$L^2$-property for algebraic stacks over local non-archimedean fields

TL;DR

This paper introduces an $L^2$-norm on Schwartz half-densities over algebraic stacks in non-archimedean settings, proving finiteness for certain stacks related to the analytic Langlands program.

Contribution

It defines an $L^2$-norm on Schwartz spaces over algebraic stacks and verifies its finiteness for specific stacks of $PGL_2$-bundles, confirming a conjecture in the analytic Langlands context.

Findings

$L^2$-norms are finite for stacks of $PGL_2$-bundles with parabolic structures at ≥3 points.

The introduced $L^2$-norm extends the analysis of algebraic stacks over non-archimedean fields.

Supports conjectures in the analytic Langlands correspondence.

Abstract

We introduce an $L^2$-norm on the space of Schwartz half-densities over algebraic stacks over local non-archimedean fields. We show that these $L^2$-norms are finite for the stacks of $PGL_2$-bundles on $\mathbb{P}^1$ with parabolic structures at $\ge 3$ points. The latter property was conjectured in the context of the analytic Langlands correspondence of arXiv:2103.01509.