The local period integrals and essential vectors

TL;DR

This paper demonstrates that certain local period integrals for irreducible admissible generic representations of GL_n over p-adic fields, when evaluated at newforms, match local formal L-functions and relate to special L-value evaluations.

Contribution

It establishes the connection between local period integrals involving essential Whittaker functions and formal L-functions, extending understanding of distinguished vectors in p-adic representation theory.

Findings

Integral representations match local formal L-functions at s=1.

Essential Whittaker functions serve as test vectors for period integrals.

Period integrals relate to special values of L-factors for certain representations.

Abstract

By applying the formula for essential Whittaker functions established by Matringe and Miyauchi, we study five integral representations for irreducible admissible generic representations of ${\rm GL}_n$ over $p$-adic fields. In each case, we show that the integrals achieve local formal $L$-functions defined by Langlands parameters, when the test vector is associated to the newform. We give the relation between local periods involving essential Whittaker functions and special values of formal $L$-factors at $s=1$ for certain distinguished or unitary representations. The period integrals are also served as standard non-zero distinguished forms.