Godement--Jacquet L-function and homological theta lifting

TL;DR

This paper studies the theta lifting of dual pairs over non-Archimedean fields, combining homological and analytic methods to determine lifts and properties of the Weil representation, with implications for L-functions and eigendistributions.

Contribution

It provides a complete characterization of big theta lifts under certain conditions and analyzes the projectivity of the Weil representation in relation to the stable range.

Findings

Complete determination of big theta lift when L-function is holomorphic at a critical point

Calculation of big theta lifts of all characters and eigendistributions

Identification of conditions for the Weil representation to be projective

Abstract

In this paper we investigate the theta lifting of type II dual pairs over a non-Archimedean local field, by combining the homological method of Adams--Prasad--Savin and the analytic method of Fang--Sun--Xue. We have three main results: 1. we determine completely the big theta lift of an irreducible representation when its Godement--Jacquet L-function is holomorphic at a critical point; 2. we compute the big theta lift of all characters, hence determine the space of eigendistributions on matrix spaces for all characters; 3. we show that the Weil representation is projective if and only if the dual pair is almost in the stable range.