Lectures on Chevalley groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 8 July 2013

§8. Variants of the Bruhat lemma

Let G be a Chevalley group, k,B as usual. We recall (Theorems 4 and 4'):

(a) G=wWbwB, a disjoint union.
(b) For each wW, BwB=BwUw, with, uniqueness of expression on the right. Our purpose is to present some analogues of (b) with applications.

For each simple root, α we set Gα=𝔛α,𝔛-α, a group of rank1, Bα=BGα, and assume that the representative of wα in N/H, also denoted wα, is chosen in Gα.

Theorem 15: For each simple root α let Yα be a system of representatives for Bα\(Gα-Bα), or more generally for B\BwαB. For each wW choose a minimal expression w=wαwβwδ as a product of reflections relative to simple roots α,β. Then BwB=BYαYβYδ with uniqueness of expression on the right.


Since Gα-Bα=BαwαBα, the second case above really
is more general than the first. We have

BwB = BwαBwαwB (by Lemma 25) = BwαB Yβ Yδ (by induction) = BYα Yβ Yδ (by the choice ofYα).

Now assume byαyβ yγyδ = byα yβ yγyδ with b,bB, etc. Then byαyγ= byα yγ yδ yδ-1. We have yδyδ-1B or BwδB. The second case can not occur since then the left side would be in BwwδB and the right side in BwB (by Lemma 25). From the definition of Yδ it follows that yδ=yδ, and then by induction that yγ=yγ,, whence, the uniqueness in Theorem 15.

Lemma 43: Let φα:SL2Gα be the canonical homomorphism (see Theorem 4', Cor. 6). Then Yα satisfies the conditions of Theorem 15 in each of the following cases.

(a) Yα= wα𝔛α.
(b) k= (resp. ) and Yα is the image under φα of the elements of SU2 (resp. SO2) (standard compact forms) of the form [a-bba] with b>0.
(c) If θ is a principal ideal domain (commutative with 1), θ* is the group of units, k is the quotient field, and Yα is the image under φα of the elements of SL2(θ) of the form [abcd] with c running through a set of representatives for (θ-0)/θ*, and for each c, a running over a set of representatives for the residue classes of θ mod c.


We have (a) by Theorem 4' applied to Gα. To verify (b) and (c) we may assume that Gα is SL2 and Bα the superdiagonal subgroup B2 since kerφαB2. Any element of SL2() can be converted to one of SU2 by adding a multiple of the second row to the first and normalizing the lengths of the rows. Thus SL2()=B2()·SU2. Then B2()\SL2()(B2()SU2)\SU2, whence (b). Now assume [pqrs]SL2(k) with k as in (c). We choose a,c in θ relatively prime and such that pa+qc=0 (using unique factorization), and then b,d in θ so that ad-bc=1. Multiplying the preceding matrix on the right by [abcd] we get an element of B2(k). Thus SL2(k)=B2(k)SL2(θ), and (c) follows.

Remarks: (a) The case (a) above is essentially Theorem 4' since wUw= wα𝔛α· wβ𝔛β wδ𝔛δ in the notation of Theorem 15, by Appendix II 25, or else by induction on the length of the expression. (b) In (c) above the choice can be made precise in the following cases:

(1) θ=; choose a,c so that 0a<c.
(2) θ=F[X] (F a field); choose so that c is monic and dga<dgc.
(3) θ=p (p-adic integers); choose c a power of p and a an integer such that 0a<c.

In what follows we will give separate but parallel developments of the consequences of (b) and (c) above. In (b) we will treat the case k= for definiteness, the case k= being similar.

Lemma 44: Let and {Xα,Hα} be as in Theorem 1.

(a) There exists an involutory semiautomorphism σ0 of 
(relative to complex conjugation of ) such that σ0Xα=-X-α and σ0Hα=-Hα for every root α.
(b) On the form {X,Y} defined by (X,σ0Y) in
terms of the Killing form is negative definite.


This basic result is proved, e.g., in Jacobson, Lie algebras, p. 147.

Theorem 16: Let G be a Chevalley group over viewed as a Lie group over .

(a) There exists an analytic automorphism σ of G such that σxα(t)=x-α(-t) and σhα(t)=hα(t-1) for all α and t.
(b) The group K=Gσ of fixed points of σ is a
maximal compact subgroup of G and the decomposition G=BK holds (Iwasawa decomposition).


Let σ1 be σ0 in Lemma 44 composed with complex conjugation, and ρ the representation of used to define G. Applying Theorem 4', Cor. 5 to the Chevalley groups (both equal to G) constructed from the representations ρ and ρ0σ1 of , we get an automorphism of G which aside from complex conjugation satisfies the equations of (a), hence composed with conjugation satisfies these equations. From Theorem 7 adapted to the present situation (see the remark at the end of §5) it follows that σ is analytic, whence (a). We observe that if G is defined by the ajoint representation of , then σ is effected by conjugation by the semiautomorphism σ0 of Lemma 44.

Lemma 45: Let K=Gσ, Kα=KGα for each simple root α.

(a) Kα=φαSU2 (see Lemma 43(b)), hence YαKα.
(b) Bσ=Hσ= { hH| |μˆ(h)|=1 for allμˆLˆ (global weights) } = { hi(ti) (see Lemma 28)||ti| =1 } = maximal torus in K.


The kernel of φα:SL2Gα is contained in {±1}, and σ pulls back to the inverse transpose conjugate, say σ2, on SL2. Since the equation σ2x=-x has no solutions we get (a).

Since σhα(t)=hα(t-1), μˆ(hα(t))= tμ(Hα) (here μ and μˆ are corresponding weights on and H), and the hα(t) generate H, we have μˆ(σh)=μˆ(h)-1 for all hH, so that σh=h if and only if |μˆ(h)|=1 for all weights μˆ. If h=hi(ti), then μˆ(h)=tiμ(Hi). Since there are linearly independent weights μ, we see that if |μˆ(h)|=1 for all μˆ, then |ti|n=1 for some n>0, whence |ti|=1, for all i. If G is universal, then Bσ is the product of the circles {hi(·)}, hence is a torus; if not, we have to take the quotient by a finite group, thus still have a torus. Now if hHσ is general enough, so that the numbers αˆ(h) (αΣ) are distinct and different from 1, then Gh, the centralizer of h in G, is H, by the uniqueness in Theorem 4', so that Hσ is in fact a maximal Abelian subgroup of Gσ, which proves the lemma.

Exercise: Check out the existence of h and the property Gh=H above.

Now we consider part (b) of Theorem 16. By Theorem 15 and Lemmas 43(b) and 45(a) we have G=BK. By the same results (BwB)σBσKαKδ, a compact set since each factor is (the compactness of tori and SU2 is being used). Thus K=Gσ is compact. (This also follows easily from Lemma 44(b)). Let K1 be a compact subgroup of G, K1K. Assume xK1. Write x=by with bB, yK, and then b=uh with uU, hH. Since K1 is compact, all eigenvalues μˆ(hn) (n=0,±1,±2,) are bounded, whence hK by Lemma 45(b). Then all coefficients of all un are bounded so that u=1. Thus xK, so that K is maximal compact.

Remark: It can be shown also that K is semisimple and that a complete set of semisimple compact Lie groups is got from the above construction.

Corollary 1: Let G be of the same type as G with a weight lattice containing that of G,K=Gσ, and π:GG the natural projection. Then πK=K.


This follows from the fact proved in Lemma 45 that K is generated by the groups φαSU2.


(a) If G=SLn(), then K=SUn.
(b) If G=SOn(), then K fixes simultaneously the forms Σxixn+1-i and Σxixi, hence equals SOn() (compact form) after a change of coordinates. Prove this.
(c) If G=Sp2n(), then K fixes the forms Σ1n ( xiy2n+1-i- x2n+1-iyi ) and Σxixi, and is isomorphic to SUn() (compact form, = quaternions). For this see Chevalley, Lie groups, p. 22.
(d) We have isomorphisms and central extensions, *= SU1() SU2() SO3(), SU2() SO5(), SU2()2 SO4(), SU4() SO6() (compact forms).

This follows from (a), (b), (c), Corollary 1 and the equivalences C1=A1=B1, C2=B2, A12=D2, A3=D3.

Corollary 2: The group K is connected.


As already remarked, K is generated by the groups φαSU2. Since SU2 is connected, so is K.

Corollary 3: If T denotes the maximal torus Hσ, then T\K is homeomorphic to B\G under the natural map.


The map KB\G, kBk, is continuous and constant on the fibres of T\K, hence leads to a continuous map of T\K into B\G which is 1-1 and onto since T=BK and G=BK. Since T\K is compact, the map is a homeomorphism.

Corollary 4:

(a) G is contractible to K.
(b) If G is universal, then K is simply connected.


Let A={hH|μˆ(h)>0for allμˆLˆ}. Then we have H=AT, so that G=BK=UAK. On the right there is uniqueness of expression. Since K is compact it easily follows that the natural map UA×KG is a homeomorphism. Since UA is contractible to a point, G is contractible to K. If also G is universal, then G is simply connected by Theorem 13; hence so is K.

Corollary 5: For wW set (BwB)σ= BwBK= Kw, and let α,β,,δ be as in Theorem 15. Then K=wKw and Kw=TYαYδ, with uniqueness of expression on the right.


This follows from Theorem 15 and Lemma 43(b).

Remark: Observe that Kw is essentially a cell since each Yα is homeomorphic to (consider the values of a in Lemma 43(b)). A true cellular decomposition is obtained by writing T as a union of cells. Perhaps this decomposition can be used to give an elementary treatment of the cohomology of K.

Corollary 6: B\G and T\K have as their Poincaré polynomials ΣwWt2N(w). They have no torsion.


We have B\BwB homeomorphic to wUw, a cell of real
dimension 2N(w). Since each dimension is even, it follows that the cells represent independent elements of the homology group and that there is no torsion (essentially because the boundary operator lowers dimensions by exactly 1), whence Cor. 6. Alternately one may use the fact that each Yα is homeomorphic to .

Remark: The above series will be summed in the next section, where it arises in connection with the orders of the finite Chevalley groups.

Corollary 7: For wW let w=wαwδ be a minimal expression as before and let S denote the set of elements of W each of which is a product of some subsequence of the expression for w. Then Kw (topological closure) =wSKw.


If Tα=TKα, we have Kα=TαYαTα by Lemma 45(a) and TαYα=Kα by the corresponding result in SU2. Now BwB=B·TαYαTδYδ by Lemma 43(b). Hence Kw=T·TαYαTδYδ, so that KwTKαKδ, and we have equality since each factor on the right is compact, so that the right side is compact, hence closed. Since KwKα Kw Kwwα if wW and α is simple, by Lemma 25, Cor. 7 follows.

Corollary 8:

(a) T=K1 is the closure of every Kw.
(b) Kw is closed if and only if w=1.

Corollary 9: The set S of Cor. 7 depends only on w, not on the minimal expression chosen, hence may be written S(w).


Because Kw doesn't depend on the expression.

Lemma 46: Let w0 be the element of W which makes all positive roots negative. Then S(w0)=W.


Assume wW, and let w=w1wm be a minimal expression as a product of simple reflections and similarly for w-1w0=wm+1wn. Then w0=w1wmwn is one for w0 since if N is the number of positive roots then m m=N(w), n=N-N(w), and m+n=N=N(w0). Looking at the initial segment of w0 we see that wS(w0).

Corollary 10: If w0 is as above and w0=wαwβwδ is a minimal expression, then

(a) K=Kw0.
(b) K=KαKβKδ.


(a) By Cor. 7 and Lemma 46.

(b) By (a) K=TKαKβKδ. We may write T=Tγ (γ simple), then absorb the Tγ's in appropriate Kγ's to get (b).

Exercise: If G is any Chevalley group and w0,α,β, are as above, show that G=BGαGβGδ.

Remarks: (a) If and SU2 are replaced by and SO2 in accordance with Lemma 43(b), then everything above goes through except for Cor. 4, Cor. 6 and the fact that T is no longer a torus. In this case each Kα is a circle since SO2 is. The corresponding angles in Cor. 10(b), which we have to restrict suitably to get uniqueness, may be called the Euler angles in analogy with the classical ease:

G=SL3(), K=SO3(), Kα,Kβ= {rotations around thez-axis,x-axis}, K=KαKβKα.

(b) If Kw is replaced by BwB=BKw in Cor. 7, the formula for BwB is obtained. (Prove this.) If (or ) is replaced by any algebraically closed field and the Zariski topology is used, the same formula holds. So as not to interrupt the present development, we give the proof later, at the end of this section.

Theorem 17: (Cartan). Again let G be a Chevalley group over or , K=Gσ as above, and A={hH|μˆ(h)>0for allμˆLˆ}.

(a) G=KAK (Cartan decomposition).
(b) In (a) the A-component is determined uniquely up to conjugacy under the Weyl group.


(a) Assume xG. By the decompositions H=AT and G=BK (Theorem 16), t here exist elements in KxKUA. Given such an element y=ua, we write a=expH (H, uniquely determined by a), then set |a|=|H|, the Killing norm in . This norm is invariant under W. We now choose y to maximize |a| (recall that K is compact). We must show that u=1. This follows from: (*) if u1, then |a| can be increased. We will reduce (*) to the rank 1 case. Write u=β>0uβ (uβ𝔛β). We may assume uα1 for some simple α: choose α of minimum height, say n, such that uα1, then if n>1, choose β simple so that (α,β)>0 and htwβα<n, then replace y by wβ(1)ywβ(1)-1 and proceed by induction on n. We write u=uuα with u𝔛P-{α} (here P is the set of positive roots). Then we write a=expH, choose c so that H=H-cHα is orthogonal to Hα, set aα=expcHα AGα, a=expH A, a=aαa. Then a commutes with Gα elementwise and is orthogonal to aα relative to the bilinear form corresponding to the norm introduced above. By (*) for groups of rank 1, there exist y,zKα such that yuαaαz=aαAGα and |aα|>|aα|. Then yuaz=yuuαaαaz=yuy-1aαa. Since Gα normalizes 𝔛P-{α} (since 𝔛α and 𝔛-α do), yuy-1U. Since |aαa|2= |aα|2+ |a|2> |aα|2+ |a|2= |aαa|2= |a|2, we have (*), modulo the rank 1 case. This case, essentially G=SL2, will be left as an exercise.

(b) Assume xG, x=k1ak2 as in (a). Then σx=k1a-1k2, so that xσx-1=k1a2k1-1. Here σa=a-1 since μˆ(σa)=μˆ(a)-1=μˆ(a-1) for all μˆLˆ.

Lemma 47: If elements of H are conjugate in G (any Chevalley group), they are conjugate under the Weyl group.

This easily follows from the uniqueness in Theorem 4'.

By the lemma x above uniquely determines a2 up to conjugacy under the Weyl group, hence also a since square-roots in A are unique.

Remark: We can get uniqueness in (b) by replacing A by A+={aA|αˆ(a)1for allαˆ>0}. This follows from Appendix III 33.

Corollary: Let P consist of the elements of G which satisfy σx=x-1 and have all eigenvalues positive.

(a) AP.
(b) Every pP is conjugate under K to some aA, uniquely determined up to conjugacy under W (spectral theorem).
(c) G=KP, with uniqueness on the right (polar decomposition).


(a) This has been noted in (b) above.

(b) We can assume p=kaKA, by the theorem. Apply σ-1: p=ak-1. Thus k commutes with a2, hence also with a. (Since a is diagonal (relative to a basis of weight vectors) and positive, the matrices commuting with a have a certain block structure which does not change when it is replaced by a2.) Then k2=1 and k=a-12pa-12P, so that k is unipotent by the definition of P. Since K is compact, k=1. Thus p=a. The uniqueness in (b) follows as before.

(c) If xG, then x=k1ak2 as in the theorem, so that x=k1k2·k2-1ak2KP. Thus G=KP. Assume k1p1=k2p2 with kiK and piP. By (b) we can assume that p2A. Then p1=k1-1k2p2. As in (b) we conclude that k1-1k2=1, whence the uniqueness in (c).

Example: If G=SLn(), so that K=SUn(), A={ positive diagonal matrices }, P={ positive-definite Hermitean matrices }, then (b) and (c) reduce to classical results.

We now consider the case (c) of Lemma 43. The development is strikingly parallel to that for case (b) just completed although the results are basically arithmetic in one case, geometric in the other. Throughout we assume that θ,θ*,k,Yα are as in Lemma 43(c) and that the Chevalley group G under discussion is based on k. We write Gθ for the subgroup of elements of G whose coordinates, relative to the original lattice M, all lie in θ.

Lemma 48: If φα is as in Theorem 4', Cor. 6, then φαSL2(θ)Gθ.


If θ is a Euclidean domain then SL2(θ) is generated by its unipotent superdiagonal and subdiagonal elements, so that the lemma follows from the fact that xα(t) acts on M as an integral polynomial in t. In the general case it follows that if p is a prime in θ and θp is the localization of θ at p (all a/bk such that a,bθ with b prime to p) then φαSL2(θ)Gθp. Since pθp=θ, e.g. by unique factorization, we have our result.

Remark: A version of Lemma 48 is true if θ is any commutative ring since φα[abcd] is generically expressible as a polynomial in a,b,c,d with integral coefficients (proof omitted). The proof just given works if θ is any integral domain for which θ=θp (p= maximal ideal), which includes most of the interesting cases.

Lemma 49: Write K=Gθ, Kα=GαK.

(a) BK=(UK)(HK).
(b) UK= {α>0xα(tα)|tαθ}.
(c) HK= {hH|μˆ(h)θ*for allμˆLˆ}= {hi(ti)|alltiθ*}.
(d) φαSL2(θ)= Kα. Hence YαKα.


(a) If b=uhBK, then its diagonal h, relative to a basis of M made up of weight vectors (see Lemma 18, Cor. 3), must be in K, hence u must also.

(b) If u=xα(tα)UK, then by induction on heights, the equation xα(t)=1+tXα+ and the primitivity of Xα in End(M) (Theorem 2, Cor. 2) we get all tαθ.

(c) If hHK, in diagonal form as above, then μˆ(h) must be in θ for each weight μˆ of the representation defining G, in fact in θ* since the sum of these weights is 0 (the sum is invariant under W). If we write h=hi(ti) and use what has just been proved, we get tinθ* for some n>0, whence tiθ* by unique factorization.

(d) Set Sα=φαSL2(θ). By Lemma 48, SαKα. Since
 Gα=BαBαYα by Lemma 43(c) and YαSα, the reverse inclusion follows from: BαKSα. Now if x=xα(t)hα(t*)BαK,
 then tθ and t*θ* by (a), (b), (c) applied to Gα, so that xxα(θ),x-α(θ)=Sα, whence (d).

Theorem 18: Let θ,k,G and K=Gθ be as above. Then G=MK (Iwasawa decomposition).


By Lemmas 43(c) and 49(d), BwB=BYαYδBK for every wW, so that G=BK.

Corollary 1: Write Kw=BwBK.

(a) K=wWKw.
(b) Kw=(BK)YαYδ, with BK given by Lemma 49, and on the right there is uniqueness of expression.

Remark: This normal form in K=Gθ has all components in Gθ whereas the usual one obtained by imbedding Gθ in G doesn't.

Corollary 2: K is generated by the groups Kα.


By Lemma 49 and Cor. 1.

Corollary 3: If θ is a Euclidean domain, then K is generated by {xα(t)|αΣ,tθ}.


Since the corresponding result holds for SL2(θ), this follows from Lemma 49(d) and Cor. 2.

Example: Assume θ=, k=. We get that G is generated by {xα(1)}. The normal form in Cor. 1 can be used to extend Nielsen's theorem (see (1) on p. 96) from SL3() to G whenever Σ has rank2, is indecomposable, and has all roots of equal length (W. Wardlaw, Thesis, U.C.L.A. 1966). It would be nice if the form could be used to handle SL3() itself since Nielsen's proof is quite involved. The case of unequal root lengths is at present in poor shape. In analogy with the fact that in the earlier development K is a simple compact group if Σ is indecomposable, we have here: Every normal subgroup of G is finite or of finite index if Σ is indecomposable and has rank2. The proof isn't easy.

Exercise: Prove that G/𝒟G is finite, and is trivial if Σ is indecomposable and not of type A1,B2 or G2.

Returning to the general set up, if p is a prime in θ, we write ||p for the p-adic norm defined by |0|p=0 and |x|p=2-r if x=pra/b with a and b prime to p.

Theorem 19: (Approximation theorem): Let θ and k be as above, a principal ideal domain and its quotient field, S a finite set of inequivalent primes in θ, and for each pS, tpk. Then for any ε>0 there exists tk such that |t-tp|p<ε for all pS and |t|q1 for all primes qS.


We may assume every tpθ. To see this write tp=pra/b as above. By choosing s-r and c and d so that a=cps+db and replacing a/b by d, we may assume b=1. If we then multiply by a sufficiently high power of the product of the elements of S, we achieve r0, for all pS. If we now choose n so that 2-n<ε, e=pSpn, ep=e/pn, then fp,gp so that fppn+gpep=1, and finally t=Σgpeptp, we achieve the requirements of the theorem.

Now given a matrix x=(aij) over k, we define |x|p=max|aij|p. The following properties are easily verified.

(1) |x+y|p max|x|p, |y|p.
(2) |xy|p |x|p|y|p.
(3) If |xi|p=|yi|p for i=1,2,,n then |xi-yi|p maxi |y1|p |yˆi|p |yn|p |xi-yi|p.

Theorem 20: (Approximation theorem for split groups): Let θ,k,S,ε be as in Theorem 19, G a Chevalley group over k, and xpG for each pS. Then there exists xG so that |x-xp|p<ε for all pS and |x|q1 for all qS.


Assume first that all xp are contained in some 𝔛α,xp=xα(tp) with tpk. If x=xα(t),tk, then |x|qmax|t|q,1 because xα(t) is an integral polynomial in t and similarly |xxp-1-1|p|t-tp|p, so that |x-xp|p |xp|p |t-tp|p by (1) and (2) above. Thus our result follows from Theorem 19 in this case. In the general case we choose a sequence of roots α1,α2, so that xp=xp1xp2 with xpi𝔛αi for all pS. By the first case there exists xi𝔛αi so that

|xi-xpi|p <|xpi|p andε|xpi|p /|xp1|p |xp2|p

if pS and |xi|q1 if qS. We set x=x1x2. Then the conclusion of the theorem holds by (3) above.

With Theorem 20 available we can now prove:

Theorem 21: (Elementary divisor theorem): Assume θ,k,G,K=Gθ are as before. Let A+ be the subset of H defined by: αˆ(h)θ for all positive roots αˆ.

(a) G=KA+K (Cartan decomposition).
(b) The A+ component in (a) is uniquely determined modHK, i.e. mod units (see Lemma 49); in other words, the set of numbers {μˆ(h)|μˆ weight of the representation defining G} is.

Example: The classical case occurs when G=SLn(k), K=SLn(θ), and A+ consists of the diagonal elements diag(a1,a2,,an) such that ai is a multiple of ai+1 for i=1,2,.

Proof of theorem.

First we reduce the theorem to the local case, in which θ has a single prime, modulo units. Assume the result ture in this case. Assume xG. Let S be the finite set of primes at which x fails to be integral. For pS, we write θp for the local ring at p in θ, and define Kp and
 Ap+ in terms of θp as K and A+ are defined for θ. By the local case of the theorem we may write x=cpapcp with cp,cpKp and aAp+, for all pS. Since we may choose ap so that μˆ(ap) is always a power of p and then replace all ap by their product, adjusting the c's accordingly, we may assume that ap is independent of p, is in A+, and is integral outside of S. We have cpacpx-1=1 with a=ap for pS. By Theorem 20 there exist c,cG so that |c-cp|p<|cp|p for pS and |c|q1 for qS, the same equations hold for c and cp, and |cacx-1-1|p1 for all pS. By properties (1), (2), (3) of ||p, it is now easily verified that |c|p1, |cp|1 and |cacx-1-1|p1, whether p is in S or not. Thus cK,cK and cacx-1K, so that xKA+K as required. The uniqueness in Theorem 21 clearly also follows from that in the local case.

We now consider the local case, p being the unique prime in θ. The proof to follow is quite close to that of Theorem 17. Let A be the subgroup of all hH such that all μˆ(h) are powers of p, and redefine A+, casting out units, so that in addition all αˆ(h) (αˆ>0) are nonnegative powers of p.

Lemma 50: For each aA there exists a unique H, the -module generated by the elements Hα of the Lie algebra , such that μˆ(a)=pμ(H) for all weights μ.


Write a=hα(cαpnα) with cαθ*,nα. Then μˆ(a)=(cαpnα)μ(Hα). Since μˆ(a) is a power of p the cα, being units, may be omitted, so that μˆ(a)=Pμ(H) with H=ΣnαHα. If H is a second possibility for H, then μ(H)=μ(H) for all μ, so that H=H.

If a and H are as above, we write H=logpa,a=pH, and introduce a norm: |a|=|H|, the Killing norm. This norm is invariant under the Weyl group. Now assume xG. We want to show xKA+K. From the definitions if T=HK then H=AT. Thus by Theorem 18 there exists y=uaKxK with uU,aA. There is only a finite number of possibilities for a: if a=pH, then {μ(H)|μ a weight in the given representation} is bounded below (by -n if n is chosen so that the matrix of pnx is integral, because {pμ(H)} are the diagonal entries of y), and also above since the sum of the weights is 0, so that H is confined to a bounded region of the lattice . We choose y=ua above so as to maximize |a|. If u=uα (uα𝔛α), we set suppu={α|uα1} and then minimize suppu subject to a lexicographic ordering of the supports based on an ordering of the roots consistent with addition (thus suppu<suppu means that the first α in one but not in the other lies in the second). We claim u=1. Suppose not. We claim (*) uαK and a-1uαaK for αsuppu. If uα were not in K, we could move it to the extreme left in the expression for y and then remove it. The new terms introduced by this shift would, by the relations (B), correspond to roots higher than α, so that suppu would be diminished, a contradiction. Similarly a shift to the right yields the second part of (*). Now as in the proof of Theorem 17 we may conjugate y by a product of wβ(1)'s (all in K) to get uα1 for some simple α, as well as (*). We write a=pH, choose c so that H=H-cHα is orthogonal to Hα, set aα=pcHα, a=pH, a=aαa. We only know that 2c=H,Hα, so that this may involve an adjunction of p1/2 which must eventually be removed. If we bear this in mind, then after reducing (*) to the rank 1 case, exactly as in the proof of Theorem 17, what remains to be proved is this:

Lemma 51: Assume y=ua= [1t01] [pcp-c] with 2c, tk, tθ and tp-2cθ. Then c can be increased by an integer by multiplications by elements of K.


Let t=ep-n with eθ*. Then n, n>0 and n+2c>0 by the assumptions, so that c+n>c, c+n>-c and |c+n|>|c|. If we multiply y on the left by [pne-e-10], on the right by [10-e-1pn+2c1], both in K, we get [pc+n00p-c-n], which proves the lemma, hence that u=1. Thus y=aA, so that xKAK. Thus G=KAK. Finally every element of A is conjugate to an element of A under the Weyl group, which is fully represented in K (every wα(1)K). Thus G=KA+K. It remains to prove the uniqueness of the A+ component. If G is the universal group of the same type as G and π is the natural homomorphism, it follows from Lemma 49(d) and Theorem 19, Cor. 2 that πK=K and from Lemma 49 that π maps A+ isomorphically onto A+. Thus we may assume that G is universal. Then G is a direct product of its indecomposable factors so that we may also assume that G is indecomposable. Let λi be the ith fundamental weight, Vi an -module with λi as highest weight, Gi the corresponding Chevalley group, πi:GGi the corresponding homomorphism, and μi the corresponding lowest weight. Assume now that x=cacG, with c,cK and aA+. Set μˆi(a)=p-ni. Each weight on Vi is μi increased by a sum of positive roots, Thus ni is the smallest integer such that pniπia is integral, i.e. such that pniπix is since πic and πic are integral, thus is uniquely determined by x. Since {μi} is a basis of the lattice of weights (μi=w0λi), this yields the uniqueness in the local case and completes the proof of Theorem 21.

Corollary 1: If θ is not a field, the group K is maximal in its commensurability class.


Assume K is a subgroup of G containing K properly. By the theorem there exists aA+K, aK. Some entry of the diagonal matrix a is nonintegral so that by unique factorization |K/K| is infinite.

Remark: The case θ= is of some importance here.

Corollary 2: If θ=p and k=p (p-adic integers and numbers) and the p-adic topology is used, then K is a maximal compact subgroup of G.


We will use the fact that p is compact. (The proof is a good exercise.) We may assume that G is universal. Let k be the algebraic closure of k and G the corresponding Chevalley group. Then G=GSL(V,k) (Theorem 7, Cor. 3), so that K=GSL(V,θ). Since θ is compact, so is End(V,θ), hence also is K, the set of solutions of a system of polynomial equations since G is an algebraic group, by Theorem 6. If K is a subgroup of G containing K properly, there exists aA+K, aK, by the theorem. Then {|an|p|n]} is not bounded so that K is not compact.

Remark: We observe that in this case the decompositions G=BK and G=KA+K are relative to a maximal compact subgroup just as in Theorems 16 and 17. Also in this case the closure formula of Theorem 16, Cor. 7 holds.

Exercise (optional): Assume that G is a Chevalley group over , or p and that K is the corresponding maximal compact subgroup discussed above. Prove the commutativity under convolution of the algebra of functions on G which are complex-valued, continuous, with compact support, and invariant under left and right multiplications by elements of K. (Such functions are sometimes called zonal functions and are of importance in the harmonic analysis of G.) Hint: prove that there exists an antiautomorphism φ of G such that φxα(t)=x-α(t) for all α and t, that φ preserves every double coset relative to K, and that φ preserves Haar measure. A much harder exercise is to determine the exact structure of the algebra.

Next we consider a double coset decomposition of K=Gθ itself in the local case. We will use the following result, the first step in the proof of Theorem 7.

Lemma 52: Let be the Lie algebra of G (the original Lie algebra of §1 with its coefficients transferred to k), N the number of positive roots, and {Y1,Y2,,Yr} a basis of N made up of products of Xα's and Hi's with Y1=α>0Xα. For xG write xY1=Σcj(x)Yj. Then xU-HU if and only if c1(x)0.

Theorem 22: Assume that θ is a local principal ideal domain, that p is its unique prime, and that k and G are as before.

(a) BI=Up-HθUθ is a subgroup of Gθ.
(b) Gθ=wWBIwBI (disjoint), if the representatives for W in G are chosen in Gθ
(c) BIwBI= BIwUw,θ with the last component of the right uniquely determined modUw,p.


Let θ denote the residue class field θ/pθ, Gθ the Chevalley group of the same type as G over θ, and Bθ,Hθ, the usual subgroups. By Theorem 18, Cor. 3 reduction mod p yields a homomorphism π of Gθ onto Gθ.

(1) π-1(Uθ-HθUθ)U-HU. We consider G acting on N as in Lemma 52. As is easily seen Gθ acts integrally relative to the basis of Y's. Now assume πxUθ-HθUθ. Then c1(πx)0 by the lemma applied to Gθ, whence c1(x)0 and xU-HU again by the lemma.

(2) Corollary: kerπU-HU.

(3) BI=π-1Bθ. Assume xπ-1Bθ. Then xU-BU by (1). From this and xGθ it follows as in the proof of Theorem 7(b) that xUθ-HθUθ, and then that xBI.

(4) Completion of proof: By (3) we have (a). To get (b) we simply apply π-1 to the decomposition in Gθ relative to Bθ. We need only remark that a choice as indicated is always possible since each wα(1)Gθ. From (b) the equation in (c) easily follows. (Check this.) Assume b1wu1= b2wu2 with biBI, uiUw,θ. Then b1-1b2= wu1u2-1w-1 BIUθ-= Up-, whence u1u2-1Uw,p and (c) follows.

Remark: The subgroup BI above is called an Iwahori subgroup. It was introduced in an interesting paper by Iwahori and Matsumoto (Pabl. Math. I.H.E.S. No. 25 (1965)). There a decomposition which combines those of Theorems 21(a) and 22(b) can be found. The present development is completely different from theirs.

There is an interesting connection between the decomposition Gθ=BIwBI above and the one, Gθ=(BwB)θ, that Gθ inherits as a subgroup of G, namely:

Corollary: Assume wW, that S(w) is as in Theorem 16, Cor. 9, and that π:GθGθ is, as above, the natural projection. Then π(BIwBI)=BθwBθ, and π(BwB)θ=wS(w)BθwBθ. Hence if θ is a topological field, e.g. , or p, then π(BwB)θ is the topological closure of π(BIwBI).


The first equation follows from π-1Bθ=BI, proved above. Write w=wαwβ as in Lemma 25, Cor. Then (*) (BwB)θ= (BwαB)θ (BwβB)θ by Theorem 18, Cor. 1. Now (BwαB)θx-α(p) and wα(1) and is a union of Bθ double cosets. Thus π(BwαB)θ Bθ Bθwα Bθ =BθGα,θ. The reverse inequality also holds since (BwαB)θ BθGα,θ by Theorem 18, Cor. 1. From this, (*), the definition of S(w), and Lemma 25, the required expression for π(BwB)θ now follows.

Appendix. Our purpose is to prove Theorem 23 below which gives the closure of BwB under very general conditions. We will write ww if wS(w) with S(w) as in Theorem 16, Cor. 9, i.e. if w is a subexpression (i.e. the product of a subsequence) of some minimal expression of w as a product of simple reflections.

Lemma 53: The following are true.

(a) If w is a subexpression of some minimal expression for w, it is a subexpression of all of them.
(b) In (a) the subexpressions for w can all be taken to be minimal.
(c) The relation is transitive.
(d) If wW and α is a simple root such that wα>0 (resp. w-1α>0), then wwα>w (resp. wαw>w).
(e) w0w for all wW.


(a) This was proved in Theorem 16, Cor. 7 and 9 in a rather roundabout way. It is a direct consequence of the following fact, which will be proved in a later section: the equality of two minimal expressions for w (as a product of simple reflections) is a consequence of the relations w1w2=w2w1 (w1,w2 distinct simple reflections, n terms on each side, n= order w1w2).

(b) If w=w1w2wr is an expression as in (b) and it is not minimal, then two of the terms on the right can be cancelled by Appendix II 21.

(c) By (a) and (b).

(d) If wα>0 and w1w2ws is a minimal expression for w, then w1wswα is one for wwα by Appendix II 19, so that wwα>w, and similarly for the other case.

(e) This is proved in Lemma 46.

Now we come to our main result.

Theorem 23: Let G be a Chevalley group. Assume that k is a nondiscrete topological field and that the topology inherited by G as a matric group over k is used. Then the following conditions on w,w are equivalent.

(a) BwB BwB.
(b) ww.


Let Y1 be as in Lemma 52 and more generally Yw=α>0Xwα for wW. For xG let cw(x) denote the
coefficient of Yw in xY1. We will show that (a) and (b) are equivalent to:

(c) cw is not identically 0 on BwB.

(a) (c). We have xβ(t)Xα=Xα+ΣtjXj with Xj of weight (0 or a root) α+jβ, and nwXα=cXwα (c0) if nw represents w in W in N/H. Thus (*) BwBY1k*Yw+ higher terms in the ordering given by sums of positive roots. Thus cw is not identically 0 on BwB, hence also not on BwB, by (a).

(c) (b). We use downward induction on N(w). If this is maximal then w=w0, the element of W making all positive roots negative, and then w=w0 by (c) and (*) above. Assume ww0. Choose α simple so that w-1>0, hence N(wαw)>N(w). Since cw(BwB)0 and BwαbwBBwBBwαwB, we see that cwαw(BwB)0 or cwαw(BwαwB)0, so that wαww or wαwwαw. In the first case w<w by Lemma 53(c) and (d). In the second case if w-1α<0 then wαw<w by Lemma 53(d) which puts us back in the first case, while if not we may choose a minimal expression for w starting with wα and conclude that ww.

(b) (a). By the definitions and the usual calculus of double cosets, this is equivalent to: if α is simple, then BwαB=BBwαB. The left side is contained in the right, an algebraic group, hence a closed subset of G. Since BwαB contains 𝔛α-1 and the topology on k is not discrete, its closure contains 1, hence also B, proving the reverse inequality and completing the proof of the theorem.

Remark: In case k above is , or p, the theorem reduces to results obtained earlier. In case k is infinite and the Zariski topology on k and G are used it becomes a result of Chevalley (unpublished). Our proof is quite different from his.

Exercise: (a) If wW and α is a positive root such that wα>0, prove that wwα>w (compare this with Lemma 53(d)), and conversely if ww then (*) there exists a sequence of positive roots α1,α2,,αr such that if wi=wαi then ww1wi-1αi>0 for all i and ww1wr=w. Thus ww and (*) are equivalent.

(b) It seems to us likely that ww is also equivalent to: there exists a permutation π of the positive roots such that wπα-wα is a sum of positive roots for every α>0; or even to: Σα>0(wα-wα) is a sum of positive roots.

Notes and References

This is a typed excerpt of Lectures on Chevalley groups by Robert Steinberg, Yale University, 1967. Notes prepared by John Faulkner and Robert Wilson. This work was partially supported by Contract ARO-D-336-8230-31-43033.

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