Ext-multiplicity theorem for standard representations of $$(\textrm{GL}_{n+1},\textrm{GL}_n)$$

Let $$\pi _1$$ be a standard representation of $$\textrm{GL}_{n+1}(F)$$ and let _2$$ the smooth dual $$\textrm{GL}_n(F)$$ . When F is non-Archimedean, we prove that $$\textrm{Ext}^i_{\textrm{GL}_n(F)}(\pi _1, \pi _2)$$ $$\cong \mathbb {C}$$ when $$i=0$$ vanishes $$i \ge 1$$ The main tool proof notion left right Bernstein–Zelevinsky filtrations. An immediate consequence result to give new on multiplicity at most one theorem. Along way, also study an application Euler–Poincaré pairing formula D. Prasad coefficients Kazhdan–Lusztig polynomials. Archimedean field, use left–right Bruhat-filtration for equal rank Fourier–Jacobi models principal series.