Enhanced Variety of Higher Level and Kostka Functions Associated to Complex Reflection Groups

Abstract

Abstract. Let V be an n-dimensional vector space over an algebraic closure of a finite field Fq, and G = GL(V ). A variety X = G × V r−1 is called an enhanced variety of level r. Let Xuni = Guni × V r−1 be the unipotent variety of X . We have a partition Xuni = λ X λ indexed by r-partitions λ of n. In the case where r = 1 or 2, X λ is a single G-orbit, but if r ≥ 3, X λ is, in general, a union of infinitely many G-orbits. In this paper, we prove certain orthogonality relations for the characteristic functions (over Fq) of the intersection cohomology IC(X λ ,Q l ), and show some results, which suggest a close relationship between those characteristic functions and Kostka functions associated to the complex reflection group Sn ⋉ (Z/rZ) n . IntroductionLet V be an n-dimensional vector space over an algebraic closure of a finite field F q , andhave a geometric interpretation in terms of the intersection cohomology associated to the closure of unipotent classes in G in the following sense. Let C λ be the unipotent class corresponding to a partition λ of n, and K = IC(C λ ,Q l ) be the intersection cohomology complex on the closure C λ of C λ . He proved that H i K = 0 for odd i, and that for partitions λ, µ of n,where x ∈ C µ ⊂ C λ , and n(λ) is the usual n-function. Kostka polynomials are polynomials indexed by a pair of partitions. In [S1], [S2], as a generalization of Kostka polynomials, Kostka functions K λ,µ (t) associated to the complex reflection group S n ⋉ (Z/rZ) n were introduced, which are a-priori rational functions in Q(t) indexed by r-partitions λ, µ of n (see 3.10 for the definition of r-partitions of n). It is known by [S2] that K λ,µ (t) are actually polynomials if r = 2. Although those Kostka functions are defined in a purely combinatorial way, and have no geometric background, recently various generalizations of Lusztig's result for those Kostka functions were found. Under the notation above, consider a variety X = G × V , which is called the enhanced variety, and its subvariety X uni = G uni × V is isomorphic to the enhanced nilpotent cone introduced by Achar-Henderson [AH] (here G uni is the unipotent variety of G). The set of G-orbits under the diagonal action of G on X uni is parametrized by double partitions of n ([AH], [T]). In [AH], they proved that Kostka polynomials indexed by double partitions have a geometric interpretation as in (*) in terms of the intersection cohomology associated to the closure of G-orbits in X uni .On the other hand, let V be a 2n-dimensional symplectic vector space over an algebraic closure of F q with ch F q = 2, and consider G = GL(V ) ⊃ H = Sp(V ). The variety As a generalization of the enhanced variety G × V or the exotic symmetric space G/H × V , we consider G × V r−1 or G/H × V r−1 for any r ≥ 1. G acts diagonally on G × V r−1 , and H acts diagonally on G/H × V r−1 . X = G × V r−1 is called the enhanced variety of level r, and a certain H-stable subvariety X of G/H × V r−1 is called the exotic symmetric space of level r. For those varieties X , one can consid...