Multiplicity one theorems for the generalized doubling method

TL;DR

This paper proves a local multiplicity one theorem for the generalized doubling method, establishing uniqueness results crucial for defining local factors and applications in covering groups and global integrals.

Contribution

It establishes the local multiplicity one theorem for the generalized doubling method over any characteristic 0 local field, enabling new applications.

Findings

Proved local multiplicity one theorem for the generalized doubling method.

Applied the theorem to local factors for covering groups.

Enhanced understanding of global unfolding in doubling integrals.

Abstract

In this work we prove the local multiplicity at most one theorem underlying the definition and theory of local $\gamma$-, $\epsilon$- and $L$-factors, defined by virtue of the generalized doubling method, over any local field of characteristic 0. We also present two applications: one to the existence of local factors for genuine representations of covering groups, the other to the global unfolding argument of the doubling integral.