Enhanced variety of higher level and Kostka functions associated to complex reflection groups

TL;DR

This paper links intersection cohomology of certain algebraic varieties related to complex reflection groups with Kostka functions, extending known results for specific cases and revealing orthogonality relations of Frobenius trace functions.

Contribution

It establishes a connection between intersection cohomology and Kostka functions for enhanced varieties associated with complex reflection groups, generalizing previous results.

Findings

Frobenius trace functions satisfy orthogonality relations

Kostka functions can be expressed via intersection cohomology in some cases

Extension of known results from r=1,2 to more general cases

Abstract

Let $V$ be an $n$ dimensional vector space over an algebraic closure of a finite field $F_q$ and put $G = GL(V)$. For a positive integer $r$, we consider the variety $X_{uni} = G_{uni} \times V^{r-1}$, on which $G$ acts diagonally. $X_{uni}$ is the "unipotent part" of the enhanced variety of level $r$. $X_{uni}$ is partitioned into finitely many pieces $X_{\lambda}$ labelled by $r$-partitions $\lambda$ of $n$, and we consider the intersection cohomology $IC_{\lambda}$ associated to $X_{\lambda}$. In this paper, we show that the Frobenius trace functions (over $F_q$) associated to those $IC_{\lambda}$ satisfy certain orthogonality relations, which are very close to the equations characterizing the Kostka functions indexed by (a pair of) $r$-partitions. Using this we show, in some special cases, that the Kostka functions can be described in terms of those intersection cohomology, which is a (partial) generalization of the known results for the case $r = 1, 2$.