Fourier Eigenspaces of Waldspurger's Basis

TL;DR

This paper studies invariant distributions on p-adic symplectic Lie algebras, identifying Fourier eigenspaces, their dimensions, and stability properties, revealing deep connections with triangular number representations.

Contribution

It characterizes Fourier eigenspaces of Waldspurger's invariant distributions, linking eigenvalues and dimensions to representations of integers as sums of triangular numbers.

Findings

Single eigenvalue when n is triangular or sum of triangular numbers

Dimension of eigenspace equals the count of noncommuting triangular representations

Stable distributions exist only when n equals twice a triangular number

Abstract

In this paper we investigate invariant distributions on $p$-adic $\mathfrak{sp}_{2n}$ defined by Waldspurger in his 2001 tome and find the Fourier eigenspaces in their span. We prove that there is a single eigenvalue if $n$ can be represented as a sum of triangular numbers or is triangular itself and that none exist otherwise. We determine that the dimension of this lone eigenspace is equal to the number of noncommuting representations of $n$ as the sum of at most two triangular numbers. Each such representation corresponds to what we call a Lusztig distribution. These distributions belong to the generating set defined by Waldspurger and form a basis for the eigenspace. Finally, we show that the eigenspace contains a 1-dimensional subspace consisting of stable distributions when $n=2\Delta$, $\Delta$ a triangular number, but otherwise consists of distributions that are not stable.