Lectures on Chevalley groups

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

Last update: 24 July 2013

§14. Representations concluded

Now we turn to the complex representations of the groups just considered. Here the theory is in poor shape. Only GLn (Green, T.A.M.S. 1955) and a few groups of low rank have been worked out completely, then only in terms of the characters. Here we shall consider a few general results which may lead to a general theory.

Henceforth K will denote the complex field. Given a (one-dimensional) character λ on a subgroup B of a group G, realized on a space Vλ, we shall write VλG for the induced module for G. This may be defined by VλG=KGKBVλ (this differs from our earlier version in that we have not switched to a space of functions), and may be realized in KG in the left ideal generated by Bλ=ΣbBλ(b-1)b (and will be used in this form). Its dimension is |G/B|.

Exercise: Check these assertions.

Lemma 84: Let B,C be subgroups of a finite group G, let λ,μ be characters on B,C, and let VλG,VμG be the corresponding modules for G.

(a) If xG, then BλxCμ in KG is determined up to multiplication by a nonzero scalar by the (B,C) double coset to which x belongs.
(b) HomG(VλG,VμG) is isomorphic as a K-space to the one generated by all BλxCμ.
(c) If B=C and λ=μ, then the isomorphism in (b) is one of algebras.
(d) The dimension of HomG(VλG,VμG) is the number of (B,C) double cosets D such that BλxCμ0 for some, hence for every x in D, or, equivalently, such that the restrictions of λ and xμ to BxCx-1 are equal.


(a) This is clear.

(b) Assume THomG(VλG,VμG). Since Bλ generates VλG as a KG-module, TBλ determines T. Let TBλ=ΣxG/CcxxCμ. Since bBλ=λ(b)Bλ, we get by averaging over B that TBλ=ΣxB\G/CcxBλxCμ. Thus T is realized on Bλ, hence on all of VλG, by right multiplication by |B|-1ΣcxBλxCμ. Conversely, any such right multiplication yields a homomorphism, which proves (b).

(c) By discussion in (b).

(d) The first statement follows from (a) and (b). Let B1=BxCx-1 and C1=x-1BxC, and {yi} and {zj} systems of representatives for B1\B and C/C1. Set Bλ=Σλ(yi-1)yi and Cμ=Σμ(zj-1)zj. Then BλxCμ=BλB1λB1,xμxCμ with (xμ)(xcx-1)=μ(c) since xC1x-1=B1. If λxμ on B1, then B1λB1,xμ=0. If λ=xμ, this product is |B1|B1λ, and then BλxCμ0 since the elements yib1xzj are all distinct.


(a) This is a special case of a theorem of G. Mackey. (See, e.g., Feit's notes.)
(b) The algebra of (c) is also called the commuting algebra since it consists of all endomorphisms of VλG that commute with the action of G.

Theorem 47: Let G be a (perhaps twisted) finite Chevalley group.

(a) If λ is a character on H extended to B in the usual way, then VλG is irreducible if and only if wλλ for every wW such that w1.
(b) If λ,μ both satisfy the conditions of (a), then VλG is isomorphic to VμG if and only if λ=wμ for some wW.


(a) VλG is irreducible if and only if its commuting algebra is one-dimensional (Schur's Lemma), i.e., by (c) and (d) of Lemma 84, if and only if λ and wλ agree on BwBw-1, hence on H, for exactly one wW, i.e. for only w=1.

(b) Since VλG and VμG irreducible, they are isomorphic if and only if dimHomG(VλG,VμG)=1, which, as above, holds exactly when λ=wμ for some (hence for exactly one) wW.


(a) dimHomG (VλG,VμG) =|Wλ| if λ=wμ for some w, dimHomG (VλG,VμG) =0 otherwise.
(b) In Theorem 47 the conclusion in (b) holds even if the condition in (a) doesn't.

Here Wλ is the stabilizer of λ in W. We see, in particular, that if λ=1 then the commuting algebra of VλG is |W|-dimensional. But more is true.

Theorem 48: Let V be the KG-module induced by the trivial one-dimensional KB-module. Then the commuting algebra EndG(V) is isomorphic to the group algebra KW.


(a) The multiplicities of the irreducible components of V are just the degrees of the irreducible KW-modules.
(b) The subalgebra, call it A, of KG spanned by the double coset sums BwXw is isomorphic to KW.
(c) The algebra of functions f:GK biinvariant under B (f(bxb)=f(x) for all b,bB) with convolution as multiplication is isomorphic to KW.


The theorem is equivalent to (a) by Schur's Lemma and to (b) by Lemma 84, while (b) and (c) are clearly equivalent. We shall give a proof of (b), due to J. Tits, wˆ will denote the average in KG of the elements of the double coset BwB. The elements wˆ form a basis of A and 1ˆ is the unit element. If a is a simple root, ca will denote |Xa|-1.

(1) A is generated as an algebra by {wˆa|asimple} subject to the relations

(α) wˆa2= ca1ˆ+ (1-ca)wˆa for all a.
(β) wˆa wˆb wˆa = wˆb wˆa wˆb, as in Lemma 83(a).


We observe that if each ca is replaced by 1 then these relations go over into a defining set for KW, by Appendix II.38. Since BBwaXa is a group, we have (BwaXa)2=rB+sBwaXa with r,sK. Since BwaX contains with each of its elements its inverse, we get r=|BwaXa|, and then from the total coefficient s=|BwaXa|-|B|. Thus (α) holds in A, and so does (β) by Lemma 25,Cor., which also shows that the wˆa generate A. Conversely, let A1 be the abstract associative algebra (with 1ˆ) generated by symbols wˆa subject to (α) and (β). For each wW choose a minimal expression w=wawb and set wˆ=wˆawˆb. By Lemma 83(a) and (β) this is independent of the expression chosen. By (α) it follows that

wˆawˆ = wawˆ ifw-1a>0, = cawawˆ+ (1-ca)wˆ ifw-1a<0.

Thus the wˆ form a basis for A1, which, having the same dimension as A, is therefore isomorphic to it.

(2) There exists a positive number c=c(G) such that. ca=cna with na a positive integer depending only on the type of G. The multiplication table of A in terms of the basis {wˆ} is given by polynomials in c depending only on the type.


Consider A42(q2), for example. Here the two possibilities for |Xa| are q2 and q3 by Lemma 63(c). If we set c=q-1, then the corresponding values of na are 2 and 3, which depend only on the type. For each of the other types the verification is similar. From the first statement of (2) and the equations of the proof of (1) the second statement follows.

(3) An associative algebra Ac with multiplication table given by the polynomials of (2) exists for every complex number c. In particular Ac(G)=A and A1=KW.


Since the type of the group G contains an infinite number of members, the multiplication table is associative for an infinite set of values of c, hence for all values.

(4) Ac is semisimple for c=c(G), for c=1, and for all but a finite number of values of c.


Ac is for c=c(G) the commuting algebra of a KG-module and for c=1 a group algebra KW, hence semisimple in both cases. The discriminant of Ac is a polynomial in c, nonzero at c=1 since then Ac is semisimple, hence nonzero for all but a finite number of values of c.

(5) Completion of proof. Since A is semisimple and K is an algebraically closed field, A is a direct sum of complete matric algebras, of certain degrees over K (see, e.g., Jacobson's Structure of Rings or Feit's notes), and similarly for KW. We have to show that the degrees are the same in the two cases. If 𝒜 is any finite-dimensional associative algebra which is separable, i.e. which is semisimple when the base field is extended to its algebraic closure, we define the numerical invariants of 𝒜 to be the degrees of the resulting matric algebras. The proof of Theorem 48 will be completed by the following lemma.

Lemma 85: Let R be an integral domain, F its field of quotients, and f a homomorphism of R onto a field K. Let 𝒜 be a finite—dimensional associative algebra over R, and 𝒜F and 𝒜K the resulting algebras over F and K. If 𝒜F and 𝒜K are separable, then they have the same numerical invariants.

In fact, from (3) and the lemma with R=K[c], 𝒜=Ac, and first f:cc(G) and then f:c1, it follows that A and KW have the same numerical invariants, hence that they are isomorphic.

Proof of the lemma.

(a) Assume that is a finite-dimensional semisimple associative algebra over an algebraically closed field L, that b1,b2,,bn form a basis for /L, that x1,x2,,xn are independent indeterminates over L, that b=Σxibi, and that P(t) is the characteristic polynomial of b acting from the left on L(x1,,xn), written as P(t)=ΠPi(t)pi with the Pi distinct monic polynomials irreducible over L(x1,,xn). Then:

(a1) The pi are the numerical invariants of .
(a2) pi=dgtPi for each i.
(a3) If P(t)=ΠQj(t)qj is any factorization over L(x1,,xn) such that qj=dgtQj for each j, then it agrees with the one above so that the qj are the numerical invariants of .


For (a1) and (a2) we may assume that is the complete matric algebra EndLp and that b=ΣxijEij in terms of the matric units Eij. If X=[xij], then P(t)=det(tI-X)p, so that we have to show that det(tI-X) is irreducible over L(xij). This is so since specialization to the set of companion matrices

[ 1 1 xp1 xp2 xpp ]

yields the general equation of degree p. In (a3) if some Qj were reducible or equal to some Qk with kj, then any irreducible factor Pi of Qj would violate (a2).

(b) Let R* be the integral closure of R in F (consisting of all elements satisfying monic polynomial equations over R), and x1,x2,,xn indeterminates over F. Then R*[x1,,xn] is the integral closure of R[x1,,xn] in F(x1,,xn).


See, e.g., Bourbaki, Commutative Algebra, Chapter V, Prop. 13.

(c) If R* is as in (b), then any homomorphism of R into K can be extended to one of R* into K.


By Zorn's lemma this can be reduced to the case R*=R[α], where it is almost immediate since K is algebraically closed.

(d) Completion of proof. Let {ai} be a basis for 𝒜/R, hence also for 𝒜F/F, and {xi} independent indeterminates over F and also over K. The given homomorphism f:RK defines a homomorphism f:𝒜𝒜K. By (c) it extends to a homomorphism of R* into K and then naturally to one of R*[x1,,xn] into K[x1,,xn]. If a=Σxiai and P(t)=ΠPi(t)pi is its characteristic polynomial, factored over F(x1,,xn) as before, then the coefficients of each Pi are integral polynomials in its roots, hence integral over the coefficients of P, hence integral over R[x1,x2,,xn], hence belong to R*[x1,,xn] by (b). Thus if f(a)=Σxif(ai), then its character polynomial has a corresponding factorization Pf(t)=ΠPif(t)pi over K(x1,,xn). By (a1) the pi are the numerical invariants of 𝒜F, and by (a2) they satisfy pi=dgtPi=dgtPif, so that by (a3) with =𝒜K they are also the numerical invariants of 𝒜K, which proves the lemma.

Exercise: If λ is a character on H extended to B in the usual way, then EndG(VλG) is isomorphic to KWλ. (Observe that this result includes both Theorem 47 and Theorem 48.)

Remark: Although A is isomorphic to KW there does not seem to be any natural isomorphism and no one has succeeded in decomposing the module V of Theorem 48 into its irreducible components, except for some groups of low rank. We may obtain some partial results, in terms of characters, by inducing from the parabolic subgroups and using the following simple facts.

Lemma 86: Let π be a set of simple roots, Wπ and Gπ the corresponding subgroups of W and G (see Lemma 30), and VπW and VπG the corresponding trivial modules induced to W and G; and similarly for π.

(a) A system, of representatives in W for the (Wπ,Wπ) double cosets becomes in G a system of representatives for (Gπ,Gπ) double cosets.
(b) dimHomG (VπG,VπG) = dimHomW (VπW,VπW) .


(a) Exercise.

(b) By Lemma 84(d) and (a).

Corollary 1: Let χπG denote the character of VπG and similarly for W. If {nπ} is a set of integers such that χW=ΣnπχπW is an irreducible character of W, then χG=ΣnπχπG is, up to sign, one of G.


Let (χ,ψ)G denote the average of χψ over G. We have (χπG,χπG)G= dimHomG (VπG,VπG) , and similarly for W. Since χW is irreducible, (χW,χW)W=1, it follows that (χG,χG)G=1, so that ±χG is irreducible, by the orthogonality relations for finite group characters.


(a) In a beautiful paper in Berliner Sitzungsberichte, 1900, Frobenius has constructed a complete set of irreducible characters for the symmetric group Sn, i.e. the Weyl group of type An-1, as a set of integral combinations of the characters χπW. Using his method and the preceding corollary one can decompose the character of V in Theorem 48 in case G is of type An-1. (See R. Steinberg T.A.M.S. 1951).
(b) The situation of (a) does not hold in general. Consider, for example, the group W of type B2, i.e. the dihedral group of order 8. It has five irreducible modules (of dimensions 1,1,1,1,2), while there are only four χπW's to work with.
(c) A result of a general nature is as follows.

Corollary 2: If the notation is as above and (-1)π is as in Lemma 66(d), then χG=Σ(-1)πχπG is an irreducible character of G and its degree in |U|.


Consider χW=Σ(-1)πχπW. By (8) on p. 142, extended to twisted groups (check this using the hints given in the proof of Lemma 66), χW=det, an irreducible character. Hence ±χG is also one by Cor. 1 above. We have χG(1)=Σ(-1)π|G/Gπ|. If G is untwisted and the base field has q elements, then by Theorem 4' applied to G and to Gπ this can be continued Σ(-1)πW(q)/Wπ(q) =qN=|U|, as in (4) of the proof of Theorem 26. If G is twisted, the proof is similar.

We continue with some remarks on the algebra A of Theorem 48(b).

Lemma 87: The homomorphisms of A onto K are given by: f(wˆa)=1 or -ca for each simple root a, subject to the condition that f(wˆa) is constant on each W-orbit.


For a and b simple, let n(a,b) denote the order of wawb in W. We claim that (*) a and b belong to the same W-orbit if and only if there exists a sequence of simple roots a=a0,a1,,ar=b such that n(ai,ai+1) is odd for every i. The equation (waiwai+1)n=1 with n odd can be rewritten to show that wai and wai+1 are conjugate, so that if the sequence exists then a and b are conjugate. If a and b are conjugate, then so are wa and wb, and this remains true when we project into the reflection group obtained by imposing on W the additional relations: (wcwd)2=1 whenever n(c,d) is even. In this new group wa and wb must belong to the same component, so that the required sequence exists. By (*) the condition of the lemma holds exactly when f(wˆa)= f(wˆb) whenever n(a,b) is odd, i.e. exactly when f preserves the relations (β) (of the proof of Theorem 48). Since f(wˆa)=1 or -ca exactly when f preserves (α) (solve the quadratic), we have the lemma.

Remark: By finding the annihilator in A of the kernel of each of the homomorphisms of Lemma 87, we get a one-dimensional ideal I in A. This corresponds to an irreducible submodule of multiplicity 1 in V, realized in the left ideal KGI of KG. By working out the corresponding idempotent, the degree of the submodule can be found.


(a) If f(wˆa)=1 for all a, show that I=KΣqwwˆ with qw=|XW|, and that the corresponding KG-module is the trivial one. (Hint: by writing W as a union of right cosets relative to {1,wa} and writing (α) in the form (wˆa-1)(wˆa+ca)=0, show that I is as indicated.)
(b) If f(wˆa)=-ca for all a, show that I=KΣ(detw)wˆ, and that in this case the dimension is |U|. (Hint: if e is the given sum and cw=qw-1, show that e2=me with m=Σcw=|G/B|/|U|.)
(c) For G=B2(q) work out all (four) cases of Lemma 87, hence obtain the degrees of all (five) irreducible components of V.
(d) Same for A2 and for G2.

Remark: It can be shown that the module of (b) is isomorphic to the one with character χG as in Lemma 86, Cor.2. If we reduce the element Σ(detw) |U/XW| Bw XW (which is |B||U|e) of I mod p, we see from the proof of Theorem 46(e) that the given module reduces to the one of that theorem for which λ=1 and every μa=-1. This latter module can itself be shown to be isomorphic to the one in Theorem 43 for which λ,α=q(α)-1 for every α. From these facts and Weyl's formula the character of the original module can be found up to sign and the result used to prove that the number of p-elements (of order a power of p) of G is |U|2. For the details see R. Steinberg, Endomorphisms of linear algebraic groups, to appear.

Finally, we should mention that the algebra A admits an involution given by wˆa1-ca -wˆa for all a (which in case ca1 and AKW reduces to w(detw)w).

The preceding discussion points up the following
Problem: Develop a representation theory for finite reflection groups and use it to decompose the module V (or the algebra A) of Theorem 48.

It is natural that in studying the complex representations of G we have considered first those induced by characters on B since for representations of characteristic p this leads to a complete set. In characteristic 0, however, this is not the case, as even the simplest case G=SL2 shows. One must delve deeper. Therefore, we shall consider representations of G induced by (one-dimensional) characters λ on U. We can not expect such a representation ever to be irreducible since its degree |G/U| is too large (larger than |G|12), but what we shall show is that if λ is sufficiently general then at least it is multiplicity-free. In other words, EndG(VλG) is Abelian, hence a direct sum of fields. (If λ is not sufficiently general, we can expect the Weyl group to play a role, as in Theorem 48.)

Before stating the theorem, we prove two lemmas.

Lemma 88: Let k be a finite field and λ a nontrivial character from the additive group of k into K*. Then every character can be written uniquely λc:tλ(ct) for some ck.


The map cλc is a homomorphism of k into its dual, and its kernel is clearly 0.

Lemma 89: For wW the following conditions are equivalent.

(a) If a and wa are positive roots and one of them is simple, then so is the other.
(b) If a is simple and wa is positive, then wa is simple.
(c) w=w0wπ for some set π of simple roots, with w0 as usual and wπ the corresponding object of Wπ.
There are 2 possibilities for w.


(c) (a) Because wπ maps π onto -π and (*) permutes the positive roots with support not in π (same proof as for Appendix I.11).

(a) (b) Obvious.

(b) (c) Let π be the set of simple roots kept positive, hence simple, by w. We claim: (**) if a>0 and suppaπ, then wa<0. Write a=b+c with suppbπ, suppcΠ-π. Then wa=wb+wc. Here wc<0 by the choice of π, and suppwcwπ suppwb. Thus wa<0. If a is a simple root not in π then wwπa<0 by (*) and (**), while if a is in π this holds by the definition of π. Thus wwπ=w0, whence (c).

Theorem 49: Let G be a finite, perhaps twisted, Chevalley group and λ:UK* a character such that λ|Xa1 if a is simple, λ|Xa=1 if a is positive but not simple. Then VλG is multiplicity-free. In other words, EndG(VλG) is Abelian, or, equivalently, the subalgebra A of KG spanned by the elements UλhwUλ (hH,wW) is Abelian.

Here Uλ=ΣuUλ(u-1)u, and we assume that the w are chosen as in Lemma 83(b).


(a) If a is not simple, then usually Xa𝒟U, so that the assumption λ|Xa=1 is superfluous, but this is not always the case, e.g. for B2 or F4 with |k|=2 or for G2 with |k|=3. In these latter groups, there are other possibilities, which because of their special nature will not be gone into here.
(b) The proof to follow is suggested by that of Gelfand and Graev, Doklady, 1963 who have given a proof for SLn and announced the general result for the untwisted groups. T. Yokonuma, C. Rendues, Paris, 1967, has also given a proof for these latter groups, but his details are unnecessarily complicated.

Proof of Theorem 49.

The fact that A is Abelian will follow from the existence of an (involutory) antiautomorphism f of G such that

(a) fU=U.

(b) λf=λ on U.

(c) For each double coset UnU such that UλnUλ0, we have fn=n (here nN=ΣHw).

For since f extended to KG and then restricted to A is an antiautomorphlsm and at the same time the identity (by (a), (b), (c)) it is clear that A is Abelian. The existence of f will be proved in several steps.

(1) If UλnUλ0 and nHw, then w=w0wπ for some set π of simple roots.


By Lemma 89 we need only prove that if a is simple and wa positive then wa is simple. Writing the first Uλ above with the Xwa component on the right, and the second with the Xa component on the left, we get Xwa,λn Xa,λn-1 0. Since λ is nontrivial on Xa it is also so on Xwa, whence wa is simple by the assumptions on λ, which proves (1).

The condition in (1) essentially forces the correct definition of f. We set a*=-w0a. If a is simple, so is a*. In order to simplify the discussion in one or two spots we assume henceforth that G (i.e. its root system) is indecomposable. If G is untwisted, we start with the graph automorphism corresponding to * (see the Corollary on p. 156), compose it with the inversion xx-1, and finally with a diagonal automorphism so that the result f satisfies, not only fU=U but also λf=λ on U. This is possible because of Lemma 89 and the assumptions on λ in the theorem. If G is twisted, then we may omit the graph automorphism, (because * is then the identity), and use the explicit isomorphism Xa/𝒟Xak of (2) of the proof of Theorem 36 in combination with Lemma 89 to achieve the second condition. We see that

(2) f is an involutory antiautomorphism which satisfies the required conditions (a) and (b). We must prove that it also satisfies (c). As consequences of the construction we have:

(3) fh=w0hw0-1 for every hH.

(4) If a*=a, then f is the identity on Xa/𝒟Xa.

(5) If a*=a, there exists a nontrivial element of Xa fixed by f.


For Xa of type A1 this follows from (4). For Xa of type A22 we choose the element (t,u) of Lemma 63(c) with t=2, u=2 if p2, and t=0, u=1 if p=2, since f(t,u)=(t,-ttθu) (check this, referring to the construction of f). For types C22 and G22 we may choose (0,1) and (0,1,0) since f is the identity on {(0,u)} and {(0,u,0)}.

(6) The elements waG may be so chosen that:

(6a) fwa=wa* for every simple root a.
(6b) If a and b are simple and nN is such that nXan-1=Xb and λ(nxn-1)=λ(x) for all xXa, then nwan-1 =wb.


Under the action of f and the inner automorphisms in by elements n as in (6b) the Xa (a simple) form orbits. From each orbit we select an element Xa. If a*=a, we choose xaXa as in (5), write it as (*) xa=x1wax2 with x1,x2X-a, and choose wa accordingly. Since f is an antiautomorphism and fixes X-a, it also fixes wa by the uniqueness of the above form. If a*a, we choose xaXa, xa1, arbitrarily. We then use the equations fwa=wa* and inwa=wb of (6a) and (6b) to extend the definition of w to the orbit of a. We must show this can be done consistently, that we always return to the same value. Let a1,a2,,an be a sequence of simple roots such that a1=an=a and for each j either aj+1=aj* or else there exists nj such that the assumptions in (6b) holds with aj,aj+1,nj in place of a,b,n. Let g denote the product of the corresponding sequence of f's and inj's. We must show that g fixes wa. We have gXa=Xa, gX-a=X-a, and in fact g acts on Xa/𝒟Xa, identified with k, by multiplication by a scalar c as follows from the definition of f and the usual formulas for in. Since λg=λ by the corresponding condition on f and each in, it follows from Lemma 89 that c=1, so that g is the identity on Xa/𝒟Xa. If 𝒟Xa=0, then g fixes the element xa above, hence also wa by (*), whether g is an automorphism or an antiautomorphism. If 𝒟Xa0, then G is twisted so that a*=a. If g is an automorphism, then by the proof of Theorem 36 from (5) on its restriction to Xa,X-a is the identity so that it fixes wa, while if g is an antiautomorphism then by the same result its restriction coincides with that of f so that it fixes wa by the choice of wa.

Remark: If G is untwisted, the above proof is quite simple.

We assume henceforth that the wa are as in (6).

(7) If w=wawb as in Lemma 83(b) and w*=w0w-1w0-1, then fw=w*.


Since wawb is minimal, wb*wa* is also. (Check this.) Since f is an antiautomorphlsm it follows from (6a) that fw= wb* wa* = wb*wa* =w*.

(8) If w is as in (1) then fw=w*.


w*=w in this case (see (7)).

(9) If n is as in (1) then fn=n.


By (1), nwH with w=w0wπ. Assume aπ. Then wa is simple and λ(nxn-1)=λ(x) for all xXa, by the inequality in the proof of (1). Thus nwan-1=wwa by (6b), from which we get, on picking a minimal expression for wπ, that (*) nwπn-1=wπ*. Since N(w)= N(w0wπ)= N(w0)- N(wπ), it follows that if we put together minimal expressions for w and wπ we will get one for w0. Thus w0=wwπ by Lemma 83(b), and similarly w0=wπ*w, so that (**) wwπw-1=wπ*. If now we write n=wh, then h commutes with wπ by (*) and (**). Hence

fn = fh·fw sincefis an antiautomorphism = w0h w0-1w by (3) and (8) = wwπh wπ-1 sincew0= wwπ = wh sincehcommutes withwπ = n.

Thus f satisfies condition (c) and the proof of Theorem 49 is complete.


(a) Prove that if {wa} is as in (6) and w as in (1), then UλwUλ0.
(b) Deduce that if Hπ denotes the kernel of the set of simple roots π then the dimension of A, hence the number of irreducible components of VλG, in Theorem 49 is Σ|Hπ|.

Remark: The natural group for the preceding theorem seems to be the adjoint group extended by the diagonal automorphisms, a group of the same order as the universal group, but with something extra at the top instead of at the bottom. For this group, G, prove that the dimension above is just Π(|Ha|+1)=Πq(α) in the notation of the exercise just before Lemma 83. Prove also that in this case VλG is independent of λ.

Remark: The problem now is to decompose the algebra A of Theorem 49 into its simple (one-dimensional) components. If this were done, it would be a major step towards a representation theory for G. As far as we know this has been done only for the group A1 (see Gelfand and Graev, Doklady, 1962). It would not, however, be the complete story. For not every irreducible G-module is contained in one induced by a character on U, i.e., by Frobenius reciprocity, contains a one-dimensional U-module, as the following, our final, example, due to M. Kneser, shows (although it is for some types of groups such as An).

As remarked earlier, reduction mod 3 yields an isomorphism of the subgroup W+ of elements of determinant 1 of the Weyl group W of type E6 onto the group G=SO5(3), the adjoint group of type B2(3). If we reverse this isomorphism and extend the scalars we obtain a representation of G on a complex space V. The assertion is that U, i.e. a 3-Sylow subgroup of G, fixes no line of V. Consider the following diagram.

α1 α2 α3 α4 α5 α6 α7

This is the Dynkin diagram of E6 with the lowest root α7 adjoined (α7 is the unique root in -D (see Appendix III.33), unique because all roots are conjugate in the present case. It is connected as shown because of symmetry and the fact that each proper subdiagram must represent a finite reflection group.) We choose as a basis for V the α's with α3 omitted, a union of three bases of mutually orthogonal planes. w1w2 acts as a rotation of 120 in the plane α1,α2 and as the identity in the other two planes, and similarly for w4w5 and w6w7. The group W+ also contains an element permuting the three planes cyclically as shown, because of the conjugacy of simple systems and the uniqueness of lowest roots, and the four elements generate a 3-subgroup of W. It is now a simple matter to prove that this subgroup fixes no line of V.

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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