Generalized Green functions associated to complex reflection groups

TL;DR

This paper develops a comprehensive framework for generalized Green functions linked to complex reflection groups, introducing new functions and algorithms, and connecting them to symplectic group representations.

Contribution

It constructs Hall-Littlewood and Kostka functions for r-symbols, introduces a multi-parameter version, and provides an algorithm for computing multi-parameter Kostka functions.

Findings

Generalized Green functions for symplectic groups described combinatorially.

Algorithm for computing multi-parameter Kostka functions established.

Connections made between Green functions and Kostka functions.

Abstract

In this paper, we consider the set of r-symbols in a full generality. We construct Hall-Littlewood functions and Kostka functions associated to those r-symbols. We also discuss a multi-parameter version of those functions. We show that there exists a general algorithm of computing multi-parameter Kostka functions. As an application, we show that the generalized Green functions of symplectic groups can be described combinatorially in terms of those (one-parameter) Kostka functions.