%0 Generic %1 chan2021extmultiplicity %A Chan, Kei Yuen %D 2021 %K Multiplicity %T Ext-Multiplicity Theorem for Standard Representations of $(GL_n+1,GL_n)$ %U http://arxiv.org/abs/2104.11528 %X Let $\pi_1$ be a standard representation of $GL_n+1(F)$ and let $\pi_2$ be the smooth dual of a standard representation of $GL_n(F)$. When $F$ is non-Archimedean, we prove that $Ext^i_GL_n(F)(\pi_1, \pi_2)$ 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.

@misc{chan2021extmultiplicity, 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.}, added-at = {2022-06-23T08:49:45.000+0200}, author = {Chan, Kei Yuen}, biburl = {https://www.bibsonomy.org/bibtex/20bae869ec60510b22bf8a36b378753ca/dragosf}, description = {Ext-Multiplicity Theorem for Standard Representations of $(\mathrm{GL}_{n 1},\mathrm{GL}_n)$}, interhash = {a6fd08aa1f01d69d3225834d1f470098}, intrahash = {0bae869ec60510b22bf8a36b378753ca}, keywords = {Multiplicity}, note = {cite arxiv:2104.11528Comment: 23 pages, v2: 25 pages, minor changes}, timestamp = {2022-06-23T08:49:45.000+0200}, title = {Ext-Multiplicity Theorem for Standard Representations of $(\mathrm{GL}_{n+1},\mathrm{GL}_n)$}, url = {http://arxiv.org/abs/2104.11528}, year = 2021 }