Gelfand Pairs of Complex Reflection Groups

TL;DR

This paper computes the zonal spherical functions for a specific Gelfand pair of complex reflection groups, provides a product formula, and introduces a discrete Laplace operator with these functions as eigenfunctions.

Contribution

It explicitly calculates spherical functions for the Gelfand pair $(G(r,d,n), S_n)$ and introduces a new discrete Laplace operator related to these functions.

Findings

Explicit formulas for spherical functions of the Gelfand pair.

A product formula for these spherical functions.

A discrete Laplace operator with spherical functions as eigenfunctions.

Abstract

In this article the zonal spherical functions of the Gelfand pair $(G(r,d,n), S_n)$ of complex reflection groups will be calculated. After this, a product formula for these spherical functions and a discrete analog of the Laplace operator which has the spherical functions as eigenfunctions will be given.