Ext-Multiplicity Theorem for Standard Representations of
$(GL_n+1,GL_n)$
(2021)cite arxiv:2104.11528Comment: 23 pages, v2: 25 pages, minor changes.Аннотация
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.
Описание
Ext-Multiplicity Theorem for Standard Representations of $(\mathrm{GL}_{n 1},\mathrm{GL}_n)$