Poincare series and miraculous duality

TL;DR

This paper explores the interplay between Poincare series and miraculous duality within the geometric Langlands framework, establishing new functorial relationships and constructing Jacquet functors for parabolic subgroups.

Contribution

It introduces a novel connection between Poincare series and duality functors, and constructs Jacquet functors for arbitrary parabolic subgroups in the geometric Langlands setting.

Findings

Miraculous duality intertwines Poincare series with Whittaker coefficients.

Constructs Jacquet functors controlling constant terms of Eisenstein series.

Establishes functorial relationships in the geometric Langlands context.

Abstract

In the setting of global geometric Langlands, we show that miraculous duality on the stack of principal bundles on a curve intertwines the functor of Poincare series with the dual functor to Whittaker coefficients. We construct, for arbitrary parabolic subgroups, Jacquet functors controlling constant terms of Poincare series and Whittaker coefficients of Eisenstein series in local terms.