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

§ 6. PIM's of the u-algebra and Humphreys' numbers dλ

6.1. The Lie algebra of an algebraic group in characteristic p0 possesses a 1-ary operation, called p-power operation commonly denoted xx[p], under which it becomes a restricted Lie algebra (also called p-Lie algebra), introduced and studied initially by Jacobson and Zassenhaus. Treating the elements of the Lie algebra as the right- or left-invariant derivations of the affine-ring, this operation arises directly from the fact that the pth power of a derivation is again a derivation. This operation is given explicitly for the basis elements hi=hi 1K𝔥K =𝔥Kand eα=eα 1K𝔤α K=𝔤Kα (for 1il and αΔ) of the Lie algebra 𝔤K of our Chevalley group G by hip= hi, eαp=0. The quotient u of the universal enveloping algebra of 𝔤K by the ideal generated by the elements of the form xp-x[p] for x𝔤K is called the restricted enveloping algebra (also called the u-algebra) of 𝔤K; it is the universal object with respect to the restricted 𝔤K-modules, viz. those representations for which the action of x[p] coincides with the pth power of the endomorphism representing x𝔤K. Thus restricted 𝔤K-modules and u-modules are synonymous. We treat 𝔤K as a subspace of u. By virtue of a Birkhoff—Witt type basis of u (with exponents in the monomials ranging from 0 to p-1 only), one has dimKu=pdimG.

A most remarkable fact about u (which does not seem to have been noticed earlier) is that, in contrast with the above construction of u as a quotient of the ordinary enveloping algebra, it can also be identified with the subalgebra of UK=UK generated by 𝔤K; this can be argued either by noticing that hip-hip· Uandeαp p·Ufor 1ilandαΔ (of which the latter is trivial), or else through more sophisticated arguments which involve identifying UK with the algebra of all (right- or) left-invariant differential operators of the affine-ring AK of G. We shall investigate this new object UK elsewhere (as we do not require it here), calling it the "hyperenveloping algebra for G", since it is the noncommutative analog of the hyperalgebra introduced by Dieudonné for a commutative algebraic group in nonzero characteristic (which is again the algebra of invariant differential operators). Suffice it to say here that UK is a Hopf algebra in some sense dual to AK, and that though it is not finitely generated as an algebra, for representation-theoretic purposes it behaves infinitely better than either the ordinary or the restricted enveloping algebra of 𝔤K, in that every finite-dimensional UK-module can be canonically lifted to a G-module (just as a finite-dimensional U-module lifts to the connected simply-connected complex Lie group with Lie algebra 𝔤C).

At any rate, the infinitesimal of a G-module is naturally defined as a restricted representation of 𝔤K. But this correspondence from the isomorphism classes of G-modules to those of (finite-dimensional) u-modules is neither one-one nor onto. Nevertheless, there are good reasons (even for someone interested purely in group representations, — for algebraic, or finite, Chevalley groups) to study the category of all u-modules; apart from the classical result of Curtis and Steinberg [Bor1970, § 7] giving a bijection between the isomorphism classes of the irreducible modules for algebras u and KG𝔽p, there have recently arisen some new indications in support of this view: in [Ver1972-3] and the forthcoming joint work [HVe1973] we provide certain structural relationships between the projective modules for the two associative algebras, whereas Humphreys has already demonstrated [Hum1971] how some of the structure of the indecomposable projective modules for u can be understood by analysing certain u-modules Zλ which cannot normally be lifted to the group G, which are the analogues of the subject-matter of [Ver1966] and [BGG1971]. (Added in Proof: That is, in the terminology of Dixmier's book quoted in Footnote (5) in § 5.4, Zλ are the u-algebra analogues of Verma modules.) We shall presently describe the modules Zλ and the said results of Humphreys in detail, one of the main objectives of this paper (Parts I and II combined) being the formulation of certain conjectures on the decomposition of these modules Zλ.

It is easy to see that the infinitesimal action of 𝔤K on the G-module MpλMλFr is trivial for all λX+; consequently, by Steinberg's theorem, for λX+ and μX one has Mpλ+μu (dimMλ)· Mμ. (6.1) It is a result of Curtis [Bor1970, 6.4-6.6] that Mμ is u-irreducible for every μX, and that every irreducible u-module is isomorphic to a member of {Mμ|μX} which are mutually nonisomorphic (as we shall see presently). Thus every irreducible u-module lifts uniquely to a G-module. It is known that there exist indecomposable u-modules which do not arise as infinitesimal of any G-module; for the case of type A1 (GSL2) it is not difficult to show that the indecomposable u-module Zλ to be defined presently for λX, has this property with one exception when (for λ=(p-1)δ) it is irreducible (giving the Steinberg module), and it is believed that the same is true for all types. In [Ver1972-3] we shall show that every projective u-module lifts to a G-module, generalizing a phenomenon for the case of SL2 that was very recently observed to follow from certain computations of A. Vincent Jeyakumar on p-modular representations of the group SL2(𝔽p). However, we do not know the answer to the crucial question: Do there exist nonisomorphic G-modules such that the associated actions of.u are indecomposable and isomorphic? Clearly, indecomposability is essential, for the left and right sides in (6.1) are nonisomorphic G-modules.

6.2. There is a natural homomorphism from X=X(T) into the dual 𝔥K* of the Lie algebra 𝔥K of T obtained by taking the infinitesimals of the irreducible representations (characters) of T. Its kernel is p·X and the image is the set of "restricted infinitesimal weights", viz. the set of all those elements of 𝔥K* which (considered as 1-dimensional representations of the Abelian Lie algebra 𝔥K) afford restricted representations of 𝔥K. Thus if 𝔥𝔽p=i𝔽p·hi denotes the canonical 𝔽p-form of 𝔥K, this image is identified with the 𝔽p-dual 𝔥𝔽p*; following Humphreys we shall use the notation Λ in place of 𝔥𝔽p*. The map XΛ, which induces an isomorphism from X|pX to Λ, gives a bisection π:XΛ on restriction to the subset X. We shall use letters λ,μ to denote typical elements of Λ as well as X; and for λΛ the u-modules Mπ-1(λ) and Vπ-1(λ) shall be written simply Mλ and Vλ (and in the same vein, for λX, the module Zπ(λ) defined in the following paragraph may also be denoted by Zλ); in spite of this notational abuse (which can hardly lead to confusion) it is important to maintain a conceptual distinction between the group ΛX/pX and the set X.

Let us note that the action of the Borel subalgebra 𝔟K=𝔥K+αΔ+𝔤Kα on the u-irreducible module Mλ for λΛ, stabilizes a unique 1-dimensional subspace, viz. the highest-weight-space of this G-module (stabilized by B), on which the action of 𝔥K is given by the linear function λ; this shows that the irreducible u-modules Mλ are mutually nonisomorphic for λΛ. Further, since each G-module Mλ is defined over 𝔽p, these u-irreducibles possess 𝔽p-forms with respect to the canonical 𝔽p-form of u, viz. the u-algebra of the "form" 𝔤𝔽p=𝔤Fp of 𝔤K. It follows that the u-modules Mλ are absolutely irreducible for λΛ (or X).

In accordance with [Hum1971], for λΛ let Zλ denote the (right) u-module 1λbu obtained by inducing from the 1-dimensional b-module 1λ for the "Borel subalgebra" b of u generated by 𝔟K, where on 1λ the action of 𝔥K is given by the restricted weight λ and that of the radical 𝔤K+= [𝔟K,𝔟K]= αΔ+ 𝔤Kα of 𝔟K is trivial. If u- denotes the subalgebra of u generated by 𝔤K-=αΔ-𝔤Kα, one finds from the vector space isomorphism ubu- (Birkhoif—Witt theorem for u) that Zλ is a free u- module of rank 1, and thus has dimension pr for r=|Δ+|. The module Zλ is generated by a unique 1-dimensional subspace of the λ-weight-space which is annihilated by 𝔤K+ (is "standard cyclic with highest weight λ" in the terminology of [Hum1971]), and is universal with this property, in that every u-module with this property is a quotient of Zλ. It is proved in [Hum1971, Proposition 1.1] that for λΛ the u-module Vλ has this property, so that one has the surjections ZλVλMλ. (For λX+, λX, the dimension of Vλ generally exceeds pr; however, it is easy to see that for no such λ can the u-module Vλ be "standard cyclic".) It should be noted that though the G-module Mλ has the λ-weight-space 1-dimensional for all λX+, in the infinitesimal situation normally the λ-weight-space of the u-irreducible Mλ for λΛ has dimension greater than one (and hence the same holds for Vλ and Zλ), on account of the merging of weights when one passes from the decomposition for the torus T to that for the Lie algebra 𝔤K. In fact, every weight-space of every Zλ has dimension pr-l. Considering the decomposition of u- as a direct sum of the subspace of scalars and the augmentation ideal (generated by 𝔤K-), one finds that every proper submodule of Zλ is contained in the image.of the 1-dimensional generating subspace under this augmentation ideal. This shows that Zλ has a unique maximal proper submodule (the kernel of ZλMλ), and that Zλ has no irreducible module other than Mλ as its quotient, and thus is indecomposable.

6.3. On account of the standard facts from the theory of Artinian rings (which applies to u because of its finite dimensionality), we know that (i) up to isomorphism there exists a unique projective u-module Qλ (called the projective cover of Mλ) such that Mλ is its only irreducible quotient, (ii) every indecomposable u-projective is isomorphic to some Qλ for λΛ, which are clearly all indecomposable, (iii) the category of projective u-modules is semisimple, i.e. every projective module is isomorphic to a direct sum λΛnλQλ, and (iv) the (right-regular) representation of u on u (which is free, and therefore projective, over u) is equivalent to the direct sum λΛ (dimMλ)· Qλ, so that every Qλ does occur in this representation. The modules Qλ are called PIM's (principal indecomposable modules). Two PIM's Qλ and Qμ are called linked if there exists a chain of PIM's beginning at Qλ and ending at Qμ such that the consecutive pairs have a common composition factor; we write QλQμ in that case. Further, one knows (from the theory of Artinian rings) that the sum of all PIM's belonging to one linkage class (i.e. equivalence class under ) forms a 2-sided ideal of u generated by a primitive central idempotent, and thus is a minimal direct summand in the associative algebra u, called a "block" of u. Since both Mλ and Qλ are defined over the prime field, one finds that the semi-simple part of each block, i.e. the quotient by its radical, is "split" in the sense that it is a direct sum of matrix algebras, — exactly one of dimension (dimMλ)2 for each λ corresponding to Qλ's appearing in the linkage class. From #the decomposition of u as a direct sum (as K-algebra) of blocks, it is clear that if Qλ and Qμ are not linked then Mλ and Mμ cannot be "linked" either (as u-modules, in a sense similar to that for G-modules: see the beginning of § 5.3), the converse being trivial. It must be noted that if Mλ and Mμ are linked as G-modules (for λ,μX) then these are also linked as u-modules, but not conversely: for in the former case one necessarily has λ-μX, whereas this is far from necessary in the case of u-modules as we shall see presently.

The infinitesimal version of the Harish-Chandra principle (cf. § 5) conjectures that for λ,μΛ if Mλ and Mμ occur as a composition factor of an indecomposable u-module (or more generally, if Mλ and Mμ are linked as u-modules) then λ+δ and μ+δ must be conjugate under the Weyl group W (which acts canonically on ΛX/pX), where δ stands for the image of δX under π. When this conclusion on λ,μ holds, let us introduce the terminology that λ and μ are W-linked; in the notation of [Hum1971] this is expressed as λμ. It must be observed that λ,μΛ are W-linked if and only if the restricted dominant (global) weights π-1(λ),π-1(μ)X are W-linked (and not just W-linked), in the sense that π-1(λ)+δ and π-1(μ)+δ are W-conjugate. (The term "W-linked" was introduced in § 5.1.)

The conjecture on the linkage of u-modules that we just mentioned was proved to be true in [Hum1971], provided that p>hΔ. In this (infinitesimal) case the converse is also easy, and follows from the fact that if a fundamental reflection RiW sends λ+δ to μ+δ=λ+δ-kαi with 1kp-1 (where αi stands for the projection of the global root αiΔX under the natural projection XΛ), then there is a nonzero homomorphism from Zμ to Zλ (if v and v are nonzero elements in the 1-dimensional generating subspaces of the highest-weight-spaces of Zλ and Zμ respectively, then this homomorphism is determined by vvfik, which shows that the indecomposable u-module Zλ has both Mλ and Mμ as composition factors. In fact, pursuing this argument further, Humphreys has found [Hum1971, Theorem 2.2] (without assuming the Harish-Chandra principle) that if λ and μ are W-linked elements of Λ then ZλuZμ, where u denotes equivalence in the Grothendieck ring of the category of u-modules, i.e. Zλ and Zμ have identical composition factors (multiplicity included). It follows that if λ,μX are W-linked then Mλ and Mμ are linked as u-modules, which corroborates the statement immediately preceding the last paragraph.

6.4. Now let b be the subalgebra of u generated by the Borel subalgebra 𝔟K=𝔥K+𝔤K- opposite to 𝔟K. It is readily seen, by looking at the regular representation of b on b, that the modules {Zλ|λΛ} exhaust the (isomorphism classes of) projective b-modules (upon restriction to b), the unique irreducible quotient of Zλ being the 1-dimensional irreducible b-module 1λ on which 𝔤K- acts trivially and 𝔥K acts through the restricted weight λ. Since u is b-free (under right-multiplication, as is clear from the Birkhoff—Witt theorem for u-algebras) and Qλ is a direct summand in u, we see that the restriction of Qλ to b is b-projective. Hence there is a decomposition in the semisimple category of projective b-modules: Qλbμ [Qλ:Zμ]b-proj ·Zμ, where the meaning of "b-proj" in subscript is clear.

We are now ready to state the most interesting consequence of the (infinitesimal) Harish-Chandra principle that Humphreys has derived [Hum1971, Theorem 4.5]:

If we define dμ=[Zμ:Mμ]u (=[Zλ:Mμ]u for every λΛ which is W-linked to μ), then Qλb dλ·(μλZμ), i.e. the multiplicity [Qλ:Zμ]b-proj is dλ or 0 according as Qλ is linked to Qμ or not.

It follows that if aλ is the cardinality of the W-linkage class of λ in Λ then dimQλ=aλ dλpr. In order to compute aλ one has only to know the stabilizer Wλ+δ of λ+δ in W, for then one has |W|=aλ·|Wλ+δ|. For p[X:X] we claim that Wλ+δ is again a reflection subgroup. Since Λ is the set of "integral points" on the torus E/pX, this fact can be obtained from the study of Steinberg of the action of Weyl groups on tori [Ste1968-2, Lemma 4.1], but the following observation not only gives this fact directly but also shows how to compute the order of this stabilizer. Let μ=π-1(λ)X, and denote by Wμ+δ the stabilizer of μ+δ in W. Then p[X:X] gives Wμ+δW by exactly the same type of argument as in the fourth paragraph of § 5. The subgroup Wμ+δ is then generated by reflections; in fact, if νXJ* is the unique element of X* which is W-linked to μ then Wμ+δ is conjugate to the stabilizer WJ in W of ν+δ (WJ is defined in § 1.4 as the subgroup by the reflections {Rj|jJ}). Since Wμ+δ contains no translations, the projection WW gives an isomorphism of Wμ+δ onto Wλ+δ. Thus to calculate aλ (for p[X:X]) one needs only determine J{0,1,,l}, for the Coxeter relations on {Rj|jJ} can be then read off from the extended Dynkin diagram and the order of the group generated by them readily computed by standard methods. This effectively solves the problem of computing aλ in any given case. (Things could be accounted for even in the more general situation when p is allowed to divide [X:X], through a slightly more detailed analysis; see § 7.2.)

6.5. In the next section we offer a conjecture on the decomposition of the u-module Zλ which implies a neat formula ((7.2)) for Humphreys' numbers dλ. This formula involves the constants bσ of Conjecture I and the infinitesimal decomposition numbers cλμ=[Vλ:Mμ]u analogous to the constants cλμ, and thus forms the raison d'être for the two parts of this paper. The numbers cλμ for λ,μΛ can be readily derived from a knowledge of cλμ for λX and μX+; we recall that the latter set of numbers were shown to be sufficient, in § 4, for determining the formal characters of all the irreducible G-modules and their dimensions. Explicitly, Vλ G μX+ cλμMμ G μX ( μY0 cλ,μ+pμ MμMμFr ) u μX ( μY0 cλ,μ+pμ ·dimMμ ) ·Mμ, which gives cλμ= μY0 cλ,μ+pμ ·dimMμ= traceCλμ, (6.2) where Cλμ is as in (4.5), and the trace of an element λnλeλ of ·eX means the sum λnλ of all the coefficients (multiplicities). Here we have repeated the earlier abuse of identifying indices in X and Λ via the bijection π. In case p is large enough to guarantee the irreducibility of Vμ for μY0 (cf. Corollary to Proposition 3 in § 5), one substitutes dimMμ=D(μ+δ) in (6.2) and obtains a cleaner formula for cλμ. Note that on has cλμ=cλμ whenever Y0={0} (as is the case for types A2 and B2; cf. Example 2, § 5).

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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