arXiv subjects
Publications and source records attributed to Zahi Hazan.
In this note, presented as a ``community service", followed by the PhD research of the author, we draw the relation between Casselman's theorem regarding the asymptotic behavior of matrix coefficients of reductive algebraic groups over $p$-adic fields and its expression as a finite sum of finite functions. In addition, we write the expansion explicitly for general linear groups.
We give a general identity relating Eisenstein series on general linear groups. We do it by constructing an Eisenstein series, attached to a maximal parabolic subgroup and a pair of representations, one cuspidal and the other a character, and express it in terms of a degenerate Eisenstein series. In the local fields analogue, we prove the convergence in a half plane of the local integrals, and their meromorphic continuation. In addition, we find that the unramified calculation gives the Godement-Jacquet zeta function. This realizes and generalizes the construction proposed by Ginzburg and Soudry in Section 3 in their aritcle "Integral derived from the doubling method".
Let $π$ be an irreducible cuspidal representation of $\mathrm{GL}_{kn}\left(\mathbb{F}_q\right)$. Assume that $π= π_θ$, corresponds to a regular character $θ$ of $\mathbb{F}_{q^{kn}}^{*}$. We consider the twisted Jacquet module of $π$ with respect to a non-degenerate character of the unipotent radical corresponding to the partition $(n^k)$ of $kn$. We show that, as a $\mathrm{GL}_{n}\left(\mathbb{F}_q\right)$-representation, this Jacquet module is isomorphic to $π_{θ\upharpoonright_{\mathbb{F}_n^*}} \otimes \mathrm{St}^{k-1}$, where $\mathrm{St}$ is the Steinberg representation of $\mathrm{GL}_{n}\left(\mathbb{F}_q\right)$. This generalizes a theorem of D. Prasad, who considered the case $k=2$. We prove and rely heavily on a formidable identity involving $q$-hypergeometric series and linear algebra.
The present paper proves a $q$-identity, which arises from a representation $π_{N,ψ}$ of $\text{GL}_n(\mathbb{F}_q)$. This identity gives a significant simplification for the dimension of $π_{N,ψ}$, which allowed the second author to obtain a description of the representation.