ENHANCED VARIETY OF HIGHER LEVEL AND KOSTKA FUNCTIONS ASSOCIATED TO COMPLEX REFLECTION GROUPS

Abstract

Let V be an n-dimensional vector space over an algebraic closure of a finite field Fq, and G = GL(V). A variety Open image in new window is called an enhanced variety of level r. Let Open image in new window be the unipotent variety of Open image in new window. We have a partition Open image in new window indexed by r-partitions λ of n In the case where r = 1 or 2, Xλ is a single G-orbit, but if r ≥ 3, Xλ 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 \( \mathrm{I}\mathrm{C}\left({\overline{X}}_{\boldsymbol{\uplambda}},{\overline{\mathbf{Q}}}_l\right) \), and show some results, which suggest a close relationship between those characteristic functions and Kostka functions associated to the complex reflection group Open image in new window.