# A characteristic map for the symmetric space of symplectic forms over a finite field

## TL;DR

This paper constructs an isomorphism linking the representation theory of symplectic forms over finite fields to symmetric functions, specifically mapping spherical functions to Macdonald polynomials with parameters (q, q^2).

## Contribution

It introduces a characteristic map for the symmetric space of symplectic forms over finite fields, extending the classical symmetric group case to a new algebraic setting.

## Key findings

- Spherical functions are mapped to Macdonald polynomials with parameters (q, q^2).
- Provides combinatorial formulas for spherical function values.
- Shows Schur-positivity of skew Macdonald polynomials with specific parameters.

## Abstract

The characteristic map for the symmetric group is an isomorphism relating the representation theory of the symmetric group to symmetric functions. An analogous isomorphism is constructed for the symmetric space of symplectic forms over a finite field, with the spherical functions being sent to Macdonald polynomials with parameters $(q,q^2)$. An analogue of parabolic induction is interpreted as a certain multiplication of symmetric functions. Applications are given to Schur-positivity of skew Macdonald polynomials with parameters $(q,q^2)$ as well as combinatorial formulas for spherical function values.