The space $L^2$ on semi-infinite Grassmannian over finite field

TL;DR

This paper constructs a $GL$-invariant measure on a semi-infinite Grassmannian over a finite field, analyzes its symmetries, decomposes the associated $L^2$ space, and relates spherical functions to orthogonal polynomials.

Contribution

It introduces a new invariant measure on the semi-infinite Grassmannian and provides a detailed spectral decomposition of the $L^2$ space.

Findings

Spectrum of the $L^2$ space is discrete.

Spherical functions are expressed via Al Salam--Carlitz orthogonal polynomials.

Invariant measures are constructed on Grassmannian and flag spaces.

Abstract

We construct a $GL$-invariant measure on a semi-infinite Grassmannian over a finite field, describe the natural group of symmetries of this measure, and decompose the space $L^2$ over the Grassmannian on irreducible representations. The spectrum is discrete, spherical functions on the Grassmannian are given in terms of the Al Salam--Carlitz orthogonal polynomials. We also construct an invariant measure on the corresponding space of flags.