The Bruhat decomposition is a basic fact about Lie groups. Remarkably for some-thing so basic, it went undiscovered for a long time. It originated in Ehresmann’s study of flag manifolds, but was not really articulated until Bruhat’s work in the 1950s.
Tits found axioms, which were slightly generalized later by Iwahori and Mat-sumoto. Let G be a group and B, N subgroups. It is assumed that T = N ∩B is normal in N. The group W = N/T will be a Weyl group. If w ∈ W, then w is actually a coset ωT, but we will write wB, Bw and BwB to denote the cosets and double cosetωB,Bω and BωB. These do not depend on the representative ω since T ⊆B.
Axiom TS1. The group T =B∩N is normal in N;
Axiom TS2. There is specified a set I of generators of the group W = N/T such that if s∈I then s2 = 1;
Axiom TS3. Let w∈W and s ∈I. Then
wBs⊂BwsB∪BwB; (34)
Axiom TS4. Let s∈I. Then sBs−1 6=B;
Axiom TS5. The group G is generated by N and B.
Then we say that (B, N, I) is a Tits’ system.
We will be particularly concerned with the double cosets C(w) = BwB with w∈W. Then Axiom TS3 can be rewritten
C(w)C(s)⊂ C(w)∪ C(ws),
which is obviously equivalent to (34). Taking inverses, this is equivalent to
C(s)C(w)⊂ C(w)∪ C(sw). (35)
Theorem 9 Let (B, N, I) be a Tits’ system within a group G, and let W be the corresponding Weyl group. Then
G= [
w∈W
BwB, (36)
and this union is disjoint.
Proof Let us show that S
w∈WC(w) is a group. It is clearly closed under inverses.
We must show that it is closed under multiplication.
So let us consider C(w1)· C(w2), wherew1, w2 ∈W. We will denote by l(w) the length of a shortest decomposition of w ∈ W into a product of elements of I. We show by induction on l(w2) that this is contained in a union of double cosets. If l(w2) = 0, then w2 = 1 and the assertion is obvious. If l(w2) > 0, write w2 = sw20 wheres ∈I and l(w02)< l(w2). Then by Axiom TS3, we have
C(w1)· C(w2) =Bw1Bsw20B ⊂Bw1Bw20B ∪Bw1sBw02B, and by induction, this is contained in a union of double cosets.
We have shown that the right side of (36) is a group, and since it clearly contains B and N, it must be all of G by Axiom TS5.
It remains to be shown that the union (36) is disjoint. Of course two double cosets are either disjoint or equal. So assume that C(w) = C(w0) where w, w0 ∈W. We will show that w=w0.
Without loss of generality, we may assume that l(w) 6 l(w0), and we proceed by induction on l(w). If l(w) = 0, then w = 1, and so B = C(w0). Thus in N/T a representative for w0 will lie in B. Since B ∩N = T, this means that w0 = 1, and we are done in this case. Assume therefore that l(w) > 0, and that whenever C(w1) =C(w01) with l(w1)< l(w) we have w1 =w01.
Write w = w00s where s ∈ I and l(w00) < l(w). Thus w00s ∈ C(w0), and since s has order 2, we have
w00 ∈ C(w0)s ⊂ C(w0)∪ C(w0s)
by Axiom TS3. Since two double cosets are either disjoint or equal, this means that either
C(w00) =C(w0) or C(w00) = C(w0s).
Our induction hypothesis implies that eitherw00 =w0 or w00=w0s. The first case is impossible since l(w00)< l(w)6l(w0). Therefore so w00=w0s. Hencew=w00s =w0,
as required.
As a first example, let G = GL(r+ 1, F), where F is any field. As in the last section, letB be the Borel subgroup of upper triangular matrices in G, let T be the standard maximal torus of all diagonal elements, and letN be the normalizer inGof T. The group N consists of the monomial matrices, that is, matrices having exactly one nonzero entry in each row and column.
Our goal is to show that N and B form a Tits system.
If α = αi,j with i 6= j is a root, let xα : F −→ G(F) be the homomorphism xα(a) = 1 +aEi,j where Ei,j is (as in the last section) the matrix with 1 in the i, j position and 0 elsewhere.
The positive roots are αi,j with i < j, and the simple roots are αi = αi,i+1 with 16i6r. Suppose thatα=αi is a simple root. The corresponding simple reflection is
Let Mα be the centralizer of Tα, and let Pα be the “parabolic subgroup” generated byB and Mα. We have a semidirect product decomposition Pα =MαUα, whereUα
where∗ indicates an arbitrary value.
Lemma 5 Let G = GL(n, F) for any field F, and let other notations be as above.
If s is a simple reflection then B∪ C(s) is a subgroup of G.
Proof First let us check this whenn = 2. In this case there is only one simple root sα whereα =α12. We check easily that
The action ofW onT by conjugation induces the action ofW on Φ. This action is such that ifω ∈N represents the Weyl group element w∈W, we have
ωxα(λ)ω−1 ∈xw(α)(F). (37) Lemma 6 Let G = GL(n, F) for any field F, and let other notations be as above.
If α is a simple root and w∈W such that w(α)∈Φ+ then C(w)C(s) = C(ws).
Proof We will show that
wBs⊆BwsB.
If this is known, then multiplying right and left by B gives C(w)C(s) = BwBsB ⊆ BwsB = C(ws). The other inclusion is obvious, so this is sufficient. Let ω and σ be representatives of w and s as cosets in N/T =W, and let b ∈ B. We may write b=txα(λ)u wheret ∈T, λ∈F and u∈Uα. Then
ωbσ=ωtω−1.ωxα(λ)ω−1.ωσ.σ−1uσ.
We haveωtω−1 ∈T ⊂Bsinceω ∈N =N(T). We haveωxα(λ)ω−1 ∈xw(α)(F)⊂B, using (37) and the fact that w(α) ∈ Φ+. We have σ−1uσ ∈ Uα ⊂ B since Mα normalizes Uα and σ ∈Mα. We see that ωbσ ∈BwsB as required.
Proposition 25 Let G = GL(n, F) for any field F, and let other notations be as above. If w, w0 ∈W are such that l(ww0) = l(w) +l(w0), then
C(ww0) = C(w)· C(w0).
Proof It is sufficient to show that ifl(w) = r, and ifw=s1· · ·srbe a decomposition into simple reflections, then
C(w) =C(s1)· · · C(sr). (38) Indeed, assuming we know this fact, let w0 = s01· · ·s0r0 be a decomposition into simple reflections with r0 = l(r0). Then s1· · ·srs01· · ·s0r0 is a decomposition of ww0 into simple reflections with l(ww0) =r+r0, so
C(ww0) =C(s1)· · · C(sr)C(s01)· · · C(s0r0) =C(w)C(w0).
To prove (38), let sr = sα, and let w1 = s1· · ·sr−1. Then l(w1sα) = l(w1) + 1, so by Proposition 13 we have w0(α)∈Φ+. Thus Lemma 6 is applicable and C(w) = C(w1)C(sr). By induction on r, we have C(w1) = C(s1)· · · C(sr−1) and so we are
done.
Theorem 10 With G= GL(n, F) and B, N, I as above, (B, N, I) is a Tits’ system in G.
Proof Only Axiom TS3 requires proof; the others can be safely left to the reader.
Letα∈Σ such that s=sα.
First, suppose that w(α) ∈ Φ+. In this case, it follows from Lemma 6 that wBs⊂BwsB.
Next suppose that w(α) 6∈ Φ+. Then wsα(α) = w(−α) = −w(α) ∈ Φ+, so we may apply the case just considered, with wsα replacing w, to see that
wsBs⊂Bws2B =BwB. (39)
By Lemma 5,B∪BsB is a group containing a representative of the coset ofs ∈N/T, soB∪BsB =sB∪sBsB and thus
Bs⊂sB∪sBsB.
Using (39),
wBs⊂wsB∪wsBsB ⊂BwsB∪BwB.
This proves Axiom TS3.
Similarly, if G is any split reductive group, T a maximal split torus, B a Borel subgroup containingG and N the normalizer of T, then B(F) and N(F) are a Tits system and so we have a Bruhat decomposition. For proofs, see Borel’s book Linear Algebraic Groups.