Asymptotics of Schwartz functions

TL;DR

This paper compares two definitions of Schwartz spaces on affine closures of certain algebraic groups, proves their equivalence for classical groups and G2, and establishes key properties and Poisson summation formulas relevant for applications.

Contribution

It demonstrates the equivalence of two definitions of Schwartz spaces for classical groups and G2, and proves several conjectured properties and Poisson summation formulas.

Findings

The two definitions of Schwartz space coincide for classical groups and G2.

Several conjectured properties of the Schwartz space are proved.

Poisson summation formulas are established in these cases.

Abstract

Let $G$ be a split, simply connected, almost simple algebraic group, and let $P$ be a maximal parabolic subgroup of $G$. Braverman and Kazhdan in \cite{BKnormalized} defined a Schwartz space on the affine closure $X_P$ of $P^{\mathrm{der}}\backslash G$. An alternate, more analytically tractable definition was given in \cite{Getz:Hsu:Leslie}, following several earlier works. When $G$ is a classical group or $G_2$, we show the two definitions coincide and prove several previously conjectured properties of the Schwartz space that will be useful in applications. Along the way, we give an alternative construction of the ring of differential operators on $X_P$ using the Fourier theory. We also establish the Poisson summation formulae in these cases.