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: 3 April 2014

§ 5. The "Harish-Chandra principle" and two conjectures on cλμ

5.1. It was conjectured by this author in 1967, in letters to C. W. Curtis and T. A. Springer, that if the irreducible G-modules Mλ and Mμ (for λ,μX+) occur as composition factors of some indecomposable G-module then λ+δ(μ+δ)σ (mod pX) must hold for some σW. This conjecture was preceded by an infinitesimal version (to be discussed in § 6) which was announced to Curtis orally in 1966 (see Footnote**). It has been dubbed "Harish-Chandra principle in nonzero characteristic", because of its close analogy with the theorem of Harish-Chandra on the infinitesimal characters of 𝔤 and because of a central role of this theorem in the study of 𝔤-modules (infinite-dimensional in general), particularly in the study of the universal 𝔤-modules generated by highest-weight-vectors investigated in [Ver1966] and [BGG1971], This analogy is clear from the detailed discussion in [Hum1971 (§ 3)], where our conjectures have been proved true for p>hΔ (Theorems 3.1, 4.1 and 5.1).

It is believed that the Harish-Chandra principle (along with its infinitesimal version) holds for all p, including the bad cases when the Lie algebra 𝔤K of G is not simple. The decompositions of Vλ (for λ ranging in a set larger than X) for the case of type G2 with p=3, given by Springer in [Spr1968, p. 103], as well as a few such decompositions for the case of type F4 with p=2 that are implied by [Vel1970, Tables I and II], are all consistent with our view.

As in Part I (§ 1), let W be the semidirect extension of W by a group D isomorphic to X. However, we find it convenient to modify the action of D on X slightly now: for λX we define the corresponding element of D to be the translation on X (or on E=X) by pλ and not λ as before. Accordingly D, W, W, Ω, and the elements wσ of W, all have modified action on X or E which is dependent on p. This does not affect the abstract structure of any of these groups, nor does it affect the partial orders on W+, W ect. Also, we find it convenient not to change the meaning of E and E*, so that it is pE (resp. pE*) and not E (resp. E*) which is now the fundamental region in E for the action of D (resp. W).

With this convention, the Harish-Chandra principle states that if Mλ and Mμ are composition factors of an indecomposable G-module then λ+δ and μ+δ must lie in the same W-orbit. Since all weights of an indecomposable G-module lie in one single coset of X, clearly λ-μX is another necessary condition. We claim that when p[X:X] these two conditions together imply (and therefore are equivalent to) the condition that λ+δ and μ+δ be conjugate under W. In view of W=WΩ, it suffices to consider the situation when λ+δ and μ+δ are conjugate under Ω={ωj|jJ0} (cf. § 1.2); then μ+δ= (λ+δ)ωj= (λ+δ)σ0σj +pλj(λ+δ) +pλj(modX) (as σ0σjW), gives pλjX; because of p[X:X] we have λjX, which gives λj=0 and ωj=id. as desired.

Let us call λ,μX, W-linked if λ+δ and μ+δ are conjugate under W. It is clear that every element of X is W-linked to a unique element of the set X* = (XpE*)-δ= {λX|p-1(λ+δ)E*} = { λX| λ+δ,αi 0for1il, andλ+δ,-α0 p } , which is partitioned into disjoint subsets XJ* for J{0,1,,l}, where XJ*= (XpEJ*) -δ= { λX*| (λ+δ)Rj =λ+δjJ } . Then μX can be written in the form μ=(λ+δ)w-δ with λXJ*, wWJ, where J,λ,w are all unique; and μX+ (resp. X) if and only if wWJ++ (resp. wWJ). We recall that WJ is empty precisely when the complement of J in {0,1,,l} is a singleton {j} with jJ0, jj0. Note that X* does not lie (entirely) in X+, and that X*X+=X* X=X* X{0}*, where this intersection (resp. the subset X*) is nonempty if and only if δ,-α0 p(resp.δ,-α0 p), i.e. phΔ-1 (resp. phΔ). Let it be noted (from the classification of root-systems) that the Coxeter number hΔ is always a composite number except in the case of type Ap-1 (when hΔ=p), and in this case X* consists of the point 0 alone. This shows that for p<hΔ every point of X has a nontrivial stabilizer in W (and hence in W), and for phΔ every point has a nontrivial stabilizer in W; for the latter statement one needs to observe that if p=hΔ (type Al, l=p-1), then the set XpE*=X*+δ consists of only one point δ which turns out to be the bary-centre of the p vertices {pλj|jJ0} of the fundamental simplex pE* and is therefore fixed by Ω (cf. § 1). We shall call λX p-regular if its stabilizer in W is the identity; for such λ the stabilizer in W is also the identity provided p[X:X].

For J={1,2,,l} the set Xj* has -δ as its solitary element, and the only element of X, W-linked to it is (p-1)δ. Since no dominant weight of V(p-1)δ is W-linked to the highest weight, the Harish-Chandra principle implies that this G-module is irreducible which gives dimM(p-1)δ =pr(wherer= |Δ+|). (5.1) This may be regarded as the first evidence in favour of our conjecture, because (5.1) is known to be true for all p. Since dimMλdim Vλ<pr forλX,λ (p-1)δ, (5.1) follows from the fact that at least one member of {Mλ|λX} has dimension pr; this fact is a consequence of Steinberg's theorem [Bor1970, Theorem 7.5] asserting that upon restriction to G𝔽p (for the form of G defined and split over 𝔽p) the set {Mλ|λX} furnishes all the irreducible KG𝔽p-modules, and another celebrated discovery of Steinberg that there is an irreducible representation of G𝔽p of degree pr (the p-Sylow order of 𝔽p) which remains irreducible upon "reduction mod p in Brauer's sense", i.e. upon passing to KG𝔽p. The module M(p-1)δ is called the Steinberg module (for the p-modular group-ring KG𝔽p, or the algebraic Chevalley group G, or for its Lie algebra 𝔤K, for each of which it is irreducible; see also § 6), and has the property that no indecomposable module other than itself can possess it as a composition factor. Curtis has generalized the existence and uniqueness of the Steinberg module to a class of Tits systems that generalizes the class of finite Chevalley groups.

5.2. The situation (5.1) above is a special case of the following more general consequence of the Harish-Chandra principle.

For a given dominant weight λ, if the set { μ=(λ+δ)w -δ|w W,μλ, μX+ } (5.2) contains no element besides λ, then the G-module Vλ is irreducible.

From this we shall deduce the following Proposition, using the fact that the hypothesis of (5.2) is satisfied when λX*X+, i.e. when λ,-α0p-δ,-α0; it would suffice for us to use this fact for p>hΔ in which case the result of [Hum1971] on the validity of the Harish-Chandra principle is applicable.

Proposition 3. If λ=imiλiX+ satisfies i=1lminip-hΔ+1 then the G-module Vλ is irreducible.

The earlier known results on the irreducibility of Vλ are to be found in [Cur1960, Theorem 1] and [Spr1968, Corollary 4.3], and are mostly contained in the above. Let us note that the Proposition is vacuous when p<hΔ-1. Postponing the values hΔ-1 and hΔ of p, let us assume p>hΔ, which implies p[X:X] as [X:X] never exceeds hΔ. Then in the definition of the set involved in the hypothesis of (5.2) W can be replaced by W. This hypothesis is fulfilled because the set of elements of X+ in the W-orbit of λ+δ has λ+δ as the smallest element (with respect to the partial order ); this latter statement is clear from the observation that the identity is the smallest element of W+ (with respect to , and hence with respect to the affine partial order, introduced in § 1). This proves the Proposition in the major case p>hΔ. For the remaining values of p, we find that the hypothesis of (5.2) is satisfied in a most trivial manner, viz. that Vλ has no dominant weight at all except λ (which guarantees the irreducibility of Vλ irrespective of the characteristic), so that (5.2) and the Harish-Chandra principle need not be invoked. This is clear from the fact that for p=hΔ-1 (resp. p=hΔ, which can hold only when Δ is of type A so that ni=1 for all i) the only value(s) of λ for which the hypothesis of the Proposition holds is λ0=0 (resp. are {λi|0il}).

Note that in the last part of the proof above we have essentially appealed to the standard fact that when GSLl+1 the fundamental representations Vλi are irreducible (irrespective of p), being precisely the exterior powers of the natural (l+1)-dimensional representation. In general, it is not true that all the Vλi's are irreducible irrespective of p; the correct generalization of the preceding phenomenon for type A is the following: For jJ0 the G-module Vλj is irreducible for every p. This follows from the fact that Vλ has no dominant weight except λ if and only if λ{λj|jJ0}, which in turn is an immediate consequence of the observation that if λX+ lies in the coset λi+XX/X (with iJ0) then λλi.

Combining (4.6) and Proposition 3 we have

Corollary to Proposition 3. If p2hΔ-3 then Vλ is irreducible for λ lying in the set Y0 defined in § 4.5.

5.3. For the rest of this section we assume that p[X:X] and the Harish-Chandra principle is valid for the prime p at hand (unless otherwise stated). Thus, whenever Mλ and Mμ are "linked" in the sense that there exists a chain of indecomposable G-modules in which each consecutive pair has a composition factor in common and Mλ (resp. Mμ) occurs as a composition factor of the first (resp. the last) in this chain, the highest weights λ and μ are W-linked, i.e. λ+δ and μ+δ are W-conjugate.

With this assumption the solving of (4.2) is greatly simplified, because now one has Y(μ0) { μX+|μ μ0andμ- (μ0+δ)w- δforwW } . The conjecture below puts an even stricter limit on Y(μ0) which could turn out to be optimal in most cases.

Conjecture II. For μ0,μX+, the nonvanishing of the decomposition number cμ0μ defined in § 4 implies that the following condition (A) holds:

(A) "there exist reflections S1,S2,,Sk such that defining μi=(μi-1+δ)Si-δ successively for 1ik, one has μi+δX+ and μ0μ1 μk-1μk =μ".

The reader who is familiar with the author's work in [Ver1966] on the universal 𝔤-modules generated by highest-weight-vectors (see Footnote5 in § 5.4), would recognize some formal similarity between the above and the Conjecture 1 of [Ver1968] which has recently been proved true by I. M. Gelfand and collaborators [BGG1971] through an elegant tensor product trick. The following recasting is immediate, and provides our main justification for introducing the partial-order on W+.

Conjecture II (Restatement). Suppose μ,μX+ are W-linked, so that μ=(λ+δ)w -δ,μ= (λ+δ)w-δ forλXj*, w,wWJ++, where J,λ,w, are all unique. Then cμμ0implies ww.

It is possible to give substantial confirmation of this Conjecture in case of type A2 i.e. GSL3, but we relegate it to a later paper. Below we give an example in this case which shows that the μi's in the original statement of Conjecture II cannot always be chosen inside X+. With this evidence it becomes reasonable to consider the G-modules Vμ and Mμ not just for μX+, but also, somewhat artificially, for μX+-δ, defining these to be 0-dimensional in case μ+δ is W-singular i.e. lies on a wall of E+, in accordance with dimVμ=D(μ+δ).

Example 1. In the adjoining diagram (Fig. 4) consider an arbitrary point λ in X{0}*, i.e. a point of X lying on the open line-segment joining -δ+pλ1 and -δ+pλ2; say λ+δ=iλ1+(p-i)λ2 for 1ip-1. Setting μ+δ= (λ+δ)R1 R2R0=2pλ1 +iλ2, one immediately sees that the set { μ=(λ+δ)w-δ |wW{0}+ ,μμ } of weights of Vμ, which lie in X+-δ and are linked to μ, consists of 4 elements μ,μ,μ and λ where μ+δ=(λ+δ) R1R0=(p+i) λ2X++δ, and μ+δ=(λ+δ) R2R0=(2p-i) λ1X++δ. In this case we can give the decomposition VμGμcμμMμ on pure dimensionality considerations: on one hand we have dimVμ= pi(2p+i), dimVλ= 12pi(p-i)= dimMλ, -δ+pλ2 R1 -δ R1R0 μ R0 λ id. R2 R1R2 (p-1)δ R1R2R0 R2R0 μ -δ+pλ1 R0R1R2R0 μ Fig. 4. Diagram for Example 1 and on the other hand Mμ= Mλ1Fr M(p-1)λ1+(i-1)λ2 gives dimMμ=3·12pi(p+i); here we have used the irreducibility of Vλ and also of Vμ for μ+δ=pλ1+iλ2, which can be seen either from (5.2) or by appealing to Braden [Bra1967] as long as p5. (Actually Braden gives the infinitesimal decomposition numbers cλμ to be defined in § 6, but in this case it happens that cλμ=cλμ, as we remark at the end of § 6.) Now verifying dimMμ=dim Vμ- dimVλ, we are forced to the exact sequence 0Mλ Vμ Mμ0, which gives cμλ=1. The Conjecture is clearly verified with k=2 (μ0=μ), and μ1 can be taken to be either μ or μ neither of which lies in X+.

5.4. Our next conjecture says that the value of cμμ in the restatement of Conjecture II depends only on w and w (and not on λ).

Conjecture III. Let J{0,1,,l}, ν1,ν2XJ* and w,wWJ++ be all given, and write μi=(νi+δ)w -δ,μi= (νi+δ)w-δ for1i2. Then we must have cμ1μ1=cμ2μ2.

This is analogous to a phenomenon for the composition factors of the universal 𝔤-modules generated by highest-weight-vectors studied in [Ver1966] and [BGG1971], the appearance of this latter phenomenon being indicated by certain information that was communicated to me orally by I. N. Bernstein during the Budapest Summer School on Group Representations, 1971; it may be mentioned at this point that, contrary to Theorems 1 and 7 of [Ver1968] and a similar announcement in the original version of [Gel1971], the composition factors of the said 𝔤-modules can have multiplicity greater than one, as was shown in [BGG1971, p. 9]. (See Footnote5.)

Example 2. Here we shall show that for the case of type B2, each of the two conjectures II and III yields that forμX, cμμ0 impliesμX, (5.3) i.e. the set Y0 defined in § 4 reduces to {0}. (The same conclusion is valid for type A2, and is easily obtained either from the Harish-Chandra principle or from Braden's determination of cλμ in this case. For the case of type G2 it is suspected that this statement is false; this is based on the fact that {wW+|ww0} properly contains W, as was noted in § 1 and § 3; the existence of this possibility is also stated in [Hum1971, p. 76].) We first consider the general case p5, under which the Harish-Chandra principle is valid in view of [Hum1971], taking up p=2 and p=3 later. That then Conjecture II implies (5.3) is straightforward from the fact that if w,w are defined as in the restatement of Conjecture II (then because of wW, which is equivalent to μX) one has www0 and hence wW.

To see that Conjecture III also implies (5.3), we look at the various possibilities for the location of μ outside X (yet satisfying cμμ0 for some μX) in the various triangles (or on their edges) in Fig. 5. The triangles shown there are -δ+3p2λ2 -δ+pλ2 -δ+pλ22 R2R1R2R0 R1R2R0 R2R0 R0 id. -δ (p-1)δ R0R1R2R0 R0R2R0 -δ+pλ1 -δ+2pλ1 Fig. 5. Diagram for Example 2 part of the tessellation of E by (pE*)w-δ where w ranges over W, the element w being shown inside the corresponding triangle. It is clear that μ must lie inside the quadrilateral with vertices -δ,-δ+32p λ2,(p-1)δ ,and-δ+2pλ1, because no μX+ outside this region satisfies μμ for μX. From the irreducibility of V(p-1)δ (the Steinberg module) it is also clear that μ has to be an interior point of this quadrilateral, which is decomposed into seven triangles three of which lie outside p·E. If μ lies in either of the triangles marked R2R1R2R0 and R0R1R2R0, including the edge common to the triangles marked R0R1R2R0 and R0R2R0, then every μX which is W-linked to μ satisfies μμ; so we must try μ in the interior of the R0R2R0-triangle. Take μ+δ=(p+i)λ1+jλ2, with i,j positive integers such that i+2j<p. Then the only μX which is W-linked to μ and does not satisfy μμ lies in the R1R2R0-triangle, and is given by μ+δ=(p-i)λ1 +(p-j)λ2; since μ-μ= (p-2i-2j)α1 +(p-2i-j)α2, one has μμ (a necessary condition for cμμ=0) if and only if i+j<12p. Thus cμμ=0, in case i+j<12p, and hence by Conjecture III for all i,j (satisfying i+2j<p). This completes the two "proofs" of (5.3) under p5.

For p=2 there is nothing to prove because the aforementioned quadrilateral has no interior point lying in X but not in X. For p=3 there is only one such point, viz. μ=3λ1; and the only value of μX, other than the Steinberg highest weight (p-1)δ, satisfying μμ is μ=λ1+2λ2, with μ-μ=α2. The points μ,μ are not in the same W-orbit; however, we do not require to invoice the Harish-Chandra principle in this case to conclude cμμ=0. We have only to show that no nonzero μ-weight-vector of Vμ is annihilated by 𝔤K+=αΔ+𝔤Kα. We claim that the 1-dimensional μ-weight-space of the 40-dimensional G-module Vμ is certainly not annihilated by 𝔤Kα2. This is immediate from the irreducibility of the 3-dimensional SL2-module spanned by the μ,μ, and μ-weight-spaces, where μ=μ-2α2, which is stable under the action of the subgroup of G corresponding to the 3-dimensional simple Lie subalgebra generated by 𝔤Kα2 and 𝔤K-α2. This completes our discussion of (5.3).

5.5. Now assuming Conjecture III, the decomposition VμGμcμμMμ for μ=(λ+δ)w-δ with λXJ* and wWJ++, assumes the form VμG wWJ++ cwwJ M(λ+δ)w-δ, where the (well-determined) constants cwwJ do not depend on λ. Writing χwJ(λ) (resp. χwJ(λ)) for χ(λ+δ)w-δ (resp. χ(λ+δ)w-δ), for λXJ*, we have χwJ (λ)= wWJ++ cwwJ χwJ(λ). (5.4) This simplifies the problem in § 4 of solving the linear equations (4.2). By virtue of Conjecture II, the matrix [cwwJ] for w, w ranging in WJ++, is again unipotent with respect to (any linear order stronger than) the partial order , since cwwJ0 implies ww, and cwwJ=1. As one is primarily interested in evaluating χμ for μX, i.e. χwJ(λ) for wWJ, it suffices to invert the submatrix [cwwJ] for w, w running over the finite subset YJ={wWJ|ww0}. This is particularly convenient for large p, since the number and size of the matrices [c··J]YJ to be inverted are independent of p; in the Epilogue (§ 8) we shall comment on how the matrix-entries themselves are expected to be largely independent of p. Denoting the inverse by [γwwJ] we have dimM(λ+δ)w-δ= wWJ++ γwwJD ((λ+δ)w). (5.5)

** The genesis of this conjecture lies in the author's attempt to stream-line the results of Braden in his Ph. D. thesis, written under the guidance of Curtis [Bra1967], determining the infinitesimal decomposition of Vλ for λX in the case of A2. I learnt of these results during the part of the summer of 1966 that I spent at the University of Oregon under an invitation from Professor Curtis, who initiated me at that time to the /»-modular representation theory of finite Chevalley groups in characteristic p and its connection with the representations of the related Lie algebras.

5 (Added in Proof.) The recently published book "Algèbres Enveloppantes" (Cahiers Scientifiques XXXVII, Gauthier-Villars, 1974) by J. Dixmier contains an exhaustive account of these 𝔤-modules which have been termed Verma modules. But the problem of determination of their composition multiplicities remains open (on account of the said counterexample in [BGG1971]); no one seems to have even a plausible guess at these numbers. (And, therefore, with extreme apology for a rather arbitrary and vague guess, I should like to suggest that these multiplicities may not be unrelated to the numbers bσ,τ mentioned in Footnote2.) There is an obvious parallel between that problem and the present problem of determination of the numbers cλμ, but we think that this (latter) problem should better be compared with the problem of constructing "resolution" of 𝔤-irreducibles in terms of the sums of (i.e. the category generated by) Verma modules which was at once proposed and solved by I. N. Bernstein, I. M. Gelfand and S. I. Gelfand in the sequel to [BGG1971] ("Differential operators on the base affine space and a study of 𝔤-modules", in the present volume, already made famous by Dixmier in his reports in Séminaire Bourbaki (Exposé 425) and the above-mentioned book). More precisely, replacing by the p-adic field and resorting to Kostant's -form U of the enveloping algebra, one can construct certain "p-adic integral forms of Verma modules"; it may be possible to obtain the long-cherished "p-analog of Weyl's character formula" (i.e. to obtain the numbers γλμ directly without having to get cλμ first) by a suitable resolution involving these forms of Verma modules in a manner parallel to that of Bernstein—Gelfand—Gelfand.

At this point it is hard to resist the temptation of musing on the appearance of affine Weyl groups in the Bruhat—Tits theory of 𝔭-adic semisimple groups (though those are "dual" to affine Weyl groups here; cf. end of § 1.1). The natural question is whether this "coincidence" could be related in some manner. Could it be that such a relationship is revealed precisely through the said p-adic integral forms?

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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