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.
As usual, and denote the sets of the integers, the nonnegative integers, and the positive integers, respectively.
1.1. Multisets. For each set we denote the semigroup that consists of the functions with finite support by The elements of are called finite multisets on (we will omit the word "finite" in the sequel). It is convenient to represent them as a collection of the elements of in which repetitions are allowed; the number is the multiplicity of occurrence of the element in the multiset Let us set in particular, if is a set, then is the number of its elements. A mapping such that for will be called an order of the multiset We will sometimes write the order in the form
Example. The elements of the set are called partitions; if and then we say that is a partition of the number
If is a disjoint union then the semigroup is the direct sum
1.2. Segments in A nonempty finite subset of that consists of several running subsequences of integers is called a segment in We write the number is called the beginning of and is denoted by and is called the end of and is denoted by Let us denote the set of all segments in by
Let We say that precedes (in symbols, if We say that and are connected if either or
1.3. Graded Partitions. The elements of the set will be called graded partitions. Thus, a graded partition is a collection of those segments in in which each segment occurs with multiplicity
Let us define a mapping by setting For each let us set we call the elements of graded partitions of the multiset Obviously, and all the sets are finite.
1.4. Representations of the Groups (See [Zel1980,Zel1977,Zel1977-2]). Let be a non-Archimedean local field. Let us set we assume that Let us denote the category of the algebraic of finite length by its Grothendieck group by the set of equivalence classes of irreducible by and the set of equivalence classes of the cuspidal representations by We set and thus, is a graded free Abelian group with a basis For each we denote its image in by the same letter and the Jordan-Hölder series of by [in other words, we have in
For all with we define the induction functor see [Zel1980, Sec. 1], With the help of these functors, becomes a graded ring. It is commutative and associative (see [Zel1980, Sec. 1]).
1.5. Classification of Irreducible Representations. For each we set Let us denote the set of all irreducible components of by [i.e., and let us set
1.5.1. Proposition (see [Zel1980, 1.10]). The ring is graded with the help of the semigroup i.e., (this means that and
For all and we define the character of the group by the equality where is the standard norm in the field The group acts freely on by the formula The orbits of this action are called lines in let us denote the set of all lines in by so that We have (see Sec. 1.1). For each we set and By Proposition 1.5.1, is a subring of
1.5.2. Proposition (see [Zel1980, 8.6]). The ring is the tensor product of its subrings More precisely, let be represented in the form where and suppose that are different lines in Then the multiplication of representations defines a bijection of with
This proposition reduces the classification of irreducible representations of the groups i.e., the description of the set to the description of each We fix a line in The choice of the point enables us to identify with By the same token, is identified with the notions from Secs. 1.2 and 1.3 are transferred from to by means of transfer of the structure. Obviously, the property of segments in of being connected or to precede one another does not depend on the choice of the point
Now we fix The classification theorem (see [Zel1980], 6.1) asserts that there exists a bijection between and that transforms the set into for each
1.6. Construction of the Bijection Step 1. Let By virtue of [Zel1980, Sec. 2.10]), the module has a unique irreducible submodule; let us denote it by
Example (see [Zel1980, 3.2]). If is the unique representation of the group then
Step 2. Let We choose an order of the multiset such that for the segment does not precede (see Secs. 1.1 and 1.2). By virtue of [Zel1980, Sec. 6.1], the representation has a unique irreducible submodule, and it does not depend on the arbitrariness in the choice of the order of This is the module
1.7. Multiplicity For all and we set Obviously, each irreducible component of has the form where and (see Sec. 1.3). Let us set i.e., is the multiplicity of occurrence of in the Jordan-Hölder series of
Hypothesis. The multiplicity does not depend on i.e., it is determined by the graded partitions and
Thus, if we accept this hypothesis, then the computation of the multiplicities reduces to the computation of the matrix for each
Thus, we have explained the left-hand side of our hypothetical formula (see Introduction). We pass to the description of the right-hand side.
1.8. Varieties Let us consider a finite-dimensional graded vector space over (many of the following results are valid over an arbitrary field, but we will not dwell on this). We say that has type if for each
Let have type Let or denote the set of the operators of degree i.e., of operators such that for this is an affine space of dimension Obviously, the operators from are nilpotent. The group acts naturally on
Proposition-Definition. Each operator has a Jordan basis that consists of homogeneous elements of The orbits of the group on are parametrized by the set to each graded partition there corresponds the orbit that consists of the operators having exactly Jordan cells of the form where for each
Proof. The first statement follows from any proof of the theorem on the Jordan normal form of an operator. The fact that the number of the Jordan cells over each is uniquely determined by the type of the operator follows, e.g., from the Krull-Remak-Schmidt theorem; an elementary proof will be given below in Sec.2.
1.9. Formulation of the Hypothesis. For each we denote the closure of in (say, in the complex topology) by It is clear that is the union of certain
Hypothesis. The sheaves (see Introduction) are equal to for odd The coefficient is nonzero for all if and only if (this will be proved in Sec. 2), and where is an arbitrary point of