Last update: 15 April 2014
This is an excerpt of the paper analog of the Kazhdan-Lusztig Hypothesis by A. V. Zelevinskii. Terrestrial Physics Institute, Academy of Sciences of the USSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 15, No. 2, pp. 9-21, April-June, 1981. Original article submitted November 27, 1980.
In [Zel1980] the author has obtained a classification of irreducible complex representations of the groups over a field (the results of [Zel1980] were announced in [Zel1977, Zel1977-2]). Let us briefly recall this classification. Irreducible representations of the groups are parametrized by collections of segments in the set of the cuspidal representations (the exact definitions are given below in Sec. 1). With each collection of segments we associate the induced module and the irreducible module that is the only irreducible submodule of Particular cases of the representations are the representations that are induced by the one-dimensional representations of parabolic subgroups (in particular, representations of the principal series).
At present the computation of the multiplicity with which the irreducible representation occurs in the Jordan-Hölder series of the module is an open problem. Analogous problems arise in the representation theory in many other situations. In particular, there exist deep analogies between our problems and the problems on the computation of multiplicities of irreducible in the Verma module, where is a complex semisimple Lie algebra.
It was not clear until recently as to in what terms the answer for multiplicities in the Verma modules must be expressed. The right language was found in [KLu1979, KLu1980] by Kazhdan and Lusztig. We describe the content of these papers that is interesting from our point of view.
Let be a Cartan subalgebra of and be the Weyl group of the algebra with respect to the sub-algebra To each character there correspond the Verma module and the irreducible module (see [Dix1977]). The modules have finite length and all of their composition factors have the form The problem is to compute the multiplicity occurrence of in the Jordan-Hölder series of The most interesting case is where and is an integral regular dominant weight. We know that the multiplicity does not depend on (see [BGG1976]); we denote it by Thus it is required to compute the function from a pair of elements of the Weyl group.
Let denote the usual partial order relation on (see [Dix1977], 7.7). We know that if and only if moreover, for all In [KLu1979] the polynomial has been defined for each pair and the hypothesis that its value for is equal to has been advanced. The polynomials satisfy recurrence relations so that they can be explicitly computed (see [KLu1979, Eq. (2.2.c]). Thus, we can assume the problem of computation of the coefficients as solved if the equality is proved.
In [KLu1980] a cohomological interpretation of the polynomials is given in terms of the new cohomology theory of Goresky and MacPherson [GMa1968] (more precisely, in terms of its algebraic version, proposed by Deligne). This theory associates with each algebraic variety over an algebraically closed field of arbitrary characteristic the groups which coincide with the usual cohomologies for nonsingular and satisfy the Poincare duality for all In Deligne's construction, a complex of sheaves is constructed on and are defined as the cohomologies of with coefficients in (if char then consists of sheaves, where but if then can also be assumed to be complex). Let be the cohomological sheaves of the complex and for let denote the stalk of the sheaf at the point See [KLu1980] (see also Sec. 3 below) for the definition and the main properties of the sheaves and
Let be the complex semisimple algebraic group corresponding to and let be its Borel subgroup. By virtue of the Bruhat decomposition, the variety decomposes into the union of the Schubert cells where is the of the point We know that the closure of the cell (it makes no difference whether the closure is taken in the complex topology or in the Zariski topology) is equal to (see, e.g., [Ste1968, Sec. 8]).
Now, we are in a position to formulate the main result [KLu1980]: For the sheaf is equal to for odd and In particular, the Kazhdan-Lusztig hypothesis takes the following form:
The main result of the present article is a (hypothetical) expression for the multiplicities analogous to In this connection, the role of the variety is played by the variety that consists of all linear operators of degree acting in a fixed graded finite-dimensional vector space over the field The automorphism group of preserving the graduation, acts naturally on and its orbits on are the analogs of the Schubert cells. It is easily seen that they are enumerated by the collection of segments in let us denote the orbit corresponding to a collection by The connection of the varieties with the irreducible representations of the groups is very natural from the point of view of the Langlands reciprocity law (see [Zel1980, Sec. 10]).
Hypothesis. All sheaves are equal to for odd and where is an arbitrary point of
In particular, this hypothesis implies that the multiplicity is nonzero if and only if This statement is proved in Sec. 2 of this article. To prove it we develop a combinatorial technique connected with collections of segments in Their properties are analogous to the properties of the ordinary partitions; emphasizing this analogy, we call collections of segments in graded partitions. We study a partial order relation on graded partitions if it is the analog of the well-known relation of domination on partitions (see [LVi1973]).
In Sec. 3 our hypothesis is verified in all the cases for which the multiplicities are known (see [Zel1980, Sec. 11]). In all these cases we compute in a very simple manner the sheaves starting from the explicit construction of the resolution of singularities of the variety In particular, we compute the sheaves on determinant varieties (i.e., on varieties of rectangular matrices whose ranks do not exceed a preassigned value). At present we have not been able to compute these sheaves for all in Sec. 3 we overcome the encountered difficulties.
Quite recently the Kazhdan—Lusztig hypothesis has been proved by J.-L. Brylinski and M. Kashiwara and, independently, by A. A. Beilinson and I. N. Bernshtein. In their works they have done a lot more; viz., they have constructed functors that give equivalence of a certain category of with a category of certain sheaves on With the help of these functors, many notions of the representation theory are reformulated in terms of the variety These functors have not yet been constructed in the case (although there is no doubt that they do exist!).
In correspondence with this ideology, notions of the representation theory of the groups must be translated into the language of the varieties In Sec. 4 we consider an interesting example of this type. In Sec. 9 of [Zel1980] an involution has been constructed on the set of irreducible representations of the groups (it is interesting in that it interchanges the positions of the identity representation and the Steinberg representation). Obviously, the existence of such an involution has a general character; recently, it has been constructed for all finite Chevalley groups (see [Cur1980, Alv1979]). Our involution gives an involution on the set of graded partitions. Its description in the combinatorial terminology is not known (it must be analogous to the operation of transposition of the Young diagrams). In Sec. 4 we give a (hypothetical) description in the terminology of the linear algebra, i.e., we construct an involution on the cells of this construction is a particular case of a general result of Pyasetskii [Pya1975].
The author is deeply thankful to A. A. Beilinson, I. N. Bernshtein, and P. Deligne for useful and fascinating discussions that significantly advanced this work. Thus, as a result of these discussions, the author understood as to how to compute the sheaves on the determinant varieties; in addition, I. N. Bernshtein indicated  to the author. The author is also grateful to A. A. Beilinson and I. N. Bernshtein for acquainting him with the results of their article before its publication.
Often (see, e.g., [KLu1979, KLu1980]) denotes the opposite relation. Under our choice, the identity element is a maximal element.