On the existence of twisted Shalika periods: the Archimedean case

TL;DR

This paper studies the existence of twisted Shalika periods for representations of d6nd6nd6nd6n(\u00d6) over archimedean fields, connecting L-parameters with functional existence and introducing new homological tools.

Contribution

It establishes criteria for twisted Shalika functional existence via L-parameters and develops new homological spectral sequences and formulas in the archimedean setting.

Findings

Criteria for existence of twisted Shalika functionals based on L-parameters.

Development of Hochschild-Serre spectral sequence for nilpotent subgroups.

Results on twisted linear periods via theta correspondence.

Abstract

Let $\K$ be an archimedean local field. We investigate the existence of the twisted Shalika functionals on irreducible admissible smooth representations of $\GL_{2n}(\K)$ in terms of their L-parameters. As part of our proof, we establish a Hochschild-Serre spectral sequence for nilpotent normal subgroups and a Kunneth formula in the framework of Schwartz homology. We also prove the analogous result for twisted linear periods using theta correspondence. The existence of twisted Shalika functionals on representations of $\GL_{2n}^{+}(\R)$ is also studied, which is of independent interest.