arXiv:2104.11528v1 [math.RT] 23 Apr 2021 EXT-MULTIPLICITY THEOREM FOR STANDARD REPRESENTATIONS OF (GL n+1 , GL n ) KEI YUEN CHAN Abstract. Let π1 be a standard representation of GLn+1(F ) and let π2 be the smooth dual of a standard representation of GLn(F ). When F is non- Archimedean, we prove that Ext i GLn(F ) (π1,π2) is ∼ = C when i =0 and vanishes when i ≥ 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-Poincaré 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. 1. Introduction Let F be a local field. One of the important results in quotient branching law is the multiplicity one phenomenon: for an irreducible representation π 1 of GL n+1 (F ) and π 2 of GL n (F ), dim Hom GLn(F ) (π 1 ,π 2 ) ≤ 1, which is a part of the Gan-Gross-Prasad problems [GGP12]. This is established by Aizenbud-Gourevitch-Rallis-Schiffman [AGRS10] and Sun-Zhu [SZ12]. Given the uniqueness of an irreducible quotient, the existence problem remains open for general situations. Partial progress is obtained for some special cases such as generic representations (e.g. [JPSS83, GGP12]), non-tempered representations from Arthur packets (e.g. [GGP20, Ch20]) in p-adic fields. Assume F is non-Archimedean. Let G n = GL n (F ). The goal of this paper is to study the branching problem for another important class of representations– standard modules and its Ext-analog (c.f. [Pr18]). Precisely, we prove that: Theorem 1.1. Let F be a non-archimedean local field. Let π 1 and π 2 be standard modules of G n+1 and G n respectively. Then dim Hom Gn (π 1 ,π ∨ 2 )=1 and, for all i ≥ 1, Ext i Gn (π 1 ,π ∨ 2 )=0. From the viewpoint of period integrals, the local uniqueness property also gives the uniqueness of global periods as well as guarantees Euler factorization of periods. In our context, the regularized periods for Eisenstein series for GL n+1 × GL n are 1