Last update: 3 April 2014
6.1. The Lie algebra of an algebraic group in characteristic possesses a operation, called operation commonly denoted under which it becomes a restricted Lie algebra (also called 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 power of a derivation is again a derivation. This operation is given explicitly for the basis elements (for and of the Lie algebra of our Chevalley group by The quotient of the universal enveloping algebra of by the ideal generated by the elements of the form for is called the restricted enveloping algebra (also called the of it is the universal object with respect to the restricted viz. those representations for which the action of coincides with the power of the endomorphism representing Thus restricted and are synonymous. We treat as a subspace of By virtue of a Birkhoff—Witt type basis of (with exponents in the monomials ranging from 0 to only), one has
A most remarkable fact about (which does not seem to have been noticed earlier) is that, in contrast with the above construction of as a quotient of the ordinary enveloping algebra, it can also be identified with the subalgebra of generated by this can be argued either by noticing that (of which the latter is trivial), or else through more sophisticated arguments which involve identifying with the algebra of all (right- or) left-invariant differential operators of the affine-ring of We shall investigate this new object elsewhere (as we do not require it here), calling it the "hyperenveloping algebra for 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 is a Hopf algebra in some sense dual to 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 in that every finite-dimensional can be canonically lifted to a (just as a finite-dimensional lifts to the connected simply-connected complex Lie group with Lie algebra
At any rate, the infinitesimal of a is naturally defined as a restricted representation of But this correspondence from the isomorphism classes of to those of (finite-dimensional) 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 apart from the classical result of Curtis and Steinberg [Bor1970, § 7] giving a bijection between the isomorphism classes of the irreducible modules for algebras and 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 can be understood by analysing certain which cannot normally be lifted to the group 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 in § 5.4, are the analogues of Verma modules.) We shall presently describe the modules 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
It is easy to see that the infinitesimal action of on the is trivial for all consequently, by Steinberg's theorem, for and one has It is a result of Curtis [Bor1970, 6.4-6.6] that is for every and that every irreducible is isomorphic to a member of which are mutually nonisomorphic (as we shall see presently). Thus every irreducible lifts uniquely to a It is known that there exist indecomposable which do not arise as infinitesimal of any for the case of type it is not difficult to show that the indecomposable to be defined presently for has this property with one exception when (for 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 lifts to a generalizing a phenomenon for the case of that was very recently observed to follow from certain computations of A. Vincent Jeyakumar on representations of the group However, we do not know the answer to the crucial question: Do there exist nonisomorphic 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
6.2. There is a natural homomorphism from into the dual of the Lie algebra of obtained by taking the infinitesimals of the irreducible representations (characters) of Its kernel is and the image is the set of "restricted infinitesimal weights", viz. the set of all those elements of which (considered as 1-dimensional representations of the Abelian Lie algebra afford restricted representations of Thus if denotes the canonical of this image is identified with the following Humphreys we shall use the notation in place of The map which induces an isomorphism from to gives a bisection on restriction to the subset We shall use letters to denote typical elements of as well as and for the and shall be written simply and (and in the same vein, for the module defined in the following paragraph may also be denoted by in spite of this notational abuse (which can hardly lead to confusion) it is important to maintain a conceptual distinction between the group and the set
Let us note that the action of the Borel subalgebra on the module for stabilizes a unique 1-dimensional subspace, viz. the highest-weight-space of this (stabilized by on which the action of is given by the linear function this shows that the irreducible are mutually nonisomorphic for Further, since each is defined over these possess with respect to the canonical of viz. the of the "form" of It follows that the are absolutely irreducible for (or
In accordance with [Hum1971], for let denote the (right) obtained by inducing from the 1-dimensional for the "Borel subalgebra" of generated by where on the action of is given by the restricted weight and that of the radical of is trivial. If denotes the subalgebra of generated by one finds from the vector space isomorphism (Birkhoif—Witt theorem for that is a free module of rank 1, and thus has dimension for The module is generated by a unique 1-dimensional subspace of the which is annihilated by (is "standard cyclic with highest weight in the terminology of [Hum1971]), and is universal with this property, in that every with this property is a quotient of It is proved in [Hum1971, Proposition 1.1] that for the has this property, so that one has the surjections (For the dimension of generally exceeds however, it is easy to see that for no such can the be "standard cyclic".) It should be noted that though the has the 1-dimensional for all in the infinitesimal situation normally the of the for has dimension greater than one (and hence the same holds for and on account of the merging of weights when one passes from the decomposition for the torus to that for the Lie algebra In fact, every weight-space of every has dimension Considering the decomposition of as a direct sum of the subspace of scalars and the augmentation ideal (generated by one finds that every proper submodule of is contained in the image.of the 1-dimensional generating subspace under this augmentation ideal. This shows that has a unique maximal proper submodule (the kernel of and that has no irreducible module other than as its quotient, and thus is indecomposable.
6.3. On account of the standard facts from the theory of Artinian rings (which applies to because of its finite dimensionality), we know that (i) up to isomorphism there exists a unique projective (called the projective cover of such that is its only irreducible quotient, (ii) every indecomposable is isomorphic to some for which are clearly all indecomposable, (iii) the category of projective is semisimple, i.e. every projective module is isomorphic to a direct sum and (iv) the (right-regular) representation of on (which is free, and therefore projective, over is equivalent to the direct sum so that every does occur in this representation. The modules are called PIM's (principal indecomposable modules). Two PIM's and are called linked if there exists a chain of PIM's beginning at and ending at such that the consecutive pairs have a common composition factor; we write 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 generated by a primitive central idempotent, and thus is a minimal direct summand in the associative algebra called a "block" of Since both and 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 for each corresponding to appearing in the linkage class. From #the decomposition of as a direct sum (as of blocks, it is clear that if and are not linked then and cannot be "linked" either (as in a sense similar to that for see the beginning of § 5.3), the converse being trivial. It must be noted that if and are linked as (for then these are also linked as but not conversely: for in the former case one necessarily has whereas this is far from necessary in the case of as we shall see presently.
The infinitesimal version of the Harish-Chandra principle (cf. § 5) conjectures that for if and occur as a composition factor of an indecomposable (or more generally, if and are linked as then and must be conjugate under the Weyl group (which acts canonically on where stands for the image of under When this conclusion on holds, let us introduce the terminology that and are in the notation of [Hum1971] this is expressed as It must be observed that are if and only if the restricted dominant (global) weights are (and not just in the sense that and are (The term was introduced in § 5.1.)
The conjecture on the linkage of that we just mentioned was proved to be true in [Hum1971], provided that In this (infinitesimal) case the converse is also easy, and follows from the fact that if a fundamental reflection sends to with (where stands for the projection of the global root under the natural projection then there is a nonzero homomorphism from to (if and are nonzero elements in the 1-dimensional generating subspaces of the highest-weight-spaces of and respectively, then this homomorphism is determined by which shows that the indecomposable has both and as composition factors. In fact, pursuing this argument further, Humphreys has found [Hum1971, Theorem 2.2] (without assuming the Harish-Chandra principle) that if and are elements of then where denotes equivalence in the Grothendieck ring of the category of i.e. and have identical composition factors (multiplicity included). It follows that if are then and are linked as which corroborates the statement immediately preceding the last paragraph.
6.4. Now let be the subalgebra of generated by the Borel subalgebra opposite to It is readily seen, by looking at the regular representation of on that the modules exhaust the (isomorphism classes of) projective (upon restriction to the unique irreducible quotient of being the 1-dimensional irreducible on which acts trivially and acts through the restricted weight Since is (under right-multiplication, as is clear from the Birkhoff—Witt theorem for and is a direct summand in we see that the restriction of to is Hence there is a decomposition in the semisimple category of projective where the meaning of 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 for every which is to then i.e. the multiplicity is or according as is linked to or not.
It follows that if is the cardinality of the class of in then In order to compute one has only to know the stabilizer of in for then one has For we claim that is again a reflection subgroup. Since is the set of "integral points" on the torus 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 and denote by the stabilizer of in Then gives by exactly the same type of argument as in the fourth paragraph of § 5. The subgroup is then generated by reflections; in fact, if is the unique element of which is to then is conjugate to the stabilizer in of is defined in § 1.4 as the subgroup by the reflections Since contains no translations, the projection gives an isomorphism of onto Thus to calculate (for one needs only determine for the Coxeter relations on 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 in any given case. (Things could be accounted for even in the more general situation when is allowed to divide through a slightly more detailed analysis; see § 7.2.)
6.5. In the next section we offer a conjecture on the decomposition of the which implies a neat formula ((7.2)) for Humphreys' numbers This formula involves the constants of Conjecture I and the infinitesimal decomposition numbers analogous to the constants and thus forms the raison d'être for the two parts of this paper. The numbers for can be readily derived from a knowledge of for and we recall that the latter set of numbers were shown to be sufficient, in § 4, for determining the formal characters of all the irreducible and their dimensions. Explicitly, which gives where is as in (4.5), and the trace of an element of means the sum of all the coefficients (multiplicities). Here we have repeated the earlier abuse of identifying indices in and via the bijection In case is large enough to guarantee the irreducibility of for (cf. Corollary to Proposition 3 in § 5), one substitutes in (6.2) and obtains a cleaner formula for Note that on has whenever (as is the case for types and cf. Example 2, § 5).
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).