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Reductive Groups Over Fields REDUCTIVE GROUPS OVER FIELDS LECTURES BY BRIAN CONRAD, NOTES BY TONY FENG August 11, 2017 These are lecture notes that Tony Feng live-TEXed from a course given by Brian Con- rad at Stanford in“winter” 2015, which Feng and Conrad edited afterwards. Two substi- tute lectures were delivered (by Akshay Venkatesh and Zhiwei Yun) when Conrad was out of town. This is a sequel to a previous course on the general structure of linear al- gebraic groups; some loose ends from that course (e.g., Chevalley’s self-normalizing theorem for parabolic subgroups and Grothendieck’s theorem on geometrically maxi- mal tori in the special case of finite ground fields) are addressed early on. The main novelty of the approach in this course to avoid the two-step process of first developing the structure theory of reductive groups over algebraically closed fields and then using that to establish the refined version over general fields. Instead, systematic use of dynamic techniques introduced in the previous course (and reviewed here) make it possible to directly establish the general structure theory over arbitrary fields in one fell swoop (building on the “geometric” theory of general linear algebraic groups from the previous course, where reductivity was not the main focus). This document sometimes undergoes minor updates to make corrections (typos, etc.) or clarifications. Please send errata/comments to [email protected]. CONTENTS 1. Basic structure of reductive groups 2 2. The unipotent radicals 9 3. Central isogeny decomposition 14 4. Grothendieck’s covering theorem 24 5. Exponentiating root spaces 29 6. Dynamic description of parabolic subgroups 44 7. Maximal split tori and minimal parabolic subgroups 56 8. Structure theory of reductive groups I 67 9. Root systems 74 10. Structure of reductive groups II 94 11. Relative root systems and applications 106 12. The -action and Tits–Selbach classification 118 References∗ 135 1 2 LECTURES BY BRIAN CONRAD, NOTES BY TONY FENG 1. BASIC STRUCTURE OF REDUCTIVE GROUPS 1.1. Linear algebraic groups. Let’s review some notions from the previous course. Definition 1.1.1. For a field k, a linear algebraic group over k is a smooth affine k-group scheme (equivalently, a smooth closed k-subgroup of GLn ). Remark 1.1.2. We allow linear algebraic groups to be disconnected. However, the iden- tity component G 0 is geometrically connected over k. This follows from a general fact (an instructive exercise) that if X is finite type over k and X (k) = , then X is connected if and only if X is geometrically connected. 6 ; One uses this fact all the time when calculating with geometric points of normalizers, centralizers, etc. to ensure that one does not lose contact with connectedness. 1.2. Reductive groups. Recall the brute-force definition of reductivity: Definition 1.2.1. A reductive k-group is a linear algebraic group G over k whose geo- metric unipotent radical (maximal unipotent normal smooth connected k-subgroup) Ru (Gk ) is trivial. Example 1.2.2. Many classical groups are reductive (verified in lecture or homework of the previous course, but some to be revisited from scratch in this course): SO(q ) for finite-dimensional quadratic spaces (V,q ) with q = 0 that are non- • degenerate. (Non-degeneracy is defined by smoothness of the6 zero-scheme of the projective quadric (q = 0). This works uniformly in all characteristics.) U h and SU h for non-degenerate finite-dimensional hermitian spaces V ,h ( ) ( ) ( 0 ) • with respect to quadratic Galois extensions k k. 0= Sp( ) for a non-degenerate finite-dimensional symplectic space (V, ). • A× for A a finite-dimensional central simple algebra over k, representing the • functor R A R on (commutative) k-algebras. ( k )× ⊗ Remark 1.2.3. We shall use throughout Grothendieck’s fundamental theorem, proved in the previous course, that maximal k-tori in a linear algebraic group G over k are geometrically maximal (i.e., remain maximal over k, or equivalently after any field ex- tension). In particular, all such tori have the same dimension, called the rank of G . Our proof of Grothendieck’s theorem applied to infinite k; the handout on Lang’s the- orem (and dynamic methods) takes care of the case of finite fields, so Grothendieck’s theorem is thereby established in general. The following was a major result near the end of the previous course, to be used a lot in this course. Theorem 1.2.4. If G is connected reductive and split (i.e. has a split maximal k-torus) and has rank 1 then G = Gm ,SL2, or PGL2 as k-groups. There are a lot more properties of reductive groups that we would like to investigate (some to be addressed in handouts of this course), such as the following. If G G is a surjective homomorphism of linear algebraic k-groups then we 0 • would like to show R G R G , so G is reductive if G is. u ( k ) u ( k0 ) 0 REDUCTIVE GROUPS OVER FIELDS 3 If char k = 0 and G is reductive, then all linear representations of G on finite- • dimensional k-vector spaces are completely reducible. (The converse is a con- sequence of the Lie-Kolchin theorem, but in characteristic p > 0 only applies to special G such as tori and finite étale groups of order not divisible by p.) Structural properties of connected reductive k-groups G , such as: • – for locally compact k, relate compactness of G (k) to properties of G as an algebraic group. For instance if G contains Gm = GL1 then G (k) is not com- pact since its closed subgroup k is non-compact; we want to show that × this is the only way compactness fails. – for general k, prove G (k)-conjugacy for maximal split k-tori and minimal parabolic k-subgroups. Build a “relative root system”. – use root systems and root data to analyze the k-subgroup structure of G (e.g., structure of parabolic k-subgroups) and the subgroup structure of G (k) (e.g., simplicity results for G (k)=ZG (k), at least for k-split G ). – for k = R, understand π0(G (R)) and proves that if G is semisimple and sim- ply connected in an algebraic sense then G (R) is connected. 1.3. Chevalley’s Theorem. Recall that a smooth closed k-subgroup P G is parabolic if the quasi-projective coset space G =P is k-proper, or equivalently projective.⊂ (This can be checked over k.) By the Borel fixed point theorem, which says that a solvable connected linear algebraic k-group acting on a proper k-scheme has a fixed point over k, Pk contains a Borel subgroup (it is a simple group theory exercise to show that Pk contains a G (k)-conjugate of a Borel B Gk if B acting on (G =P )k has a fixed point). Here is the key result which enabled⊂ Chevalley to get his structure theory over alge- braically closed fields off the ground (as Chevalley put it, once the following was proved “the rest follows by analytic continuation”): Theorem 1.3.1 (Chevalley). Let G be a connected linear algebraic k-group and P G a ⊂ parabolic k-subgroup. Then P is connected and NG (K )PK = P (K ) for any extension K =k. Remark 1.3.2. One could ask if P = NG (P ), the scheme-theoretic normalizer. [See HW3, Exercise 3 of the previous course for the notion and existence of scheme-theoretic nor- malizers.] The answer is yes, but the proof uses a dynamic description of P . We’ll ad- dress this in Corollary 6.3.12. To prove Chevalley’s Theorem (stated without proof in the previous course but used crucially there, such as to prove Theorem 1.2.4!), first we want to pass to an algebraically closed field. For connectedness it is harmless to do this; what about the normalizer property? Note that N P N P G K G (K )( K ) = G (K )( K ) ( ) \ so if N P P K then the right side is P K . Thus, without loss of generality we G (K )( K ) = ( ) ( ) may assume that K = k = k. Next note that the normalizer property implies the connectedness. Indeed, if P con- tains a Borel, then P 0 contains a Borel (by definition Borel subgroups are connected), so P 0 is parabolic. Therefore, if the normalizer property is proved in general, then we can 4 LECTURES BY BRIAN CONRAD, NOTES BY TONY FENG apply this to P 0. Any group normalizes its own identity component, so we immediately get that P is connected. We claim that it suffices to show that NG (k)(B) = B(k) for one Borel subgroup B. Any two such B are G (k)-conjugate, so it is the same for that to hold for all Borel subgroups. Grant this equality. For general P , choose B P . Consider n NG (k)(P ); we want to show that n P . Consider the conjugation action⊂ of n on B. The2 element n doesn’t nec- essarily conjugate2 B into itself, but for our purposes it is harmless to change n by P (k)- translation. Note that nBn 1 is a Borel subgroup of P . But any two Borel subgroups of − any linear algebraic group are conjugate, so nBn 1 p Bp 1 for some p P k . This − = − ( ) implies that p 1n N B B k P k , so n P k . 2 − G (k)( ) = ( ) ( ) ( ) 2 ⊂ 2 Now we focus on the assertion NG (k)(B) = B(k). We proceed by induction on dimG . If G is solvable then the result is easy (G = B), so the case dimG 2 is fine.
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