The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras
Part II
Representations and decompositions in characteristic p0

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

Last update: 2 April 2014

§ 4. Fundamental constants cλμ and the character formula

As in Part I, let Δ be an indecomposable reduced root-system, and 𝔤 be the simple Lie algebra over (determined up to isomorphism) possessing Δ as its root-system.

4.1. Let G be an algebraic Chevalley group, i.e. a connected semisimple (affine) algebraic group, of type 𝔤 (or of type Δ) defined over a field K of characteristic p0; one denotes by GL the group of L-rational points of G for LK. Thus, depending on one's approach, G itself may be identified either with GK for the algebraic closure K of K, or with GL for a very large algebraically closed (universal) field L, or else with the functor LGL; G is characterised by its affine-ring A=AK (being K[G] in the notation of [Bor1970], our canonical reference for standard facts summarized here too briefly), which is a Hopf algebra over K, such that GL is identified with the maximal-ideal spectrum of AL=AKL for every field LK, the group-operation on GL coming from the coproduct in AL in the standard manner. By a G-module we shall always mean an algebraic G-module defined over K, using the words "module" and "representation" interchangeably. Thus K is the underlying field of every representation space V, and concretely the representation affords a homomorphism of GL into AutLVL for every overfield L of K, where VL=VKL.

Let us assume that G is simply-connected, and split over K. The former assumption means that every projective representation (i.e. algebraic homomorphism into PGL(n)) of G comes from a G-module (i.e. factors through GL(n)); this is particularly suitable for representation-theoretic purposes, and there is no loss in this assumption because of the existence and uniqueness of simply-connected (universal) coverings. The latter assumption means that G contains a maximal torus T which is defined and split over K, i.e. is isomorphic over K to (𝔾m)l where 𝔾m=GL(1) is the "multiplicative group" (which, as a group-functor, assigns to LK the multiplicative group of L). One knows that under these assumptions G is uniquely determined (up to isomorphism) by K and Δ, and that K may be taken to be the prime field 𝔽p; this means existence of an 𝔽p-form of the affine-ring, which was described explicitly by Kostant purely in terms of the root-system Δ (via the -form U of the enveloping algebra U of 𝔤 which we shall describe below in some detail, without discussing the affine-ring further as it is not required by us here); cf. [Bor1970, § 4]. We find it convenient not to specify K, but treat it as a floating (working) field. Thus the Lie algebra 𝔤K of G is a vector space over K of dimension dimG=dim𝔤=|Δ|+l. The elements of 𝔤K may be identified with either the right or the left-invariant derivations of AK. It may be remarked that unlike the situation in characteristic 0, the object 𝔤K is not entirely "local"; in particular, the Lie algebra of two isogenous groups (in the sense of algebraic groups) may not be isomorphic. In fact, one knows that this pathology arises if and only if p divides the "connexion number" [X:X] of Δ where X,X are as in § 1; in this case 𝔤K is nonsimple, and the only time that happens again is for types F4 with p=2 and G2 with p=3; but we shall not require this fact. An explicit description of 𝔤K in terms of Δ, essentially due to Chevalley, is as follows.

Let 𝔤=𝔥+αΔ𝔤α be a Chevalley lattice in 𝔤. This means the following: (i) 𝔥=𝔥 is a Cartan subalgebra of 𝔤 on which elements of Δ are considered as linear functions and 𝔥 consists of all h𝔥 such that λ(h) for all λ in the weight-lattice X of Δ (as defined in § 1, with ΔX𝔥*= vector space dual of 𝔥); thus if hα𝔥 is defined by λ(hα)=λ,α for all λ𝔥*, then putting hi=hαi one has 𝔥=i=1lhi; (ii) 𝔤α=𝔤α is the α-rootspace of 𝔤 with respect to 𝔥, and if eα denotes a generator of the (additive) infinite-cyclic group 𝔤α, then [eα,e-α]=±2hα; and (iii) if U denotes the subring of the universal enveloping algebra U of 𝔤 generated by {eαnn!|αΔ,n+}, then 𝔤 is stable under the adjoint action of U on 𝔤. It is customary to choose eα's such that [eα,e-α]=2hα, and we write ei for eαi and fi for e-αi, for 1il. It may be remarked that the set { einn!, finn! |1il, n+ } generates U as a ring [Ver1972-2].

The Lie algebra 𝔤K of G can be identified with 𝔤ZK in such a way that 𝔥K=𝔥K the Lie algebra of the maximal torus T, and identifying X with the character group X(T) of T (i.e. the group of algebraic homomorphisms of T into 𝔾m with the group-operation given by tensoring these 1-dimensional T-modules, which is written additively), the subspace 𝔤K=𝔤αK of 𝔤K becomes its α-rootspace with respect to the adjoint action of T. Thus, we are identifying Δ with the root-system of G with respect to T, as is common practice, and this should lead to no confusion. In general, for any (algebraic) T-module M and for λX(T)=X we denote by M(λ) its λ-weight-space consisting of all vectors in M on which T acts by λ, so that one has M=λXM(λ) in view of the complete reductibility of T-modules; thus 𝔤K(α) (for the adjoint action of T on 𝔤K) is the same as 𝔤Kα. The structure of a T-module is determined by the dimensions of the various weight-spaces, i.e. by the element charM of the group-ring of X over , called the formal character of M, which assigns λdimM(λ). More precisely, denoting by ·eX the set of all -linear combinations λXnλ·eλ of formal symbols {eλ|λX} where all but a finite number of nλ's are zero, and making it into a ring under the usual coordinatewise addition and convolution multiplication with eλ·eμ=eλ+μ, we have charM=λX (dimM(λ)) ·eλ·eX. The formal character of a G-module M shall mean simply that of M as a T-module upon restriction. From the standard fact that the Weyl group W of Δ can be identified with the quotient N/T in a canonical manner, where N is the normalizer of T in G, it is immediate that the formal character of a G-module is a W-invariant element of ·eX under the natural action of W.

Let B be the Borel subgroup of G having as its Lie algebra 𝔟K=𝔥K+αΔ+𝔤Kα, where Δ+ is a given choice of positive roots in Δ. One knows that every irreducible G-module contains a unique 1-dimensional highest-weight-space, in the sense that it is a weight-space for T whose elements remain fixed under the unipotent radical [B,B] of B. If this weightspace corresponds to the character λX, we say λ is the highest weight of the given module. Necessarily λX+ (the dominant elements of X with respect to the fixed choice of positive roots), and if μX is a "weight" of the given G-module, in the sense that the μ-weight-space is nonzero, then λ-μ·Δ+, which we rewrite as μλ by setting up the canonical partial-order on X compatible with the addition under which +·Δ+ is the set of all elements 0. Conversely, to every λX+ there corresponds an irreducible G-module Mλ with highest weight λ, which is unique up to isomorphism. As stated in the Introduction, one of the long-standing unsolved problems in the study of G-modules is to give a formula for charMλ in terms of λX+.

4.2. A famous theorem of Steinberg reduces the study of all irreducible G-modules Mλ for λX+, to that of the pl basic ones with λ lying in the set X= { imiλiX |0mip- 1for1il } ; the elements of X will be called the "restricted dominant characters". The formulation of this theorem hinges on the fact that G is defined over 𝔽p (as stated earlier) and that all the Mλ's are definable over 𝔽p; we will be able to see the latter fact from the construction of Mλ by the standard technique of "reduction modp", which we shall need to describe presently in some detail. Now if M is any G-module defined over the prime field, one defines its Frobenius-twist MFr to be the G-module whose underlying space is M itself, such that the new action of gGL on ML=MKL is obtained as follows: take a basis of the 𝔽p-form M𝔽p of M=MK and write the matrix of (the old action of) g with respect to this basis of ML; then the new matrix obtained by replacing each entry by its pth power gives the new (Frobenius-twisted) action of g on ML with respect to the same basis. It is immediate that the μ-weight-space of M coincides with the pμ-weight-space of MFr, which gives charMFr=(charM)Fr, where forχ=λ nλeλeX ,we defineχFr =λnλepλ. Steinberg's tensor-product theorem then essentially says that for λX+ and μX the module MλFrMμ is irreducible and hence isomorphic to Mpλ+μ; in particular, MλFrMpλ. It follows that if λX+ has the p-expansion λ=μ0+μ1p++ μkpk,with μjXfor 0jk, then MλMμ0 Mμ1Fr MμkFrk, where Frj denotes the j-fold iteration of Fr, and consequently, charMλ= j=0k (charMμj)Frj. (4.1)

4.3. In order to define the constants cλμ in the title of this section, we need to quickly introduce certain G-modules Vλ,K which are modelled after the irreducible 𝔤-module Vλ, with highest weight λX+, so that the formal character of Vλ,K coincides with that of Vλ,. We caution in advance that Vλ,K is not uniquely determined by λ alone; however, there are only a finite number of choices (up to isomorphism). The formal character of Vλ,K is determined by λ, and is given by Weyl's character formula as in characteristic 0; thus charVλ,K is independent of p. Also (by Proposition 2 below) the composition factors of Vλ,K are determined by λ alone; these are far from being independent of p though, and in fact for every λ one can find a bound for p beyond which Vλ,K is irreducible. This is not a new result, and follows from the "discriminant criterion" described below.

Let Vλ, be a U-stable lattice in Vλ, ("admissible -form" in the terminology of [Bor1970]). One sees that up to dilatation, i.e. multiplication by a nonzero scalar, there are only a finite number of choices for Vλ,; this follows from the fact that if an arbitrary nonzero element v0 in the highest-weightspace of Vλ, is required to be a primitive (= nondivisible) vector in Vλ, (as can always be ensured upon dilatation, as the highest-weight-space is 1-dimensional), then there exist two uniquely determined extremes for Vλ,: Vλ,min Vλ, Vλ,max. It is clear that the minimal lattice Vλ,min has to be simply the image v0U of v0 under the action of U on Vλ,. Let v0*(Vλ,)*= the vector space dual to Vλ,, be the linear function having value 1 at v0 and vanishing at all the weight-spaces of Vλ, except the highest one. Then v0* is a lowest-weight-vector of the contragredient 𝔤-module (Vλ,)*=Vλ*, with highest weight λ*=λτ0=-λσ0. Then, setting Vλ*,min=v0*U, one has Vλ,max= { xVλ,| y(x)for all yVλ*,min } . An alternate way of describing this is in terms of the "contragredient bilinear form" introduced by W. J. Wong [Won1971]:

Let θ denote the (unique) anti-automorphism of U which interchanges ei and fi for 1il (and hence leaves the hi's fixed); the existence of θ is classical and follows, for example, from the now well-known Harish-Chandra—Jacobson—Serre presentation of the Lie algebra 𝔤 (equivalently, of the associative algebra U) in terms of the generators {e1,e2,,el,f1,f2,,fl}. Then Vλ, possesses a unique symmetric bilinear form (,) such that (v0,v0)=1 and (x·ei,y)=(x,y·fi) for 1il and x,yVλ,; more generally (x·u,y)=(x,y·uθ) for uU, and (,) is determined by the fact that (v0·u,v0·u)=c if the projection of v0uuθ on the highest-weight-space along the sum of all other weight-spaces is cv0. The restriction of (,) to Vλ,min takes integer values, and one has Vλ,max= { xVλ, |(x,y) for ally Vλ,min } .

The "reduction mod p" Vλ,K=Vλ,K of any U-stable lattice Vλ, in Vλ,, becomes a G-module in a natural manner [Bor1970] (here simply-connectedness of G is required, just as in characteristic 0), such that the associated infinitesimal action of 𝔤K coincides with that of 𝔤K (which acts naturally on Vλ,K, since 𝔤U), with which 𝔤K has already been identified. The module Vλ,K need not be irreducible as we shall see presently, and its internal structure may vary with the choice of Vλ,. Perhaps Vλ,K is always indecomposable; this is certainly true of the two extreme choices of Vλ,. In fact Vλ,Kmin=Vλ,minK has a unique maximal proper G-submodule (contained in the sum of all the weight-spaces except the highest one), the quotient by which is the irreducible module Mλ; and on the other extreme Vλ,maxK= Vλ,Kmax= (Vλ*,Kmin)* has Mλ as its unique irreducible submodule, on account of the duality given by the last equality. The injection iλ,: Vλ,min Vλ,max gives, on reduction mod p, only a homomorphism iλ,K: Vλ,Kmin Vλ,Kmax, whose kernel is the maximal proper submodule of Vλ,min and image is the unique copy of Mλ in Vλ,Kmax; in other words, iλ,K factors through Mλ. Hence we have the "discriminant criterion": Vλ,K is irreducible if and only if p does not divide the index [Vλ,max:Vλ,min]; this index equals the discriminant of Wong's contragredient form (,) on Vλ,min described above (cf. [Bor1970, Theorem 2J). It does not seem easy to give a formula for this number. (See Footnote3.) It would be particularly interesting to determine all "small" prime divisors of this index for each λ; prime divisors not exceeding the Coxeter number hΔ of Δ are of special interest, since the ideas of the next section provide an alternate method towards deciding the irreducibility of Vλ,K particularly when p>hΔ (cf. Proposition 3).

It may be mentioned at this point that for the case of type A1, i.e. for GSL2, all Vλ,K for λX are irreducible. But this is an exceptional situation; it will be shown in a later paper that when Δ is not of type A1, there are values of λX (in fact, for large p, a sizeable percentage of values exist) such that Vλ,K is not irreducible. The first example of a non-irreducible Vλ,K with λX was given by Curtis [Cur1960, p. 859], for type A2 (GSL3).

4.4. Let us write Vλ=Vλ,Kmin, and for μX+ define cλμ to be the multiplicity [Vλ:Mμ]G, of the occurrence of Mμ as a composition factor of the G-module Vλ. (See Footnote4.) It is clear that cλλ=1, and that cλμ0 implies (μ is a weight of Vλ and therefore) μλ; in particular, λ-μX is a necessary condition for cλμ0. We have VλGμX+cλμMμ, where G denotes equivalence in the Grothendieck ring of the category of G-modules, and thus μcλμ ·charMμ= charVλ= σWdetσ·e((λ+δ)σ) σWdetσ·e(λσ) (4.2) where the second equality comes from the Weyl's character formula for Vλ,.

The linear equations (4.2) for the "unknowns" χμ=charMμ can be solved in terms of the "known" quantities χμ=charVμ, in the form χλ=μγλμχμ, where the matrix [γλμ]X+ is the inverse of the infinite matrix [cλμ]X+, the outer subscript X+ denoting the set over which the matrix indices λ,μ run. With respect to any enumeration of the set X+ which is stronger than the partial order (definition at the end of § 4.1), the matrix [cλμ]X+ is unipotent in the sense that it is triangular with diagonal entries all 1. As a matter of fact, in order to compute χμ0 for given μ0X+, one needs to consider (4.2) only for λ ranging over the finite set Y=Y(μ0) consisting of those λX+ such that there exists a sequence μ1,μ2,,μk-1,μk=λ with cμj-1μj0 for 1jk. Since for λY, μX+, cλμ0implies μY(λ)Y(μ0) =Y, we have for λY, μYcλμ χμ=χλ, i.e. only the finite segment [cλμ]Y of [cλμ]X+ needs to be inverted.

Proposition 1 (Character formula). There exist unique integers γλμ for λ,μX+, such that for given λ only finitely many γλμ are nonzero and charMλ= μ(σWdetσ·γλμ·e((μ+δ)σ)) σWdetσ·e(δσ) . (4.3)

By virtue of the standard fact that the set {χλ|λX+} is a -free basis of the W-invariants in the group-ring ·eX (see [Bou1968]: VI. 3.4, Proposition 3 and Example 2, on p. 187-188), and the fact that the matrix [γλμ]X+ in μγλμχμ=χλ is unipotent, one finds that {χλ|λX+} is also a -free basis of the formal characters of G-modules. Hence we have

Proposition 2. The composition factors of a G-module are determined by its formal character.

Because of (4.1) one is primarily interested in computing the formal characters of the pl basic irreducible G-modules with highest weight in X. We claim that these are completely determined by our knowledge of the "decomposition" VλG μX+ cλμMμ for λ ranging in X alone. To see this, let us impose the new partial-order on X, defined to be the weakest one such that whenever μ0+μ1p++μkpkλ with λ,μ0,μ1,,μk all in X, one has μiλ for 0ik; and let denote any linear order on X stronger than . Then χλ can be computed, recursively with respect to , i.e. assuming that χμ is known for all μλ, simply by substitution in χλ=χλ- μX+μλ cλμχμ, where the χμ's appearing on the right-hand side are known by virtue of (4.1) and the definition of . Our claim is proved; and as a corollary we note that a knowledge of [cλμ]λX,μX+ determines, by virtue of Proposition 2, the entire matrix [cλμ]X+. It would be very interesting to devise a formula (short of a computer algorithm) which enables one to write down, efficiently and explicitly, the decompositions of Vλ for all λX+ in terms of those for λX alone. (Here and elsewhere in this paper, by the "decomposition" of a module we mean that in the Grothendieck group of the category of modules at hand.)

4.5. When we substitute χμ=j(χμj)Frj for μ=jμjpj, with all μj's in X, in μcλμχμ=χλ, what we get is a set of nonlinear equations in {χν|νX}. However, under certain circumstances, it is possible to replace these by a set of linear equations μX Cλμχμ= χλforλ X, (4.4) where the coefficients Cλμ are no longer integers, but instead lie in the group-ring ·eX itself; in fact Cλμ is a W-invariant element of the subring ·epX consisting of {χFr|χeX}. To this end let us define Y0= { νX+| cλ,μ+νp0 for someλ,μX } ; then χλ = μXνY0 cλ,μ+νp· χμ+νp = μX ( νY0 cλ,μ+νp ·χνFr ) χμ, which gives (4.4) with Cλμ= (νY0cλ,μ+νp·χν)Fr. (4.5) Our "coefficients" in (4.4) till now are themselves polynomials in the unknowns {χν|νX}. If for a given value of p it can be shown that Vν is irreducible for νY0, one can replace χν in (4.5) by the known quantity χν; in Corollary to Proposition 3 in § 5, we shall see that such is indeed the case at least for p2hΔ-3, using the fact that Y0 { imiλi X+| i=1l mini hΔ-2 } , (4.6) where hΔ=(ini)+1 is the Coxeter number as in § 1.

To see (4.6) we note that Y0 { νX+| μ+pνλ for someλ,μX } { νX+| μ+pν,-α0 λ,-α0 for someλ,μX } { νX+| pν,-α0 (p-1)δ,-α0 } = { νX+| ν,-α0 p-1p (hΔ-1) } , where the second (resp. the third) inclusion follows in view of the fact that -α0 has non-negative inner product with all the fundamental roots (resp. all the fundamental weights), and hence also with all elements of +·Δ+ (resp. all elements of XX+, and so also all elements of the form (p-1)δ-λ for λX). The inclusion (4.6) follows in view of the fact that hΔ=2rl an integer. This also shows that Y0X if phΔ-2. It is shown in § 5.4 (Example 2) that Y0={0} for types A2 and B2 (and of course also for type A1). It seems likely that for all other types Y0{0}.

Once again, the linear equations (4.4) can be solved (in those cases where Cλμ's are known) in the same way as (4.2) earlier, because the matrix [Cλμ]X is unipotent with respect to (any linear-order on X stronger than ): { Cλλ=1= the identity element of·eX , and Cλμ0implies (μ+νpλ for someνX+ and hence)μλ. The fact that Cλμ's are themselves formal characters is no obstacle to inverting the unipotent matrix as the inverse can be explicitly written down as a polynomial in the Cλμ's. In fact there is no need to handle the entire matrix of size as large as |X|=pl; in order to compute χμ0 for μ0X one need consider (4.4) simply for λ ranging in the set Y=Y(μ0) consisting of λX such that there exists a sequence μ1,μ2,,μk with Cμj-1μj0 for 1jk. The validity of the "Harish-Chandra principle" to be discussed in § 5 implies that the cardinality of Y never exceeds |W|, which is quite comfortable for large p. Note that Y(μ)Y(μ0)+pY0; the set Y0 can also be "relativized", i.e. replaced by a subset Y0(μ0) of cardinality bounded by a number independent of p, which would show that |Y| is also bounded by a number independent of p.

3 (Added in Proof.) This problem does not seem intractable now. First note that the required index Nλ=[Vλ,max:Vλ,min] for λX+, is the product of the partial indices Nλ,μ=[Vλ,(μ)max:Vλ,(μ)min] as μ ranges over all the weights (since every U-stable -form is compatible with the weight-space decomposition); clearly it suffices to determine Nλ,μ for μX+. I have solved this problem only when μ is "sufficiently close" to λ in this special sense: the dimension of the μ-weightspace of Vλ, equals P(λ-μ) (which is the leading term in Kostant's multiplicity formula) where P() is Kostant's partition function. Based on that result it seems plausible that the value of Nλ,μ is given in general by Nλ,μ= αΔ+ j+ ((λ+δ)(hα)-1j) mj(α) ((··) being the usual binomial symbol) where mj(α)=mjλ;μ(α) is the dimension of the subspace of Vλ,(μ) annihilated by eαj+1 but not by eαj; when μ is sufficiently close to λ it is easily seen that mj(α)=P(λ-μ-jα)-P(λ-μ-(j+1)α). The suggested expression for Nλ,μ is of particular interest for μ=0, since it appears in a crucial manner in the work of K. R. Parthasarathy, R. Ranga Rao and V. S. Varadarajan (Ann. Math. 85 (1967), Theorem 4.2 on page 424) on the principal series representations of complex semi-simple Lie groups; cf. § 8.5.10(b) in Dixmier's book cited in Footnote 5. My proof of this formula (for μ sufficiently close to λ) is obtained by refining certain ideas of N. N. Sapovalov (cf. § 7.8.23 in Dixmier's book) that are not unrelated to the computations of the 3 authors quoted above; it will be published in due course.

In this connection, reader's attention may also be drawn to N. Burgoyne's ideas in § 1 of "Modular representations of some finite groups" in Proc. of Symposia in Pure Mathematics, Vol. XXI, AMS, Providence, 1971.

4 (Added in Proof.) In a recent preprint (titled 'On the modular representations of the general linear and symmetric groups') R. W. Carter and G. Lusztig have introduced the imaginative term "Weyl Module" for Vλ, albeit in the special situation of the general (or the special) linear groups. The allusion to Weyl is to emphasize the fact that the dimension is given by Weyl's formula as in characteristic zero: dimVλ=D(λ+δ); in the special situation of Carter and Lusztig it has an additional association with the treatment given by H. Weyl (in his classic monograph "Classical groups") to the problem of decomposing the tensor spaces in accordance with certain symmetries. Our notation Vλ conforms that in [Hum1971].

Notes and References

This is an excerpt of the paper The Role of Affine Weyl groups in the representation theory of algebraic Chevalley groups and their Lie algebras by Daya-Nand Verma. It appeared in Lie Groups and their Representations, ed. I.M.Gelfand, Halsted, New York, pp. 653–705, (1975).

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