Spherical representations of unitary groups at ramified places and the arithmetic inner product formula

TL;DR

This paper investigates admissible representations of ramified unitary groups over local fields and improves the arithmetic inner product formula to include ramified places with local root number -1.

Contribution

It introduces new methods to handle ramified places in the arithmetic inner product formula for unitary groups, expanding its applicability.

Findings

Enhanced the arithmetic inner product formula to include ramified places with local root number -1.

Analyzed properties of admissible representations of ramified unitary groups over local fields.

Connected the study of integral models of unitary Shimura varieties with representation theory.

Abstract

In this article, we study admissible representations of even unitary groups over local fields, where the quadratic extension is ramified, with invariant vectors under the action of the stabilizer of a unimodular lattice and some properties of the corresponding integral model of unitary Shimura varieties. As a direct application, we are able to improve the arithmetic inner product formula so that the places with local root number \((-1)\) are allowed to be ramified.