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

Let ${\pi }_1$ be a standard representation of ${\mathrm{G}\mathrm{L}}_{n+1}(F)$ and let ${\pi }_2$ be the smooth dual of a standard representation of ${\mathrm{G}\mathrm{L}}_n(F)$. When $F$ is non-Archimedean, we prove that ${\mathrm{E}\mathrm{x}\mathrm{t}}_{{\mathrm{G}\mathrm{L}}_n(F)}^i({\pi }_1,{\pi }_2)$ is $\cong \mathbb{C}$ when $i=0$ and vanishes when $i\ge 1$. 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.