Math 249B. Geometric Bruhat decomposition


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1 Math 249B. Geometric Bruhat decomposition 1. Introduction Let (G, T ) be a split connected reductive group over a field k, and Φ = Φ(G, T ). Fix a positive system of roots Φ Φ, and let B be the unique Borel ksubgroup of G containing T such that Φ(B, T ) = Φ. Let W = W (G, T )(k) = N G(k) (T )/T (k), and for each W let n N G(k) (T ) be a representative of W. For each a Φ e let r a W be the associated involution, and e let denote the base of Φ. Under the action of B B on G by (b, b ).g = bgb 1, the orbit BgB through any g G(k) is a locally closed subvariety (as for orbits of linear algebraic groups in general). The double coset C() = Bn B (a Bruhat cell) depends only on, not n, and in class e proved that C() decomposes as a direct product scheme under multiplication: for the closed subset the multiplication map Φ = Φ ( Φ ) Φ, U Φ n B C() is an isomorphism of kschemes. Equivalently, the left translate n 1 C() is identified ith the closed subscheme U 1 (Φ ) B in the open cell Ω = U Φ B G. (Thus, C() is open in G if and only if Φ = Φ, hich is to say that (Φ ) = Φ. There is exactly one such 0, and C( 0 ) is the unique open Bruhat cell.) Remark 1.1. The kscheme U Φ is a direct product (under multiplication, in any order) of the root groups U a for a Φ, so it is an affine space of dimension #Φ. This cardinality has a simple combinatorial interpretation: it is the length l() relative to the generating set {r a } a of W. (That is, this is the minimal length of a ord in the r a s hose product is equal to.) In particular, l() #Φ ith equality if and only if C() is the unique open Bruhat cell. See Corollary 2 to Proposition 17 in 1.6 of Chapter VI of Bourbaki for a proof. By Corollary 3 to Proposition 17 in 1.6 of Chapter VI of Bourbaki there is a unique longest element 0 W relative to {r a } a, it is given by the product a Φ r a taken in any order, and 0 (Φ ) = Φ. In particular, 0 2 = 1 since 2 0 (Φ ) = Φ. We call 0 the long Weyl elements. The purpose of this handout is to prove that the subvarieties C() are a pairise disjoint covering of G; e call this the geometric Bruhat decomposition since it concerns covering the entire scheme G (and since each C() is locally closed, this is equivalent to a covering statement at the level of kpoints). In the setting of possibly nonsplit connected reductive groups there ill be an analogous covering result (called the Bruhat decomposition) that is only valid at the level of kpoints, recovering the geometric version over algebraically closed fields. Remark 1.2. The Zariski closure of each C() is a union of Bruhat cells C( ) for some subset of elements W. Indeed, to prove this it suffices to ork over k (since the formation of the Zariski closure of a locally closed subscheme commutes ith extension of the ground field), and this closure is visibly stable under left and right multiplication by B. Hence, the closure of C()(k) inside G(k) is a union of B(k) double cosets, so it is indeed a union of Bruhat cells. But hich C( ) lie in the closure of C()? This has a nice combinatorial anser: if e fix a reduced decomposition = r a1 r al() (i.e., a minimal length expression for as a product of the r a s, alloing some repetition in the sequence {a j }) then C( ) C() if and only if is obtained as a subproduct of r aj by removing some terms: = r ai1 r ain ith 1
2 2 1 i 1 < < i n l(). We ill not use this important result; see in Springer s book for a proof. 2. Bruhat decomposition We begin by proving the disjointness of the Bruhat cells. This ill ultimately reduce to a general result concerning torus actions on unipotent groups. Proposition 2.1. If then C() C( ) =. Proof. Since any to B(k) double cosets in G(k) are either equal or disjoint, e just have to rule out the possibility that n C( )(k) = U Φ (k)n B(k). That is, for, W e shall prove that an equality n = un b ith u U Φ (k) and b B(k) forces =. Let U = U Φ, and define the closed set of roots Φ = Φ 1 (Φ ) inside Φ, so Φ = Φ Φ and U = U Φ U Φ under multiplication. Similarly define Φ. For u as above, e claim that (1) U Φ = uu Φ u 1. To establish this equality, e first prove the formula U Φ = U n Bn 1 for any W. A point u of U lies in n Bn 1 if and only if n 1 un B. But if e rite u = u u under the decomposition U = U Φ U Φ (via multiplication inside G) then n 1 un = (n 1 u n )(n 1 u n ) ith n 1 u n n 1 U Φ n = U 1 (Φ U ) Φ and likeise n 1 un U Φ. Hence, the direct product structure of the open cell Ω = U Φ B implies that n 1 un B if and only if u = 1, hich is to say u U Φ. But e can compute U n Bn 1 in another ay: since n = un b, clearly n Bn 1 = un n 1 u 1, so Thus, e have proved (1). No comes the key point: U n Bn 1 = u(u n Bn 1 )u 1 = uu Φ u 1. Lemma 2.2. Let U be a unipotent smooth connected kgroup equipped ith an action by a split k torus T such that all T eights on Lie(U) are nontrivial, pairise linearly independent in X(T ) Q, and have a 1dimensional eight space. If V, V U are T stable smooth connected ksubgroups that are U(k)conjugate then they are equal inside U. Proof. We may and do assume k = k, and e argue by induction on dim U (the case dim U = 0 being trivial). For each a Φ(U, T ), let T a = (ker a) 0 red. The centralizer U a = Z U (T a ) of the T a action on U is smooth and connected, ith Lie(U a ) = u a since Φ(U, T ) Qa = {a} by the hypotheses. Thus, U a is 1dimensional, so U a G a. The 1dimensionality implies by dynamic considerations (!) that U a is the unique nontrivial smooth connected T stable ksubgroup of U ith T eight a on its Lie algebra. The ksubgroups U a for a Φ(U, T ) generate U since their Lie
3 algebras span Lie(U). These conclusions also apply to any T stable smooth connected ksubgroup of U in place of U. The unipotent U is nilpotent (see the handout Nilpotence of unipotent groups ), so (Z U ) 0 red is nontrivial and T stable. The above reasoning can be applied to (Z U ) 0 red in place of U, so (Z U) 0 red contains U a for each T eight a on Lie((Z U ) 0 red ). Fix such a eight a, and consider the central quotient U/U a. The images of V and V in this quotient coincide (by dimension induction), so V U a = V U a. If the central U a U is contained in one of V or V then it is contained in both (as V and V are U(k)conjugate), in hich case V = V U a = V U a = V as desired. Hence, e can assume that U a is not contained in V nor in V. The T eight a cannot occur on Lie(V ) or Lie(V ) (as otherise the construction of U a could be carried out inside V or V, a contradiction), so the intersections V U a and V U a have vanishing Lie algebra and thus are étale. But U a = G a on hich T acts as t.x = a(t)x for the nontrivial character a of T, so visibly U a has no nontrivial T stable finite étale ksubgroup. In other ords, the surjective homomorphisms V U a V U a and V U a V U a are isomorphisms. Passing to Lie algebras and comparing T eights, e see that Φ(V, T ) = Φ(V, T ) inside Φ(U, T ). But the unipotent smooth connected V is generated by the groups U a for a Φ(V, T ), and similarly for V, so V = V inside U as desired. It follos from the lemma that U Φ = U Φ inside U = U Φ, so Φ = Φ inside Φ. Passing to complements in Φ, e also have Φ = Φ. Thus, Φ Φ = Φ Φ. Denoting this set as Ψ, e have (Ψ) = (Φ ) ( Φ ) = (Φ (Φ )) (Φ ( Φ )) = Φ and similarly (Ψ) = Φ. Thus, 1 (Φ ) = Φ, so by simple transitivity of the W action on the set of positive systems of roots in Φ e have =. Proposition 2.3 (Geometric Bruhat decomposition). The locally closed subvarieties {C()} W cover G. In particular, G(k) is covered by the disjoint subsets C()(k) = U Φ (k)n B(k) and the natural map W (G, T )(k) B(k)\G(k)/B(k) is bijective. Proof. We first treat the case hen the split connected reductive G has semisimplerank 1 (i.e., D(G) is kisomorphic to SL 2 or PGL 2 ). Let W be the unique nontrivial element, so C() = Un B for U = U Φ = G a. Thus, C()/B is an affine line that is open in G/B P 1 k, so its complement in G/B is a single krational point. The extra point C(1)/B accounts for this, so C() C(1) = G. In general, for every a Φ(G, T ) and the associated codimension1 torus T a = (ker a) 0 red, G a := Z G (T a ) is split connected reductive ith semisimplerank 1 and W (G a, T )(k) = {1, r a }. A Borel subgroup of G a containing T is B a = T U a B, so for n a N Ga(k)(T ) representing r a e have (2) G a = B a Ua n a B a. For each a, e claim that (3) G a Bn B Bn B Bn a n B, or equivalently G a C() C() C(r a ). Once this is proved, it follos that W C()(k) is stable under left multiplication by the subgroups G a (k) for a. But G(k) is generated by the subgroups G a (k) for a since T, U ±a G a and the elements n a U a, U a generate W (and W = Φ, so every root group is in the W orbit of some U a ith a ). Thus, W C()(k) is 3
4 4 stable under left multiplication by G(k) and hence it coincides ith G(k). This implies the Bruhat decomposition (conditional on (3)), as each C() is locally closed in G. To prove (3), e fix a, so Ψ = Φ {a} is a closed set of roots contained in Φ. The associated unipotent smooth connected ksubgroup U Ψ U Φ = U satisfies under multiplication. Thus, U = U Ψ U a = U a U Ψ G a Bn B = G a UT n B = G a U a U Ψ n B = G a U Ψ n B. Lemma 2.4. For each a and Ψ = Φ {a}, G a U Ψ = U Ψ G a. Proof. Since G a is generated by T, U a, U a, it suffices to prove that U Ψ is normalized by each of T, U a, and U a. Normalization by T is obvious, and the cases of U a and U a are equivalent upon replacing the positive system of roots Φ = Ψ {a} in Φ ith a (Φ ) = Ψ { a}. Thus, e focus on U a conjugation. Fix isomorphisms u c : G a U c for c Φ, so for b ±a in Φ e have u a (x)u b (y)u a (x) 1 u b (y) 1 U ( a b ) Φ. Thus, u a (x)u b (y)u a (x) 1 U ( a b ) Φ U b U Ψ since U Ψ is directly spanned in any order by the root groups U c for c Ψ. This gives that u a (x) conjugates U b into U Ψ for all b Ψ, so U a normalizes U Ψ. The preceding lemma implies (via (2)) that G a Bn B = U Ψ G a n B = U Ψ (B a U a n a B a )n B Bn B Un a B a n B = Bn B Un a U a n B since B a n = U a T n = U a n T ith T B. But Un a U a = UU a n a, so We claim that G a Bn B Bn B BU a n a n B. (4) G a U a n Bn 1 U a n a n Bn 1, from hich it ould follo that U a n a n U a n B U a n a n B, so BU a n a n B Bn B Bn a n B and hence G a Bn B Bn B Bn a n B as desired for (3). To prove (4) it is harmless to replace the Borel subgroup n Bn 1 of G containing T ith its intersection G a (n Bn 1 ). This intersection is a Borel subgroup of G a = Z G (T a ) containing T, so it is equal to one of the groups B ±a = T U ±a. Hence, (4) is equivalent to G a? = Ua B ±a U a n a B ±a for each of the roots ±a. For the case of B a this is the Bruhat decomposition (2) of the semisimplerank 1 group G a relative to (B a, T ), and for the case of B a this is the left n a translate of the Bruhat decomposition of G a relative to the pair (B a, T ). Here is an important application of the Bruhat decomposition. Proposition 2.5 (Chevalley). Let (G, T ) be a split connected semisimple group over a field k, and assume that G is simply connected. Then G(k) is generated by the subgroups U c (k) for c Φ(G, T ).
5 Proof. Let B be a Borel ksubgroup containing T, Φ = Φ(B, T ) a positive system of roots in Φ = Φ(G, T ), and the base of Φ. By the Bruhat decomposition, G(k) is generated by the subgroups U c (k), T (k), and n a N G(k) (T ) for representatives n a of the generating set {r a } a of W = W (G, T )(k). Since G is simply connected, e have an isomorphism G m T via (λ a ) a a a (λ a ). This direct product structure implies that even on rational points, T (k) is generated by its subgroups a (k ) for a. Hence, G(k) is generated by: the subgroups {U c (k)} c Φ, the subgroups a (k ) for a, and representative elements n a for a. But Φ = W, so in this generating list e can limit the root groups considered to just those associated to roots in. But U a, U a SL 2 in hich U ±a go over to the standard unipotent ksubgroups U ±, a (k ) goes over to the subgroup of diagonal elements, and n a goes over to the standard Weyl element. It is classical that SL 2 (k) is generated by U ± (k), so e are done. The simply connected hypothesis in Proposition 2.5 cannot be dropped. To prove this, consider a split connected semisimple kgroup G. Let f : G G be the simply connected central cover and T the split maximal ktorus preimage of T in G. We have seen in class (Example 5.2.6) that the bijection Φ( G, T ) = Φ(G, T ) induced by X(f) : X(T ) Q X( T ) Q yields isomorphisms U c U c for corresponding roots c Φ(G, T ) and c Φ( G, T ). Proposition 2.5 implies that the subgroups U c (k) G(k) generate the image of G(k) G(k), a normal subgroup for hich the cokernel is a generally nontrivial commutative group. (For example, SL n (k) PGL n (k), has cokernel k /(k ) n, and in general the cokernel is a subgroup of H 1 (k, ker f).) More explicitly, the commutator morphism G G G factors uniquely through a morphism G G G since G is a central quotient of G, and this induced morphism is a lift of the commutator morphism of G. Hence, the commutator subgroup of G(k) is contained in the image of G(k) G(k). 3. The long Weyl element As e noted in Remark 1.1, there is a unique longest element 0 W relative to {r a } a, and it is characterized by the property that 0 (Φ ) = Φ. It is therefore natural to onder if perhaps 0 = 1. This holds if and only if 1 W inside GL(X(T )), since if 1 W then it satisfies the property uniquely characterizing 0 W through its effect on Φ. But does 1 belong to W? Inspection of the Bourbaki Plates shos that 1 W precisely for the root systems A 1, B n (n 2), C n (n 3), D 2m (m 2), E 7, E 8, F 4, and G 2. In the other cases (i.e., A n for n 2, D 2m1 for m 2, and E 6 ), the Bourbaki Plates sho that 0 = ι here ι on Q arises from the unique diagram involution of Φ. Remark 3.1. Don t confuse 0 = a Φ r a (independent of the order of multiplication) ith the Coxeter element a r a that does depend on the order of multiplication but has conjugacy class independent of all choices (see Bourbaki Chapter IV, 1.11). Whereas 0 has order 2, the order of the Coxeter element is the Coxeter number h that is generally larger than 2. We finish by describing the long Weyl element and the Coxeter element (up to conjugacy) for the root system A n ith n 1. Recall that the eight lattice is X = Z n1 /diag. Letting ε 1,..., ε n1 be the standard basis of Z n1, a root basis is given by a i = ε i ε i1 modulo the diagonal, for 1 i n. For 1 i n, the effect of the reflection r ai is given by negation of a i, a i1 a i1 a i if i < n, a i 1 a i 1 a i if i > 1, and no effect on any other a j. This is induced by the linear automorphism of Z n1 (preserving the diagonal!) given by sapping ε i and ε i1 and leaving all other ε j unaffected. In terms of the indexed set of residue classes of ε j s in the eight lattice, it follos that the subgroup 5
6 6 S n1 GL(Z n1 ) arising as permutations of the standard basis (preserving the diagonal) maps onto W GL(X). The resulting map S n1 W is an isomorphism since the finite kernel consists of unipotent elements. Under this description of W, the long Weyl element 0 is induced by the sapping of ε i and ε n2 i for all 1 i n 1; i.e., it saps a i and a n1 i for all 1 i n. This shos quite explicitly that 0 1 hen n 2. Likeise, the Coxeter element (up to conjugacy) is induced by the left shift ε i ε i 1 using indexing modulo n 1 (so ε 1 ε n1 ). In terms of, this is induced by a i a i 1 for 1 < i n and a 1 (a 1 a n ). Clearly this latter operation is not an involution hen n > 2.
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