Modulation groups

TL;DR

This paper introduces the concept of modulation groups, generalizing the metaplectic representation, and explores their role in Schwartz spaces for spherical varieties, with detailed examples and conjectural links to algebraic groups.

Contribution

It defines modulation groups associated with Schwartz spaces of spherical varieties, extending the metaplectic group's representation, and proposes a method to connect them to ind-algebraic groups.

Findings

Schwartz spaces are naturally representations of modulation groups

Examples include vector spaces and quadric cones where modulation groups relate to algebraic groups

Discussion on linking modulation groups to ind-algebraic groups and their relation to Poisson summation

Abstract

Conjectures of Braverman and Kazhdan, Ng\^o and Sakellaridis have motivated the development of Schwartz spaces for certain spherical varieties. We prove that under suitable assumptions these Schwartz spaces are naturally a representation of a group that we christen the modulation group. This provides a broad generalization of the defining representation of the metaplectic group. The example of a vector space and the zero locus of a quadric cone in an even number of variables are discussed in detail. In both of these cases the modulation group is closely related to algebraic groups, and we propose a conjectural method of linking modulation groups to ind-algebraic groups in general. At the end of the paper we discuss adelization and the relationship between representations of modulation groups and the Poisson summation conjecture.