Ext-Multiplicity Theorem for Standard Representations of $(\mathrm{GL}_{n+1},\mathrm{GL}_n)$

TL;DR

This paper establishes a multiplicity-one theorem for certain Ext groups between standard representations of GL(n+1) and GL(n) over non-Archimedean fields, and extends results to Archimedean fields using Bruhat-filtrations.

Contribution

It introduces Bernstein-Zelevinsky filtrations to prove Ext group vanishing and multiplicity-one results for standard representations over non-Archimedean fields, and applies Bruhat-filtrations for Archimedean cases.

Findings

Ext^i groups vanish for i ≥ 1 over non-Archimedean fields.

Ext^0 groups are isomorphic to complex numbers, indicating multiplicity one.

New proof of the multiplicity at most one theorem.

Abstract

Let $\pi_1$ be a standard representation of $\mathrm{GL}_{n+1}(F)$ and let $\pi_2$ be the smooth dual of a standard representation of $\mathrm{GL}_n(F)$. When $F$ is non-Archimedean, we prove that $\mathrm{Ext}^i_{\mathrm{GL}_n(F)}(\pi_1, \pi_2)$ is $\cong \mathbb C$ when $i=0$ and vanishes when $i \geq 1$. The main tool of the proof is a notion of left and right Bernstein-Zelevinsky filtrations. An immediate consequence of the result is to give a new proof on the multiplicity at most one theorem. Along the way, we also study an application of an Euler-Poincar\'e pairing formula of D. Prasad on the coefficients of Kazhdan-Lusztig polynomials. When $F$ is an Archimedean field, we use the left-right Bruhat-filtration to prove a multiplicity result for the equal rank Fourier-Jacobi models of standard principal series.