immediate consequence of Lemma 2.2.8, cα (u, v) ∈ ker πR . Our definition of Steinberg symbols agrees with Deodhar’s description of elements bβ (λ, µ) in [Deo78, §2.32]. One can also compare with the unitary Steinberg symbols in Hahn and O’Meara in [HO89, §5.5A, §5.5F, §5.6A]. Deodhar [Deo78] shows that, in the case where R = k is a field (and G is any quasi-split group), πk : G(k) e → G(k) is the universal central extension (Theorem 1.9) and the kernel of πk is generated by symbols cα (u, v) where α is a single fixed long root and u, v ∈ k × (Theorem 2.1). Also see [HO89, 5.6.4] for a result of similar flavor, regarding generation of a unitary Steinberg group by unitary Steinberg symbols. This leads us to formulate the following conjecture concerning the Steinberg group of SU2n (L, h). Conjecture 4.2.1. Let R be a k-algebra which is a product of local rings (e.g. the ring A from 2.4.6). Then ker πR is generated by symbols cα (u, v) for α ∈ Φk and u, v ∈ Vα (R)× . A stronger version of the conjecture might assert that ker πR is generated by symbols cα (u, v) for some particular fixed long root α, but this stronger statement is unnecessary for our applications. More generally, one might conjecture that for G a quasi-split group and R as in the conjecture that ker πR is generated by Steinberg symbols. In the remainder of this section, we show how to obtain a similar result to Theorem 2.0.1 in which we remove the technical hypothesis on commutativity of the unipotent radical Ru (H) but replace it by the assumption that ker πR is generated by symbols. The result is as follows. √ Theorem 4.2.2. Let L = k( d) be a quadratic extension of a field k of characteristic zero, and for n ≥ 2, set G = SU2n (L, h) to be the special unitary group of a (skew-)hermitian form h : L2n × L2n → L of maximal Witt index. Let K be an algebraically closed field of 68 characteristic zero and consider an abstract representation ρ : G(k) → GLm (K). Set H = ρ(G(k)) to be the Zariski closure of the image of ρ. If Conjecture 4.2.1 holds, then there exists a commutative finite-dimensional K-algebra A, a ring homomorphism f : k → A with Zariski-dense image, and a morphism of algebraic K-groups σ : G(A) → H such that ρ = σ ◦ F , where F : G(k) → G(A) is the group homomorphism induced by f . To prove Theorem 4.2.2, we reuse results used for Theorem 2.0.1 through the end of §2.4. Then we use Conjecture 4.2.1 to descend σ e to a map on G(A). The arguments for rationality of σ are essentially the same as in Proposition 2.5.3. We recall the notation and setup of §2.1-§2.4. We have the special unitary group G = SU2n (L, h) with root subgroup maps Xα , an algebraically closed field K of characteristic zero, an abstract homomorphism ρ : G(k) → GLm (K), and set H = ρ(G(k)). Associated to G is the Steinberg group G e and its generators X eα (v). In Proposition 2.4.6, we constructed a K-algebra A, a ring homomorphism f : k → A, and regular maps ψα : Vα (A) → H, and in Proposition 2.4.8 we constructed a group homomorphism σ e : G(A) e → H making the following diagram commute. πk G(k) e G(k) Fe ρ (4.1) σ G(A) e e H The map Fe is induced by f ; explicitly Fe is given by X eα (v) 7→ X eα (f (v)). Lemma 4.2.3. Retain the notation above and let a, b ∈ Vα (A)× . Then cα (a, b) ∈ ker σ e. Proof. As in the proof of Proposition 2.4.8, we denote Vα (f ) = f˜. Consider u, v ∈ Vα (k)× ,       and write out cα f u, f v as a product of factors of the form Xα f u , Xα f˜v , then apply ˜ ˜ e ˜ e 69   the definition of Fe to obtain cα f˜u, f˜v = Fe (cα (u, v)). Applying the commutative square (4.1), we obtain       e cα (f u, f v) = τe ◦ F cα (u, v) = ρ ◦ πk cα (u, v) = ρ(1) = 1 σ e   ˜ ˜ Thus cα f u, f v ∈ ker σ e. Now consider the regular map   θ : Vα (A× ) × Vα (A× ) → H (a, b) 7→ σ e cα (a, b) Since θ vanishes on the dense open subset f (Vα (k)× ) × f (Vα (k)× ) ⊂ Vα (A)× × Vα (A)× and is regular, θ vanishes everywhere, hence σ e vanishes on symbols cα (a, b) for all a, b ∈ Vα (A× ). Note that the previous lemma does not depend on Conjecture 4.2.1. However, combined with the conjecture, it allows us to prove the following, which we view as a unitary version of [Rap13, Proposition 3.7]. Proposition 4.2.4. Retain the setup above, and assume Conjecture 4.2.1 holds. Then there exists an abstract group homomorphism σ : G(A) → H making the diagram commute. Fe G(k) e G(A) e πk πA σ F e G(k) G(A) σ ρ H Proof. By Lemma 4.2.3, σ e vanishes on unitary Steinberg symbols cα (u, v) and by Conjecture 4.2.1, these symbols generate G(A). e So σ e vanishes on all of ker πA , and hence induces a map on the quotient G(A)/ e ker πA → H. Since πA is surjective, the quotient is exactly G(A), and we obtain the claimed map σ. Commutativity of the lower triangle is a consequence of the 70 fact that the rest of the diagram commutes and that πk is surjective, just as in the proof of Theorem 2.5.6. To complete the proof of Theorem 4.2.2, we use similar arguments as in Proposition 2.5.3 to show that σ is algebraic. Proposition 4.2.5. The map σ of Proposition 4.2.4 is a morphism of algebraic groups. Proof. We write G(A) as a product of root subgroups m Y e G(A) = Ui i i=1 with ei = ±1, and let Ym X= Ai i=1 Then define s exactly as in Proposition 2.5.3, and define t:X→H t(a1 , . . . , am ) = ψα1 (a1 )e1 · · · ψαm (am )em where ψαi are the maps from Proposition 2.4.6. It is easy to verify that σ ◦ s = t; then apply [Rap11, Lemma 3.10] with Y = G(A), Z = H to conclude that σ is rational. Finally, apply [Rap11, Lemma 3.12] to conclude that σ is a morphism of algebraic groups. 71 APPENDICES 72 Appendix A Algebraic rings In this appendix, we briefly summarize the theory of algebraic rings necessary for our analysis in §2.4 and §2.5. We also include some material expanding the discussion of the algebraic ring Bα introduced in §2.4. Historically, algebraic rings were first studied systematically by Greenberg [Gre64]. Much of the notation and terminology here draws from the later development by I. Rapinchuk [Rap11, §2]. For this whole section, K denotes an algebraically closed field. An algebraic ring is an affine K-variety A with a ring structure, such that addition and multiplication are regular maps1 . In particular, (A, +) is a commutative algebraic group. A homomorphism of algebraic rings is a ring homomorphism which is also a regular morphism of varieties. If the source and target are algebraic rings with identity, we require a homomorphism to send the identity to the identity. Example A.1. Let B be a finite-dimensional K-algebra. Giving B the Zariski topology on K n (where n = dimK B), addition and multiplication in B are regular so B is an algebraic ring. (One culmination of the theory is that in characteristic zero that all connected algebraic rings are of this form.) An algebraic ring is commutative if multiplication is commutative. When we want to emphasize that a particular ring lacks an algebraic ring structure or we wish to temporarily 1 The affine assumption can be omitted, but this adds very little to the theory, since by [Rap11, Theorem 2.21] any irreducible variety with a regular ring structure must be affine. 73 ignore the algebraic structure on a given algebraic ring, we will call it an abstract ring. By [Rap11, Lemma 2.8], every commutative algebraic ring is semilocal as an abstract ring. A commutative algebraic ring A is connected if it is connected as a variety2 . By [Rap11, Lemma 2.9], in a connected algebraic ring A, every right and left ideal is connected and Zariski-closed, hence A is artinian as an abstract ring. We denote the connected component of 0 ∈ A by A◦ , and note that this is an ideal of A. Following [Rap11], we denote the following property by (FG). (FG) A◦ is finitely generated as an ideal of A If A satisfies (F G), then A◦ is an artinian algebraic ring with identity, and A decomposes as a direct sum of algebraic rings A = A◦ ⊕C where C is a finite ring isomorphic to A/A◦ ([Rap11, Lemma 2.12]). We also have the following criterion not depending on the characteristic: if R is an abstract Noetherian ring and f : R → A is an abstract ring homomorphism such that f (R) = A, then A satisfies (FG). In the same situation, if R is not only a Noetherian ring but an infinite field, then the finite ring C in the decomposition of A must be trivial; in other words, if R is an infinite field then A = A◦ is connected. We also have a more general decomposition statement, not depending on (FG). Any algebraic ring A is a direct sum A0 ⊕ C, where A0 is an algebraic subring of A consisting of all unipotent elements in (A, +) and C is a finite ring consisting of semisimple elements ([Rap11, Lemma 2.15]). In particular, this decomposition shows that if A is connected, it consists entirely of unipotent elements. We already mentioned the simplest examples of algebraic rings – finite dimensional K- algebras with the usual Zariski topology on K n . An algebraic ring A is said to come from 2 Note that in some contexts, a ring A is called connected if spec A with the Zariski topology is connected, or equivalently A contains no nontrivial idempotents. This notion of connectedness is not equivalent to the one we use. 74 an algebra if there exists a finite-dimensional K-algebra B and an isomorphism of algebraic rings A ∼ = B. More generally, an algebraic ring A virtually comes from an algebra if there is a finite-dimensional K-algebra B and a finite ring C such that A ∼ = B ⊕ C. If char K = 0 (as always, algebraically closed), then every algebraic ring A virtually comes from an algebra ([Rap11, Proposition 2.14]). Let A be a connected algebraic ring over an algebraically closed field K of characteristic zero, so A virtually comes from an algebra. Since A is connected, the finite ring summand C in the decomposition A ∼ = A0 ⊕ C must be trivial, so in fact A comes from an algebra. In other words, there is a equivalence of categories     Obj : connected algebraic rings over K     Obj : finite-dimensional K-algebras  ∼ =  Mor : homomorphisms of algebraic rings   Mor : K-algebra homomorphisms   This categorical description is due to Greenberg [Gre64, Proposition 5.1]. To finish this discussion, we mention another statement which uses an abstract ring homomorphism f : R → A to gain information about A, coming from [Rap11, Lemma 3.2]. Let A be an affine variety with two regular maps α : A × A → A and µ : A × A → U , such that (A, α) is a commutative algebraic group. Let R be an abstract commutative unital ring and f : R → A a map such that f (R) = A and     f (t1 + t2 ) = α f (t1 ), f (t2 ) f (t1 t2 ) = µ f (t1 ), f (t2 ) for all t1 , t2 ∈ R. In other words, f is an additive group homomorphism (R, +) → (A, α) and a multiplicative map (R, ×) → (A, µ). If such f exists, then (A, α, µ) is a commutative algebraic ring with identity. 75 Agreement of algebraic ring structures In the remainder of this section, we return to the situation of §2.4, immediately following Lemma 2.4.2. We describe an algebraic ring structure on Bα1 −α3 and verify that the map π from the lemma is an isomorphism of algebraic rings. √ The setup is as follows: L/k = k( d)/k is a quadratic extension, G is the special unitary group SU2n (L, h) with relative root system Φk = Cn , and ρ : G(k) → GLm (K) is an abstract representation with K algebraically closed of characteristic zero. Let H = ρ(G(k)) be the √ Zariski closure. For a short root α ∈ Φk , the root space is Vα (k) = k 2 ∼= k⊕k d∼ = L and the root subgroup Xα (Vα (k)) is 2-dimensional (over k). We define √ Bα = ρ(Xα (k d)) √ and a map gα : k → Bα by gα (u) = ρ(Xα (u d)). We focus on the particular short root α = α1 − α3 and denote B = Bα . Let β = −2α3 , and recall that in Lemma 2.4.2 we gave an isomorphism of varieties π : B → Aβ such that π ◦ gα = fβ . We wish to make B into an algebraic ring. Addition in B is straightforward to define; it is just matrix multiplication inside GLm (K). More precisely, define add : B × B → H add(x, y) = xy where xy is matrix multiplication, and note that for all s, t ∈ k we have   add gα (s), gα (t) = gα (s + t) Since gα (k) is Zariski-dense in B, this shows that add(B × B) ⊂ B hence (B, add) is an abelian algebraic group. It remains to describe multiplication in B. Let     1 1 h = h2α−β h0 = h2α−β − w = w2α−β (1) 2 2d 76 For b ∈ B, set b0 = ρ(h0 ) · b · ρ(h0 )−1 b00 = ρ(w) · b0 · ρ(w)−1 Then define  00   mult : B × B → H mult(a, b) = ν a, b where ν = π −1 is the inverse map of π from Lemma 2.4.2. −1 · ρ(h)−1    ν : Aβ → B, ν(y) = ρ(h) · y, gα−β (−1) · y, gα−β (1) We claim that for all s, t ∈ k,   mult gα (s), gα (t) = gα (st) (A.1) Assuming this holds, it follows that mult(gα (k) × gα (k)) ⊂ gα (k), and since gα (k) is Zariski- dense in Bα , it further follows that mult(B × B) ⊂ B, so mult is a regular map B × B → B. Then applying [Rap11, Lemma 3.2] to gα : k → B makes (B, add, mult) into an algebraic ring. So to complete the description of the algebraic ring structure on B, it suffices to verify equation (A.1). Let s, t ∈ k and u, v ∈ L. We want to simplify mult gα (s), gα (t) = ν gα (s), gα (t)00 ,    so we begin with gα (t)00 . We have the following relation, which appeared in a less general form in the proof of Lemma 2.4.2. h2α−β (u) · Xα (2v) · h2α−β (u)−1 = Xα (2vu) In that lemma, we used the case u = 1 which gives h · Xα (2v) · h−1 = Xα (v) (A.2) 77 We will need this relation again, and also the case u = − 2d 1 which gives √  √ ! t d h0 · Xα t d · (h0 )−1 = Xα − (A.3) 2d One particular instance of equation (2.2) is w · Xα (v) · w−1 = Xβ−α (−v) (A.4) Combining equations (A.3) and (A.4), we obtain  √  gα (t)00 = ρ(w) · ρ(h0 ) · ρ Xα (t d) · ρ(h0 )−1 · ρ(w)−1  √  = ρ w · h0 · Xα (t d) · (h0 )−1 · w−1 √ !! t d = ρ Xβ−α − 2d Our next goal is to simplify the commutator gα (s), gα (t)00 . From Lemma 2.2.4 (2), we have   the commutator relation h i  Xα (u), Xβ−α (v) = Xβ − Tr(uv) for u, v ∈ L. Note that √ ! √ t d Tr s d, − = −st 2d so "  √  √ !# t d Xα s d , Xβ−α − = Xβ (st) (A.5) 2d Using equation (A.5), we can simplify gα (s), gα (t)00 .   "  √  √ !!#  t d gα (s), gα (t)00 = ρ Xα s d , ρ Xβ−α −   2d "  √  √ !# t d = ρ Xα s d , Xβ−α − 2d  = ρ Xβ (st) 78 Using the above expression for gα (s), gα (t)00 , we can write mult gα (s), gα (t) as      =ν gα (s), gα (t)00   = ν ρ Xβ (st) −1 · ρ(h)−1      = ρ(h) · ρ Xβ (st) , gα−β (−1) · ρ Xβ (st) , gα−β (1) h   √ i h   √ i−1  −1  = ρ (h) · ρ Xβ (st) , ρ Xα−β (− d) · ρ Xβ (st) , ρ Xα−β ( d) ·ρ h  h √ i h √ i−1 −1  = ρ h · Xβ (st), Xα−β (− d) · Xβ (st), Xα−β ( d) ·h The product of two commutators appearing in this expression simplifies via the following relation from the proof of Lemma 2.4.2. h  √ i h √ i−1  √  Xβ (st), Xα−β − d · Xβ (st), Xα−β d = Xα 2st d (A.6) Using equations (A.6) and (A.2) we can complete the calculation.  √   √  mult gα (s), gα (t) = ρ h · Xα (2st d) · h−1 = ρ Xα (st d) = gα (st)  This completes the description of the algebraic ring structure on B = Bα = Bα1 −α3 . Remark A.2. In §2.4, we used the results of [Rap11] for the split group G0 (k) to put an algebraic ring structure on Aβ = A−2α1 , and the ring structure on Bα was not relevant for any of the results of that section. However, this raises the natural question: what is the relationship between the ring structures on Aβ and Bα , in view of the isomorphism of varieties in Lemma 2.4.2? Denote addition and multiplication in Bα by + , × respectively and similarly denote Bα Bα the operations in Aβ by + , × . We claim that π is, in fact, an isomorphism of algebraic Aβ Aβ 79   ! rings π : Bα , + , × → Aβ , + , × . Recall that for s, t ∈ k, Bα Bα Aβ Aβ gα (s) + gα (t) = gα (s + t) fβ (s) + fβ (t) = fβ (s + t) Bα Aβ gα (s) × gα (t) = gα (st) fβ (s) × fβ (t) = fβ (st) Bα Aβ Also recall that π(gα (s)) = fβ (s) for all s ∈ k. Consider the following two regular maps. σ1 : Bα × Bα → Aβ σ1 (x, y) = π(x + y) Bα σ2 : Bα × Bα → Aβ σ2 (x, y) = π(x) + π(y) Aβ Then       σ1 gα (s), gα (t) = π gα (s) + gα (t) = π gα (s + t) = fβ (s + t) Bα       σ2 gα (s), gα (t) = π gα (s) + π gα (t) = fβ (s) + fβ (t) = fβ (s + t) Aβ Aβ Thus σ1 and σ2 coincide on gα (k) × gα (k), which is a dense subset of Bα × Bα , so σ1 = σ2 . In other words, π is compatible with the addition operations. Similarly, consider regular maps θ1 : Bα × Bα → Aβ θ1 (x, y) = π(x × y) Bα θ2 : Bα × Bα → Aβ θ2 (x, y) = π(x) × π(y) Aβ Then       θ1 gα (s), gα (t) = π gα (s) × gα (t) = π gα (st) = fβ (st) Bα       θ2 gα (s), gα (t) = π gα (s) × π gα (t) = fβ (s) × fβ (t) = fβ (st) Aβ Aβ 80 So θ1 , θ2 coincide on the dense subset gα (k) × gα (k) ⊂ Bα × Bα and we conclude that θ1 = θ2 , so π is compatible with the multiplication operations. Hence π is an isomorphism of algebraic rings. 81 Appendix B Computations This appendix contains various technical computations involving the group SU2n (L, h) and its Steinberg group. Commutator coefficients Nijαβ (u, v) This section contains the computations for Lemma 2.2.4. Throughout, Φk is the root system of type Cn and R is a fixed k-algebra. Given two roots α, β ∈ Φk , following [PS08] we denote by (α, β) the set of all roots consisting of positive integral linear combinations of α and β.  (α, β) = iα + jβ ∈ Φk : i, j ∈ Z≥0 This set serves as the indexing set for the right hand side of the Steinberg commutator relation given in Theorem 2.2.1(3), i.e. for α 6= ±β, h i   αβ Y Xα (u), Xβ (v) = Xiα+jβ Nij (u, v) (α,β) αβ We wish to describe the maps Nij (u, v) as concretely as possible. First, we begin with a lemma which completely describes the possibilities for (α, β) in the Cn root system. In any reduced root system, (α, β) = ∅ if α = ±β, so we ignore this case (the commutator formula does not apply when α = ±β anyway). Lemma B.1. Let α, β ∈ Φk and assume α 6= ±β. 82 (1) If α, β are both long roots, then (α, β) = ∅. (2) Suppose α, β are short roots.  (2a) If α+β is a short root, then {α, β} = αi − αj , αj − α` for three distinct indices i, j, `.  (2b) If α + β is a long root, then {α, β} = ε(αi + αj ), ω(αi − αj ) for some i < j and signs ε = ±1, ω = ±1. In particular, if α + β ∈ Φk , then (α, β) = {α + β}. (3) Suppose α is a short root and β a long root. The following are equivalent. (3a) α + β is a root. (3b) α + β is a short root. (3c) 2α + β is a root. (3d) 2α + β is a long root. (3e) (α, β) = {α + β, 2α + β} (3f ) α = εαi + ωαj and β = −ε2αi with i 6= j and independent signs ε = ±1, ω = ±1. Proof. (1) It’s clear that if α, β are long roots, then α+β is not a root, nor is any linear combination with larger coefficients. (2) Part (a) is clear from structure of the type An root system. For part (b), if two short roots ±αi ± αj and ±αk ± α` add up to a long root ±2αm , then there cannot be three distinct indices, so {i, j} = {k, `} and then α 6= ±β forces {α, β} to be as claimed. (3) The following implications are immediate. 83 (3d) (3f ) (3b) (3c) (3e) (3a) To complete the equivalence it suffices to show (3a) =⇒ (3f ) and (3c) =⇒ (3f ). (3a) =⇒ (3f ) We can write α = εαi + ωαj and β = δ2αk with ε = ±1, ω = ±1, δ = ±1. Since α + β ∈ Φk , we must have k ∈ {i, j}. The sign δ must be −ε or −ω since otherwise α + β would have a 3αk term, so relabelling if necessary we can make k = i and β = −ε2αi . (3c) =⇒ (3f ) Again write α = εαi + ωαj and β = δ2αk . Since 2α + β ∈ Φk , as before k ∈ {i, j}. Again δ must be −ε or −ω, since otherwise 2α + β would have a 4αk term, and we can relabel to write α, β as claimed. Let α, β ∈ Φk and u ∈ Vα (R), v ∈ Vβ (R). Lemma B.1 tells us what the commutator formula looks like in all possible cases. 1. If α, β are both long, then [Xα (u), Xβ (v)] = 1. 2. If α, β are both short and α + β ∈ Φk , then h i   αβ Xα (u), Xβ (v) = Xα+β N11 (u, v) 3. If α, β are different lengths, assume α is short by relabelling if necessary, and then h i     αβ αβ Xα (u), Xβ (v) = Xα+β N11 (u, v) · X2α+β N21 (u, v) Note that the two factors on the right hand side commute because (α + β, 2α + β) = ∅. Lemma B.2. Let α, β ∈ Φk be short roots such that α + β ∈ Φk , and let u ∈ Vα (R), v ∈ Vβ (R). βα αβ (1) Interchanging α, β changes N11 by a sign1 . That is, N11 (v, u) = −N11 (u, v). 1 This interchange also reverses the order of the arguments, but eventually we show that N 11 is symmetric so this is irrelevant. 84 −α,−β αβ (2) Negating both roots changes N11 by a sign. That is, N11 (u, v) = −N11 (u, v). Proof. (1) We compute the commutator of Xβ (v) and Xα (u) two different ways.   βα [Xβ (v), Xα (u)] = Xα+β N11 (v, u)  −1   αβ αβ [Xβ (v), Xα (u)] = [Xα (u), Xβ (v)]−1 = Xα+β N11 (u, v) = Xα+β − N11 (u, v) Since Xα+β is injective, the claimed equality follows. (2) Recall that X−α (u) = Xα (u)t by 2.2. We have the relation [xt , y t ] = [y −1 , x−1 ]t . Now we calculate the commutator of X−α (u) and X−β (v) in two different ways.   −α,−β [X−α (u), X−β (v)] = X−α−β N11 (u, v) [X−α (u), X−β (v)] = [Xα (u)t , Xβ (v)t ] = [Xβ (v)−1 , Xα (u)−1 ]t  t βα = [Xβ (−v), Xα (−u)]t = Xα+β N11 (−v, −u)   βα = X−α−β N11 (−v, −u) −α,−β βα By injectivity of X−α−β , we get N11 (u, v) = N11 (−v, −u). Since N11 is linear in both βα βα βα αβ variables, N11 (−v, −u) = N11 (v, u), and from part (1) we have N11 (v, u) = −N11 (u, v). Combining these we get the claimed equality. Lemma B.3. Let α, β ∈ Φk be roots with α short, β long, and α + β ∈ Φk , and let u ∈ Vα (R), v ∈ Vβ (R). 1. Interchanging α, β changes N11 and N21 by a sign. βα αβ N11 (v, u) = −N11 (u, v) βα αβ N12 (v, u) = −N21 (u, v) 85 2. Negating both α, β changes N11 by a sign and does not change N21 . −α,−β αβ N11 (u, v) = −N11 (u, v) −α,−β αβ N21 (u, v) = N21 (u, v) Proof. (1) This is the same as the argument as for Lemma B.2(1). We compute the commutator of Xα (u) and Xβ (v) two different ways.     αβ αβ [Xα (u), Xβ (v)] = Xα+β N11 (u, v) · X2α+β N21 (u, v)  −1 !    βα βα [Xα (u), Xβ (v)] = [Xβ (v), Xα (u)]−1 = Xα+β N11 (v, u) · X2α+β N12 (v, u)  −1  −1 βα βα = X2α+β N12 (v, u) · Xα+β N11 (v, u)     βα βα = X2α+β − N12 (v, u) · Xα+β − N11 (v, u)     βα βα = Xα+β − N11 (v, u) · X2α+β − N12 (v, u) So we conclude that         αβ αβ βα βα Xα+β N11 (u, v) · X2α+β N21 (u, v) = Xα+β − N11 (v, u) · X2α+β − N12 (v, u) We can rearrange this to     αβ βα αβ βα Xα+β N11 (u, v) + N11 (v, u) = X2α+β − N21 (u, v) − N12 (v, u) Since α + β and 2α + β are distinct roots, these lie in distinct root subgroups, so equality is only possible if both are the identity. Since both Xα+β , X2α+β are injective, this implies that both inputs are zero. Hence βα αβ N11 (v, u) = −N11 (u, v) βα αβ N21 (v, u) = −N12 (u, v) 86 (2) This is the same argument as for Lemma B.2(2). We compute the commutator of X−α (u) and X−β (v) two different ways.     −α,−β −α,−β [X−α (u), X−β (v)] = X−α−β N11 (u, v) · X−2α−β N21 (u, v) h i h it  t t t [X−α (u), X−β (v)] = Xα (u) , Xβ (v) = Xβ (v) , Xα (u) −1 −1 = Xβ (−v), Xα (−u)     t βα βα = Xα+β N11 (−v, −u) · X2α+β N12 (−v, −u)  t  t βα βα = X2α+β N12 (−v, −u) · Xα+β N11 (−v, −u)     βα βα = X−2α−β N12 (−v, −u) · X−α−β N11 (−v, −u)     βα βα = X−α−β N11 (−v, −u) · X−2α−β N12 (−v, −u) so we conclude     −α,−β −α,−β X−α−β N11 (u, v) · X−2α−β N21 (u, v)     βα βα = X−α−β N11 (−v, −u) · X−2α−β N12 (−v, −u) By the same kind of argument in (1), we can conclude that the respective inputs are equal. −α,−β βα N11 (u, v) = N11 (−v, −u) −α,−β βα N21 (u, v) = N12 (−v, −u) Finally, we do some rearranging using part (1), along with the fact that N11 is linear in both arguments while N12 is linear in the first argument and quadratic in the second. −α,−β βα βα αβ N11 (u, v) = N11 (−v, −u) = N11 (v, u) = −N11 (u, v) −α,−β βα βα αβ N21 (u, v) = N12 (−v, −u) = −N12 (v, u) = N21 (u, v) 87 Lemma B.4 (Repeat of Lemma 2.2.4). Let α, β ∈ Φk be relative roots such that α + β ∈ Φk , and let u ∈ Vα (R), v ∈ Vβ (R). (1) Suppose α, β are both short and α + β is short. Then by Lemma B.1 (2a) we have α = αi − αj , β = αj − α` for distinct indices i, j, ` (relabelling α, β if necessary), and αβ βα N11 (u, v) = uv N11 (u, v) = −uv (2) Suppose α, β are both short and α + β is long. Then by Lemma B.1 (2b), relabelling α, β if necessary we have α = ε(αi − αj ), β = ω(αi + αj ) for some ε = ±1, ω = ±1, with i < j, and αβ βα N11 (u, v) = ω Tr(u−εω v) N11 (v, u) = −ω Tr(u−εω v) (3) Suppose α is short and β long. Then by Lemma B.1 (3) we have α = εαi + ωαj and β = −ε2αi for some ε = ±1, ω = ±1 and i 6= j, and αβ βα N11 (u, v) = ωu−cij v N11 (v, u) = −ωu−cij v αβ βα N21 (u, v) = −εωvuu N12 (v, u) = εωvuu where    1  ij αβ In particular, whenever it is defined, the map N11 is surjective. Proof. (1) This is known from the classical Steinberg relations for type An . (2) The second equation follows from the first using Lemma B.2 (1), so we only need to prove αβ the formula for N11 . By part (2) of the same lemma, it suffices to verify the formula for 88 αβ N11 (u, v) in the two cases ε = ω = 1 and ε = 1, ω = −1. First we verify the case ε = ω = 1. By a direct computation (Example B.5) we verify that [Xαi −αj (u), Xαi +αj (v)] = X2αi (uv + uv) αβ so N11 (u, v) = uv + uv = ω Tr(uv) = ω Tr(u−εω v) as claimed in the first case. Now we verify the case ε = 1, ω = −1, again by direct computation verifying that [Xαi +αj (u), X−αi −αj (v)] = X−2αi (−uv − uv) αβ so N11 (u, v) = −uv − uv = − Tr(uv) = ω Tr(u−εω v) as claimed. (3) The strategy is similar to that in part (1). By Lemma B.3(2), it suffices to verify just αβ αβ the formulas for N11 and N21 in the following four cases. (a) ε = ω = cij = 1 (b) ε = cij = 1 and ω = −1 (c) ε = ω = 1 and cij = −1 (d) ε = 1 and ω = cij = −1 Each of these cases is verified by direct computation. For example, case (a) requires verifying that when i < j we have [Xαi +αj (u), X−2αi (v)] = X−αi +αj (uv) · X2αj (−vuu) Example B.5. To illustrate, we work through one of the direct computations asserted in part (2) of the previous lemma. We will verify the relation [Xαi −αj (u), Xαi +αj (v)] = X2αi (uv + uv) 89 in the case i < j. The relevant root subgroup maps are Xαi −αj (u) = 1 + Eij (u) − Ej+1,i+1 (u) Xαi +αj (v) = 1 + Ei,j+1 (v) − Ej,i+1 (v) Using Xα (u)−1 = Xα (−u), we expand the commutator bracket.     [Xαi −αj (u), Xαi +αj (v)] = 1 + Eij (u) − Ej+1,i+1 (u) · 1 + Ei,j+1 (v) − Ej,i+1 (v)     · 1 − Eij (u) + Ej+1,i+1 (u) · 1 − Ei,j+1 (v) + Ej,i+1 (v) Recall the identity Eij (x)Ek` (y) = δjk Ei` (xy) (δjk is the Kronecker delta function). In partilar, whenever j = k, the product vanishes. So when we distribute the products on the right side, multiple terms vanish.   : = 1 + Ei,j+1 (v) − Ej,i+1 (v) + Eij (u) + E ij (u)E i,j+1 (v) − Eij (u)Ej,i+1 (v)      : :   (v) · − Ej+1,i+1 (u) − Ej+1,i+1  (u)E   i,j+1 (v) + E j+1,i+1  (u)E  j,i+1     : · 1 − Ei,j+1 (v) + Ej,i+1 (v) − Eij (u) − E (u)E (v) ij  i,j+1   + Eij (u)Ej,i+1 (v)   :  :   + Ej+1,i+1 (u) − Ej+1,i+1 (u)Ei,j+1 (v) − Ej+1,i+1   (u)Ej,i+1 (v)       = 1 + Ei,j+1 (v) − Ej,i+1 (v) + Eij (u) − Ei,i+1 (uv) − Ej+1,i+1 (u)   · 1 − Ei,j+1 (v) + Ej,i+1 (v) − Eij (u) + Ei,i+1 (uv) + Ej+1,i+1 (u) 90 Then we distribute again, and again many terms vanish.   = 1 − Ei,j+1 (v) + Ej,i+1 (v) − Eij (u) + Ei,i+1 (uv) + Ej+1,i+1 (u)   *  : + Ei,j+1 (v) 1 −  − :  :  Ei,j+1 (v)  + E j,i+1 (v)  E (u) + E (uv) + E j+1,i+1 (u) ij i,i+1         : *  :   − Ej,i+1 (v) 1 −  j,i+1 (v) −  :  :  Ei,j+1 (v) +     E    Eij (u) +   E  (uv) + E i,i+1   j+1,i+1    (u)  : * :  + Eij (u) 1 −  − :  E i,j+1 (v)  + E j,i+1 (v) E (u) + E (uv) + E (u) ij i,i+1 j+1,i+1           *  :  : − Ei,i+1 (uv) 1 −   (v) − Eij :  :  E  (v) +  E (u) +  E  (uv) + E (u)        i,j+1 j,i+1 i,i+1 j+1,i+1       *   : : − Ej+1,i+1 (u) 1 −  − :  :  Ei,j+1  (v)  + E  j,i+1  (v)  E ij  (u)  + E  i,i+1  (uv) + E j+1,i+1   (u)  = 1 − Ei,j+1 (v) + Ej,i+1 (v) − Eij (u) + Ei,i+1 (uv) + Ej+1,i+1 (u) + Ei,j+1 (v) + Ei,j+1 (v)Ej+1,i+1 (u) − Ej,i+1 (v) + Eij (u) + Eij (u)Ej,i+1 (v) − Ei,i+1 (uv) − Ej+1,i+1 (u) From here, it is just a matter of reorganizing the terms and recognizing various cancellations in pairs. *  : = 1 − − Eij :  :  E  i,j+1 (v)  + E  j,i+1 (v)  (u)  +E i,i+1 (uv) +  E (u) j+1,i+1  + Ei,i+1 (vu) −  :  :  +E  i,j+1 (v)  E  j,i+1 (v)  *  :  :  : + Eij(u) +   Ei,i+1  (uv)  − E i,i+1  (uv)  − E (u) j+1,i+1  = 1 + Ei,i+1 (uv + uv) = X2αi (uv + uv) Thus we have the claimed equality. [Xαi −αj (u), Xαi +αj (v)] = X2αi (uv + uv) 91 Conjugation by w eα (1) As in §2, let L/k be a quadratic extension in characteristic zero, with nontrivial Galois automorphism τ (x) = x. We denote  v δ=1 vδ = v δ = −1 Let R be a k-algebra, and denote RL = R ⊗k L, and extend τ to RL by acting on the L part. Let Φk be the root system of type Cn , which we write as  Φk = ±βi ± βj : 1 ≤ i, j ≤ n \ {0} For α, β ∈ Φk , let  (α, β) = iα + jβ ∈ Φk : i, j ∈ Z≥1 We believe that the relation (2.2) from §2.2 should lift to the Steinberg group, and that this should be possible to prove directly using the Steinberg relations and Lemma 2.2.4. However, we have not yet worked this out. More precisely, our conjecture is the following. Conjecture B.6. Let α, β ∈ Φk such that α + β ∈ Φk . Let v ∈ Vα (R). Then weβ (1) · X eβ (1)−1 = X eα (v) · w ew α (ϕv) β where ϕ : RL → RL is a function of the form v 7→ ±v±1 . It is not clear exactly what relationship this bears to [Deo78, Proposition 1.11, Corollary 1.12]. It is possible we have misunderstood Deodhar, but Deodhar’s result does not appear imply the above. Regardless, we have the following computation which covers at least one case of the conjecture above. Note that in the situation of the next lemma, wβ α = α + β. 92 Lemma B.7. Let α, β ∈ Φk with α short and β long, such that α + β ∈ Φk . Let v ∈ Vα (R). Then eβ (1) · X w eβ (1)−1 = X eα (v) · w eα+β (ϕv) where ϕ : RL → RL is a function of the form v 7→ ±v±1 . Proof. This is simply a long calculation involving the Steinberg relations (R1), (R2) from Definition 2.3.1, followed by application of Lemma 2.2.4. First, for α short and β long with α + β ∈ Φk , (α, β) = {β + α, β + 2α} (−β, β + α) = {α, β + 2α} (α, β + α) = {β + 2α} We repeatedly apply (R1) and (R2) in a long calculation, the end result of which is weβ (1) · X eβ (1)−1 = X eα (v) · w eβ+2α (N6 + N7 + N4 + 2N2 ) · X eβ+α (N5 + 2N1 ) · X eα (N3 + v) where Ni are various commutator coefficients that arise along the way. To complete the proof, we show that N6 + N7 + N4 + 2N2 = 0 N5 + 2N1 = ±v±1 N3 + v = 0 Now we do the calculation. Let w = w eβ (1) and X = X eα (v). First we expand the left hand side. wXw−1 = X eβ (1) · X e−β (−1) · X eβ (1) · X eα (v) · X eβ (−1) · X e−β (1) · X eβ (−1) 93 Apply the commutator formula for X eβ (1) and X eα (v). There are two new terms because (β, α) = {β + α, β + 2α}. wXw−1 = X eβ (1) · Xe−β (−1) · X eβ (1) · X eα (v) · X eβ (−1) · Xe−β (1) · X eβ (−1) =X eβ (1) · Xe−β (−1) · X eβ+α (N1 ) · X eβ+2α (N2 ) · X eα (v) · X eβ (1) ·X eβ (−1) · X e−β (1) · Xeβ (−1) where β,α N1 = N1,1 (1, v) β,α N2 = N1,2 (1, v) The adjacent factors X eβ (1) and X eβ (−1) cancel. wXw−1 = X eβ (1) · X e−β (−1) · X eβ+α (N1 ) · X eβ+2α (N2 )  : ·Xeα (v) · X e · Xβ (−1) · X−β (1) · Xβ (−1) eβ (1)  e e  =X eβ (1) · X e−β (−1) · X eβ+α (N1 ) · X eβ+2α (N2 ) · X eα (v) · X e−β (1) · Xeβ (−1) Apply the commutator formula for X e−β (−1) and X eβ+α (N1 ). There are two new terms because (−β, β + α) = {α, β + 2α}. wXw−1 = X eβ (1) · X e−β (−1) · X eβ+α (N1 ) · X eβ+2α (N2 ) · X eα (v) · X e−β (1) · Xeβ (−1) =X eβ (1) · X eα (N3 ) · Xeβ+2α (N4 ) · X eβ+α (N1 ) · X e−β (−1) ·Xeβ+2α (N2 ) · X eα (v) · X e−β (1) · Xeβ (−1) where −β,β+α N3 = N1,1 (−1, N1 ) −β,β+α N4 = N1,2 (−1, N1 ) 94 Both −β and β + 2α are long roots so X e−β (−1) and X eβ+2α (N1 ) commute. Also −β + α is not a root, so X e−β (−1) also commutes with X eα (v). wXw−1 = X eβ (1) · X eα (N3 ) · X eβ+2α (N4 ) · Xeβ+α (N1 ) ·Xe−β (−1) · X eβ+2α (N2 ) · Xeα (v) · X e−β (1) · X eβ (−1) =X eβ (1) · X eα (N3 ) · X eβ+2α (N4 ) · Xeβ+α (N1 ) ·Xeβ+2α (N2 ) · Xeα (v) · X e−β (−1) · X e−β (1) · X eβ (−1) The adjacent factors X e−β (−1) and X e−β (1) cancel. wXw−1 = X eβ (1) · X eα (N3 ) · X eβ+2α (N4 ) · Xeβ+α (N1 ) :   ·X eβ+2α (N2 ) · Xeα (v) · X  · X−β (1) · Xβ (−1) e−β (−1) e e  =X eβ (1) · X eα (N3 ) · X eβ+2α (N4 ) · Xeβ+α (N1 ) · Xeβ+2α (N2 ) · Xeα (v) · X eβ (−1) Since (β + 2α) + (β + α) = 2β + 3α is not a root, the X eβ+α (N1 ) and X eβ+2α (N4 ) terms commute. wXw−1 = X eβ (1) · X eα (N3 ) · X eβ+2α (N4 ) · Xeβ+α (N1 ) · Xeβ+2α (N2 ) · Xeα (v) · X eβ (−1) =X eβ (1) · X eα (N3 ) · X eβ+α (N1 ) · Xeβ+2α (N4 ) · Xeβ+2α (N2 ) · Xeα (v) · X eβ (−1) We can combine the adjacent X eβ+2α terms. wXw−1 = X eβ (1) · X eα (N3 ) · X eβ+α (N1 ) · Xeβ+2α (N4 ) · Xeβ+2α (N2 ) · Xeα (v) · X eβ (−1) =X eβ (1) · X eα (N3 ) · X eβ+α (N1 ) · Xeβ+2α (N4 + N2 ) · X eα (v) · X eβ (−1) Apply the commutator formula for X eβ (1) and X eα (N3 ). There are two factors introduced 95 because (β, α) = {β + α, β + 2α}. wXw−1 = X eβ (1) · X eα (N3 ) · X eβ+α (N1 ) · X eβ+2α (N4 + N2 ) · X eα (v) · Xeβ (−1) =Xeβ+α (N5 ) · Xeβ+2α (N6 ) · Xeα (N3 ) · Xeβ (1) ·Xeβ+α (N1 ) · Xeβ+2α (N4 + N2 ) · X eα (v) · Xeβ (−1) where β,α N5 = N1,1 (1, N3 ) β,α N6 = N1,2 (1, N3 ) Since 2β + α and 2β + 2α are not roots, X eβ (1) commutes with X eβ+α (N1 ) and Xeβ+2α (N4 + N2 ). wXw−1 = X eβ+α (N5 ) · Xeβ+2α (N6 ) · X eα (N3 ) · X eβ (1) ·X eβ+α (N1 ) · Xeβ+2α (N4 + N2 ) · X eα (v) · Xeβ (−1) =X eβ+α (N5 ) · Xeβ+2α (N6 ) · X eα (N3 ) ·Xeβ+α (N1 ) · Xeβ+2α (N4 + N2 ) · X eβ (1) · Xeα (v) · X eβ (−1) Apply the commutator lemma again for X eβ (1) and X eα (v). We get the same commutator coefficients N1 , N2 as before. e −1 = X wXw eβ+α (N5 ) · X eβ+2α (N6 ) · Xeα (N3 ) · Xeβ+α (N1 ) ·Xeβ+2α (N4 + N2 ) · Xeβ (1) · X eα (v) · X eβ (−1) =X eβ+α (N5 ) · X eβ+2α (N6 ) · Xeα (N3 ) · Xeβ+α (N1 ) · X eβ+2α (N4 + N2 ) ·Xeβ+α (N1 ) · X eβ+2α (N2 ) · Xeα (v) · Xeβ (1) · Xeβ (−1) 96 Then the Xeβ terms cancel. wXw−1 = X eβ+α (N5 ) · Xeβ+2α (N6 ) · X eα (N3 ) · Xeβ+α (N1 ) · Xeβ+2α (N4 + N2 )  : ·Xeβ+α (N1 ) · Xeβ+2α (N2 ) · X eα (v) · X e · Xβ (−1) eβ (1)   =X eβ+α (N5 ) · Xeβ+2α (N6 ) · X eα (N3 ) · Xeβ+α (N1 ) ·Xeβ+2α (N4 + N2 ) · X eβ+α (N1 ) · X eβ+2α (N2 ) · X eα (v) Since (β + α) + (β + 2α) = 2β + 3α is not a root, we can commute these terms. wXw−1 = X eβ+α (N5 ) · X eβ+2α (N6 ) · X eα (N3 ) · Xeβ+α (N1 ) ·Xeβ+2α (N4 + N2 ) · X eβ+α (N1 ) · X eβ+2α (N2 ) · Xeα (v) =X eβ+α (N5 ) · X eβ+2α (N6 ) · X eα (N3 ) · Xeβ+α (N1 ) ·Xeβ+α (N1 ) · X eβ+2α (N4 + N2 ) · X eβ+2α (N2 ) · Xeα (v) We can combine the adjacent X eβ+α terms. Similarly, we can combine the adjacent X eβ+2α terms. wXw−1 = X eβ+α (N5 ) · X eβ+2α (N6 ) · Xeα (N3 ) · X eβ+α (N1 ) · X eβ+α (N1 ) ·Xeβ+2α (N4 + N2 ) · X eβ+2α (N2 ) · Xeα (v) =X eβ+α (N5 ) · X eβ+2α (N6 ) · Xeα (N3 ) · X eβ+α (2N1 ) · X eβ+2α (N4 + 2N2 ) · X eα (v) Since (β + 2α) + α = β + 3α is not a root, we can commute the last two terms. wXw−1 = X eβ+α (N5 ) · X eβ+2α (N6 ) · Xeα (N3 ) · X eβ+α (2N1 ) · X eβ+2α (N4 + 2N2 ) · X eα (v) =X eβ+α (N5 ) · X eβ+2α (N6 ) · Xeα (N3 ) · X eβ+α (2N1 ) · X eα (v) · X eβ+2α (N4 + 2N2 ) Now (α, β + α) = (β + 2α), so when we apply the commutator formula for X eα (v) and 97 Xeβ+α (2N1 ) there is one new factor introduced. wXw−1 = X eβ+α (N5 ) · Xeβ+2α (N6 ) · Xeα (N3 ) · X eβ+α (2N1 ) · Xeα (v) · X eβ+2α (N4 + 2N2 ) =X eβ+α (N5 ) · Xeβ+2α (N6 ) · Xeβ+2α (N7 ) · X eβ+α (2N1 ) · X eα (N3 ) ·Xeα (v) · Xeβ+2α (N4 + 2N2 ) where α,β+α N7 = N1,1 (N3 , 2N1 ) We combine the adjacent X eα terms into a single term. Similarly, we combine the adjacent Xeβ+2α terms. wXw−1 = X eβ+α (N5 ) · Xeβ+2α (N6 ) · Xeβ+2α (N7 ) · Xeβ+α (2N1 ) ·Xeα (N3 ) · Xeα (v) · X eβ+2α (N4 + 2N2 ) =X eβ+α (N5 ) · Xeβ+2α (N6 + N7 ) · X eβ+α (2N1 ) · X eα (N3 + v) · Xeβ+2α (N4 + 2N2 ) Since (β + α) + (β + 2α) is not a root, we can commute these terms. wXw−1 = X eβ+α (N5 ) · Xeβ+2α (N6 + N7 ) · X eβ+α (2N1 ) · X eα (N3 + v) · Xeβ+2α (N4 + 2N2 ) =X eβ+2α (N6 + N7 ) · X eβ+α (N5 ) · X eβ+α (2N1 ) · X eα (N3 + v) · Xeβ+2α (N4 + 2N2 ) We combine the adjacent X eβ+α terms. wXw−1 = X eβ+2α (N6 + N7 ) · X eβ+α (N5 ) · X eβ+α (2N1 ) · X eα (N3 + v) · Xeβ+2α (N4 + 2N2 ) =X eβ+2α (N6 + N7 ) · X eβ+α (N5 + 2N1 ) · X eα (N3 + v) · Xeβ+2α (N4 + 2N2 ) Since (β+2α)+α and (β+2α)+(β+α) are both not roots, we can move the X eβ+2α (N4 +2N2 ) term on the far right through the two terms to its left. wXw−1 = X eβ+2α (N6 + N7 ) · X eβ+α (N5 + 2N1 ) · X eα (N3 + v) · Xeβ+2α (N4 + 2N2 ) =X eβ+2α (N6 + N7 ) · X eβ+2α (N4 + 2N2 ) · X eβ+α (N5 + 2N1 ) · X eα (N3 + v) 98 We combine the adjacent X eβ+2α terms. wXw−1 = X eβ+2α (N6 + N7 ) · X eβ+2α (N4 + 2N2 ) · Xeβ+α (N5 + 2N1 ) · X eα (N3 + v) =X eβ+2α (N6 + N7 + N4 + 2N2 ) · X eβ+α (N5 + 2N1 ) · X eα (N3 + v) In summary, we showed that eβ (1) · X w eβ (1)−1 = X eα (v) · w eβ+2α (N6 + N7 + N4 + 2N2 ) · X eβ+α (N5 + 2N1 ) · Xeα (N3 + v) To complete the proof, we need to show that N6 + N7 + N4 + 2N2 = 0 N5 + 2N1 = ±v±1 N3 + v = 0 This is done in Lemma B.8 below. Lemma B.8. Let α, β ∈ Φk with α a short root and β a long root, such that α + β ∈ Φk . Let v ∈ Vα (R) = RL . Using Lemma 2.2.4 as β = ε2βi and α = −εβi + ωβj using independent signs ε, ω, and let cij be as in Lemma 2.2.4. Then β,α N1 = N1,1 (1, v) = −ωvcij β,α N2 = N1,2 (1, v) = −εωvv −β,β+α N3 = N1,1 (−1, N1 ) = −v −β,β+α N4 = N1,2 (−1, N1 ) = −εωvv β,α N5 = N1,1 (1, N3 ) = ωvcij β,α N6 = N1,2 (1, N3 ) = −εωvv α,β+α N7 = N1,1 (N3 , 2N1 ) = 4εωvv 99 In particular, N6 + N7 + N4 + 2N2 = 0 N5 + 2N1 = −ωvδ ∈ {±v, ±v} N3 + v = 0 Proof. Using Lemma 2.2.4 we can write the N1 through N6 in terms of ε, ω, δ and vδ . β,α N1 = N1,1 (1, v) = −ωvδ β,α N2 = N1,2 (1, v) = −εωvv To calculate N3 and N4 , temporarily denote β 0 = −β = ε0 2βi α0 = β + α = −ε0 βi + ω 0 βj To calculate N3 and N4 using the corollary, we use the signs ε0 = −ε, ω 0 = ω, δ 0 = δ. We get (N1 )δ 0 = (−ωvδ )δ = −ωv N1 N 1 = (−ωvδ )(−ωvδ ) = vδ vδ = vv −β,β+α β 0 ,α0 N3 = N1,1 (−1, N1 ) = N1,1 (−1, N1 ) = −ω(−1)(−ωv) = −ω 2 v = −v −β,β+α β 0 ,α0 N4 = N1,2 (−1, N1 ) = N1,2 (−1, N1 ) = −ε0 ω 0 (−1)N1 N 1 = −εωvv Now we calculate N5 and N6 . β,α N5 = N1,1 (1, N3 ) = −ω(N3 )δ = −ω(−v)δ = ωvδ β,α N6 = N1,2 (1, N3 ) = −εωN3 N3 = −εω(−v)(−v) = −εωvv To calculate N7 , we use Lemma 2.2.4. The roots in question are α = −εβi + ωβj β + α = εβi + ωβj 100 To apply the lemma, we need to know which one of these roots has the form ±(βi + βj ) and which has the form ±(βi − βj ), but these depend on whether ε = ω or ε = −ω. So we consider these cases separately. (Case 1, ω = ε) Let α = β + α = ε(βi + βj ) = ε0 (βi + βj ), and let β = α = −ε(βi − βj ) = ω 0 (βi − βj ). So our signs are ε0 = ε and ω 0 = −ε. We still have δ 0 = δ = δ ij . By Lemma 2.2.4 (2),   α,β+α β,α α,β N7 = N1,1 (N3 , 2N1 ) = N1,1 (N3 , 2N1 ) = −N1,1 (N3 , 2N1 ) = ε0 Tr (N3 )ε0 ω0 δ 0 · 2N1     = ε Tr (−v)−ε2 δ · (−2ωvδ ) = εω Tr 2v−δ vδ = 2εω Tr(vv) = 4εωvv (Case 2, ω = −ε) Let α = α = −ε(βi + βj ) = ε0 (βi + βj ) and let β = β + α = ε(βi − βj ) = ω 0 (βi − βj ), so our signs are ε0 = −ε and ω 0 = ε. We still use the same δ 0 = δ = δ ij . By Lemma 2.2.4 (2),   α,β+α α,β N7 = N1,1 (N3 , 2N1 ) = N1,1 (N3 , 2N1 ) = −ε0 Tr (N3 )ε0 ω0 δ 0 · 2N1     = ε Tr (−v)−ε2 δ · (−2ωvδ ) = 2εω Tr v−δ vδ = 2εω Tr(vv) = 4εωvv So in both cases we can write N7 as 4εωvv. This completes our calculation of the N values. The relations follow immediately. Generalization of Lemma 2.4.2 In Remark 2.4.3, we noted that the calculations in Lemma 2.4.2 can be done in slightly more generality, so we do this here. Lemma B.9. Let α, β ∈ Φk with α short, β long, and such that β − α ∈ Φk . Then there is an isomorphism of algebraic varieties π : Bα → Aβ such that π ◦ gα = fβ . 101 Proof. This is just a slight generalization of the computation for Lemma 2.4.2. By Lemma 2.2.4 (2), we can write α and β − α as α = ε(αi − αj ) β − α = ω(αi + αj ) for some signs ε = ±1, ω = ±1 and indicies i, j with i < j, and futhermore α,β−α N11 (u, v) = ω Tr (u−εω v) Now define π : Bα → H by h  ω i π(x) = x, gβ−α 2d Note that 2d is invertible because char k = 0. This commutator occurs inside GLm (K) and multiplication is regular, so π is regular. Now let s ∈ k. Then "  √  √ !!#   √      ω d ω π gα (s) = ρ Xα s d , ρ Xβ−α = ρ Xα s d , Xβ−α √ 2d 2 d √     α,β−α ω    ωs   = ρ Xβ N11 s d, √ = ρ Xβ ω Tr = ρ Xβ (s) = fβ (s) 2 d 2 This shows that π ◦ gα = fβ , and that π maps gα (k) into fβ (k). Since π is regular, it follows that π(Bα ) ⊂ Aβ . It remains to show that π is invertible (with regular inverse). By Lemma 2.2.4(3), we can write α − β = ε0 αi0 + ω 0 αj 0 β = −ε0 2αi0 for some signs ε0 = ±1, ω 0 = ±1, and indices i0 , j 0 , and furthermore β,α−β β,α−β N11 (v, u) = −ω 0 u−c 0 0 v N12 (v, u) = ε0 ω 0 vuu ij Let c = −ci0 j 0 and let h = h2α−β (1/2) be the element introduced in Definition 2.2.6. Then define −1 ν(y) = ρ(h) · y, gα−β (−ω 0 c) · y, gα−β (ω 0 c) · ρ(h)−1 ,    ν : Aβ → Bα , 102 It is clear that ν is a regular map; we claim it is an inverse for π. Let s, t ∈ k, and note that √  √ d = c d. Using the commutator relation from Theorem 2.2.1, we have c h  √ i   √    √  β,α−β β,α−β Xβ (t), Xα−β s d = Xα N11 t, s d · X2α−β N12 t, s d   √     √   √  0 = Xα − ω t s d 0 0 · X2α−β ε ω t s d s d c  √    = Xα − ω 0 cts d · X2α−β − ε0 ω 0 ts2 d . Specializing the above to the case s = −ω 0 c and noting that (ω 0 c)2 = 1, we get h  √ i h  √ i−1 Xβ (t), Xα−β −ω 0 c d · Xβ (t), Xα−β ω 0 c d √   0 : 0   0   0 −1 :  √ −1  √  = Xα t d · X2α−β  − ε ω td · X2α−β   − ε  ω td · X α − t d = X α 2t d .     We also have the relation h · Xα (2v) · h−1 = Xα (v) for all v ∈ L. Putting everything together, we obtain √ i h  √ i−1  h    0 ν fβ (t) = ρ h · Xβ (t), Xα−β −ω c d · Xβ (t), Xα−β ω c d 0 ·h −1   √     √  = ρ h · Xα 2t d · h−1 = ρ Xα t d = gα (t). Thus, ν ◦ π and π ◦ ν are the respective identity maps on dense subsets of Aβ and Bα . Since they are regular, it follows that they are the identity on the whole space, so ν is the inverse of π as claimed. Lemma 2.4.2 is the special case of Lemma B.9 where α = α1 − α3 and β = −2α3 . In this case, the various signs and indices are i=1 j=3 ε=1 ω = −1 i0 = 3 j0 = 1 ε0 = 1 ω0 = 1 103 Appendix C Logical dependency chart In this appendix we depict logical dependencies among various lemmas, propositions, and theorems from this document. Arrows in these diagrams do not indicate logical implication, only that the proof of the target depends on the source. Key results from outside sources are included as well. Section 2.2 and Appendix B Theorem 2 [PS08] Relative root subschemes Theorem 2.2.1 Relative root subgroups for SU2n (L, h) Lemmas B.1, B.2, B.3, B.5 αβ Various properties of maps Nij Lemma 2.2.4/B.4 αβ Explicit formula for Nij Lemma 2.3.3 Lemma 2.4.2 G(R) e is perfect Bα ∼ = Aβ 104 Section 2.3 Theorem 1.3 [Sta20] ker πA is central for local A Lemma 2.3.3 Lemma 2.3.4 Proposition 2.3.2 G(R) e is perfect G e respects products ker πA is central for semilocal A Proposition 2.5.2 Existence of σ 105 Section 2.4 Lemma 2.2.4 αβ Explicit formula for Nij Lemma 2.4.5 Lemma 2.4.2 Theorem 3.1 [Rap11] Reduction step Bα ∼= Aβ Algebraic ring associated to an abstract representation of a Chevalley group Proposition 2.4.6 Existence of ψα Lemma 2.3.3 Proposition 2.4.8 Proposition 5.1 [Gre64] G(R) e is perfect Existence of σe Connected algebraic rings in characteristic zero are equivalent to algebras Proposition 2.3.2 Lemma 2.5.1 Remark 2.4.7 ker πA is central H is connected A is a K-algebra Proposition 2.5.2 Existence of σ Proposition 2.5.3 σ is algebraic 106 Section 2.5 Proposition 2.4.6 Existence of ψα Proposition 2.3.2 Lemma 2.5.1 ker πA is central H is connected Lemma 3.10 [Rap13] Proposition 2.5.2 Theorem 6.3 [Hum72] Lemma 2.3.3 Lemma 3.12 [Rap13] Existence of σ Weyl’s theorem G(R) e is perfect Proposition 2.5.3 Lemma 2.5.5 Lemma 2.5.4 σ is algebraic Property (Z) Levi decomposition of G(A) Theorem 2.0.1/2.5.6 (BT) conjecture for SU2n (L, h) Section 3.1 Theorem 1.1 [Rap19] Proposition 5.1[Rap13] (BT) conjecture for Remarks 3.1.4, 3.1.5, 3.1.8 Refined (BT) conjecture for elementary groups Lemmas 3.1.6, 3.1.7 elementary groups over rings with (D1 ) over rings of S-integers Theorem 3.1.2 Theorem 3.1.3 Rigidity for biregular action Rigidity for biregular action of elementary groups of elementary groups over rings with (D1 ) over rings of S-integers on affine varieties on affine varieties 107 Section 3.2 Theorem 2 [CdC19] Theorem 1.1 [EJZK17] Theorem 1.1 [Rap19] Criterion for Elementary group has (BT) conjecture for conjϕ (Γ) ⊂ Aut(Y ) Kazhdan’s property (T) elementary groups over rings with (D1 ) Theorem 3.2.1 Rigidity for birational action of elementary groups over rings with (D1 ) on projective surfaces Section 4.2 Lemma 2.2.8 Proposition 2.4.8 Symbols in ker πA Existence of σ e Conjecture 4.2.1 Lemma 4.2.3 ker πA generated by symbols Symbols in ker σ e Lemma 3.10 [Rap13] Proposition 4.2.4 Lemma 3.12 [Rap13] Existence of σ Proposition 4.2.5 σ is algebraic Theorem 4.2.2 (BT) conjecture for SU2n (L, h) 108 BIBLIOGRAPHY 109 BIBLIOGRAPHY [AM16] M. 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