Schubert Eisenstein series and Poisson summation for Schubert varieties

TL;DR

This paper proves a Poisson summation formula for schemes related to Schubert varieties, advancing the understanding of Schubert Eisenstein series and confirming their meromorphic continuation in many cases.

Contribution

It establishes the Poisson summation formula for schemes connected to Schubert varieties, supporting the conjecture on meromorphic continuation of Schubert Eisenstein series.

Findings

Proved Poisson summation formula for certain schemes related to Schubert varieties.

Refined and confirmed the conjecture on meromorphic continuation of Schubert Eisenstein series in many cases.

Connected the problem to broader program of generalizing Poisson summation in automorphic forms.

Abstract

The first author and Bump defined Schubert Eisenstein series by restricting the summation in a degenerate Eisenstein series to a particular Schubert variety. In the case of $\mathrm{GL}_3$ over $\mathbb{Q}$ they proved that these Schubert Eisenstein series have meromorphic continuations in all parameters and conjectured the same is true in general. We revisit their conjecture and relate it to the program of Braverman, Kazhdan, Lafforgue, Ng\^o, and Sakellaridis aimed at establishing generalizations of the Poisson summation formula. We prove the Poisson summation formula for certain schemes closely related to Schubert varieties and use it to refine and establish the conjecture of the first author and Bump in many cases.