Ext-multiplicity theorem for standard representations of (GLn+1,GLn)

Abstract

Let π1 be a standard representation of GL n+1(F) and let π2 be the smooth dual of a standard representation of GL n(F). When F is non-Archimedean, we prove that ExtGLn(F)i(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.