Last update: 2 April 2014
As in Part I, let be an indecomposable reduced root-system, and be the simple Lie algebra over (determined up to isomorphism) possessing as its root-system.
4.1. Let be an algebraic Chevalley group, i.e. a connected semisimple (affine) algebraic group, of type (or of type defined over a field of characteristic one denotes by the group of points of for Thus, depending on one's approach, itself may be identified either with for the algebraic closure of or with for a very large algebraically closed (universal) field or else with the functor is characterised by its affine-ring (being in the notation of [Bor1970], our canonical reference for standard facts summarized here too briefly), which is a Hopf algebra over such that is identified with the maximal-ideal spectrum of for every field the group-operation on coming from the coproduct in in the standard manner. By a we shall always mean an algebraic defined over using the words "module" and "representation" interchangeably. Thus is the underlying field of every representation space and concretely the representation affords a homomorphism of into for every overfield of where
Let us assume that is simply-connected, and split over The former assumption means that every projective representation (i.e. algebraic homomorphism into of comes from a (i.e. factors through 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 contains a maximal torus which is defined and split over i.e. is isomorphic over to where is the "multiplicative group" (which, as a group-functor, assigns to the multiplicative group of One knows that under these assumptions is uniquely determined (up to isomorphism) by and and that may be taken to be the prime field this means existence of an of the affine-ring, which was described explicitly by Kostant purely in terms of the root-system (via the of the enveloping algebra of 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 but treat it as a floating (working) field. Thus the Lie algebra of is a vector space over of dimension The elements of may be identified with either the right or the left-invariant derivations of It may be remarked that unlike the situation in characteristic 0, the object 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 divides the "connexion number" of where are as in § 1; in this case is nonsimple, and the only time that happens again is for types with and with but we shall not require this fact. An explicit description of in terms of essentially due to Chevalley, is as follows.
Let be a Chevalley lattice in This means the following: (i) is a Cartan subalgebra of on which elements of are considered as linear functions and consists of all such that for all in the weight-lattice of (as defined in § 1, with vector space dual of thus if is defined by for all then putting one has (ii) is the of with respect to and if denotes a generator of the (additive) infinite-cyclic group then and (iii) if denotes the subring of the universal enveloping algebra of generated by then is stable under the adjoint action of on It is customary to choose such that and we write for and for for It may be remarked that the set generates as a ring [Ver1972-2].
The Lie algebra of can be identified with in such a way that the Lie algebra of the maximal torus and identifying with the character group of (i.e. the group of algebraic homomorphisms of into with the group-operation given by tensoring these 1-dimensional which is written additively), the subspace of becomes its with respect to the adjoint action of Thus, we are identifying with the root-system of with respect to as is common practice, and this should lead to no confusion. In general, for any (algebraic) and for we denote by its consisting of all vectors in on which acts by so that one has in view of the complete reductibility of thus (for the adjoint action of on is the same as The structure of a is determined by the dimensions of the various weight-spaces, i.e. by the element of the group-ring of over called the formal character of which assigns More precisely, denoting by the set of all combinations of formal symbols where all but a finite number of are zero, and making it into a ring under the usual coordinatewise addition and convolution multiplication with we have The formal character of a shall mean simply that of as a upon restriction. From the standard fact that the Weyl group of can be identified with the quotient in a canonical manner, where is the normalizer of in it is immediate that the formal character of a is a element of under the natural action of
Let be the Borel subgroup of having as its Lie algebra where is a given choice of positive roots in One knows that every irreducible contains a unique 1-dimensional highest-weight-space, in the sense that it is a weight-space for whose elements remain fixed under the unipotent radical of If this weightspace corresponds to the character we say is the highest weight of the given module. Necessarily (the dominant elements of with respect to the fixed choice of positive roots), and if is a "weight" of the given in the sense that the is nonzero, then which we rewrite as by setting up the canonical partial-order on compatible with the addition under which is the set of all elements Conversely, to every there corresponds an irreducible with highest weight which is unique up to isomorphism. As stated in the Introduction, one of the long-standing unsolved problems in the study of is to give a formula for in terms of
4.2. A famous theorem of Steinberg reduces the study of all irreducible for to that of the basic ones with lying in the set the elements of will be called the "restricted dominant characters". The formulation of this theorem hinges on the fact that is defined over (as stated earlier) and that all the are definable over we will be able to see the latter fact from the construction of by the standard technique of "reduction which we shall need to describe presently in some detail. Now if is any defined over the prime field, one defines its Frobenius-twist to be the whose underlying space is itself, such that the new action of on is obtained as follows: take a basis of the of and write the matrix of (the old action of) with respect to this basis of then the new matrix obtained by replacing each entry by its power gives the new (Frobenius-twisted) action of on with respect to the same basis. It is immediate that the of coincides with the of which gives where Steinberg's tensor-product theorem then essentially says that for and the module is irreducible and hence isomorphic to in particular, It follows that if has the then where denotes the iteration of Fr, and consequently,
4.3. In order to define the constants in the title of this section, we need to quickly introduce certain which are modelled after the irreducible with highest weight so that the formal character of coincides with that of We caution in advance that is not uniquely determined by alone; however, there are only a finite number of choices (up to isomorphism). The formal character of is determined by and is given by Weyl's character formula as in characteristic 0; thus is independent of Also (by Proposition 2 below) the composition factors of are determined by alone; these are far from being independent of though, and in fact for every one can find a bound for beyond which is irreducible. This is not a new result, and follows from the "discriminant criterion" described below.
Let be a lattice in ("admissible 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 this follows from the fact that if an arbitrary nonzero element in the highest-weightspace of is required to be a primitive nondivisible) vector in (as can always be ensured upon dilatation, as the highest-weight-space is 1-dimensional), then there exist two uniquely determined extremes for It is clear that the minimal lattice has to be simply the image of under the action of on Let the vector space dual to be the linear function having value 1 at and vanishing at all the weight-spaces of except the highest one. Then is a lowest-weight-vector of the contragredient with highest weight Then, setting one has An alternate way of describing this is in terms of the "contragredient bilinear form" introduced by W. J. Wong [Won1971]:
Let denote the (unique) anti-automorphism of which interchanges and for (and hence leaves the fixed); the existence of is classical and follows, for example, from the now well-known Harish-Chandra—Jacobson—Serre presentation of the Lie algebra (equivalently, of the associative algebra in terms of the generators Then possesses a unique symmetric bilinear form such that and for and more generally for and is determined by the fact that if the projection of on the highest-weight-space along the sum of all other weight-spaces is The restriction of to takes integer values, and one has
The "reduction mod of any lattice in becomes a in a natural manner [Bor1970] (here simply-connectedness of is required, just as in characteristic 0), such that the associated infinitesimal action of coincides with that of (which acts naturally on since with which has already been identified. The module need not be irreducible as we shall see presently, and its internal structure may vary with the choice of Perhaps is always indecomposable; this is certainly true of the two extreme choices of In fact has a unique maximal proper (contained in the sum of all the weight-spaces except the highest one), the quotient by which is the irreducible module and on the other extreme has as its unique irreducible submodule, on account of the duality given by the last equality. The injection gives, on reduction mod only a homomorphism whose kernel is the maximal proper submodule of and image is the unique copy of in in other words, factors through Hence we have the "discriminant criterion": is irreducible if and only if does not divide the index this index equals the discriminant of Wong's contragredient form on described above (cf. [Bor1970, Theorem 2J). It does not seem easy to give a formula for this number. (See It would be particularly interesting to determine all "small" prime divisors of this index for each prime divisors not exceeding the Coxeter number of are of special interest, since the ideas of the next section provide an alternate method towards deciding the irreducibility of particularly when (cf. Proposition 3).
It may be mentioned at this point that for the case of type i.e. for all for are irreducible. But this is an exceptional situation; it will be shown in a later paper that when is not of type there are values of (in fact, for large a sizeable percentage of values exist) such that is not irreducible. The first example of a non-irreducible with was given by Curtis [Cur1960, p. 859], for type
4.4. Let us write and for define to be the multiplicity of the occurrence of as a composition factor of the (See It is clear that and that implies is a weight of and therefore) in particular, is a necessary condition for We have where denotes equivalence in the Grothendieck ring of the category of and thus where the second equality comes from the Weyl's character formula for
The linear equations (4.2) for the "unknowns" can be solved in terms of the "known" quantities in the form where the matrix is the inverse of the infinite matrix the outer subscript denoting the set over which the matrix indices run. With respect to any enumeration of the set which is stronger than the partial order (definition at the end of § 4.1), the matrix is unipotent in the sense that it is triangular with diagonal entries all 1. As a matter of fact, in order to compute for given one needs to consider (4.2) only for ranging over the finite set consisting of those such that there exists a sequence with for Since for we have for i.e. only the finite segment of needs to be inverted.
Proposition 1 (Character formula). There exist unique integers for such that for given only finitely many are nonzero and
By virtue of the standard fact that the set is a basis of the in the group-ring (see [Bou1968]: VI. 3.4, Proposition 3 and Example 2, on p. 187-188), and the fact that the matrix in is unipotent, one finds that is also a basis of the formal characters of Hence we have
Proposition 2. The composition factors of a are determined by its formal character.
Because of (4.1) one is primarily interested in computing the formal characters of the basic irreducible with highest weight in We claim that these are completely determined by our knowledge of the "decomposition" for ranging in alone. To see this, let us impose the new partial-order on defined to be the weakest one such that whenever with all in one has for and let denote any linear order on stronger than Then can be computed, recursively with respect to i.e. assuming that is known for all simply by substitution in where the 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 determines, by virtue of Proposition 2, the entire matrix 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 for all in terms of those for 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 for with all in in what we get is a set of nonlinear equations in However, under certain circumstances, it is possible to replace these by a set of linear equations where the coefficients are no longer integers, but instead lie in the group-ring itself; in fact is a element of the subring consisting of To this end let us define then which gives (4.4) with Our "coefficients" in (4.4) till now are themselves polynomials in the unknowns If for a given value of it can be shown that is irreducible for one can replace in (4.5) by the known quantity in Corollary to Proposition 3 in § 5, we shall see that such is indeed the case at least for using the fact that where is the Coxeter number as in § 1.
To see (4.6) we note that where the second (resp. the third) inclusion follows in view of the fact that has non-negative inner product with all the fundamental roots (resp. all the fundamental weights), and hence also with all elements of (resp. all elements of and so also all elements of the form for The inclusion (4.6) follows in view of the fact that an integer. This also shows that if It is shown in § 5.4 (Example 2) that for types and (and of course also for type It seems likely that for all other types
Once again, the linear equations (4.4) can be solved (in those cases where are known) in the same way as (4.2) earlier, because the matrix is unipotent with respect to (any linear-order on stronger than The fact that 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 In fact there is no need to handle the entire matrix of size as large as in order to compute for one need consider (4.4) simply for ranging in the set consisting of such that there exists a sequence with for The validity of the "Harish-Chandra principle" to be discussed in § 5 implies that the cardinality of never exceeds which is quite comfortable for large Note that the set can also be "relativized", i.e. replaced by a subset of cardinality bounded by a number independent of which would show that is also bounded by a number independent of
(Added in Proof.) This problem does not seem intractable now. First note that the required index for is the product of the partial indices as ranges over all the weights (since every is compatible with the weight-space decomposition); clearly it suffices to determine for I have solved this problem only when is "sufficiently close" to in this special sense: the dimension of the of equals (which is the leading term in Kostant's multiplicity formula) where is Kostant's partition function. Based on that result it seems plausible that the value of is given in general by where is the dimension of the subspace of annihilated by but not by when is sufficiently close to it is easily seen that The suggested expression for is of particular interest for 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 My proof of this formula (for sufficiently close to 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.
(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 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: 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 conforms that in [Hum1971].
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).