On Extending Unipotent Representations to their Stabilizers

Extending real-valued characters of finite general linear and unitary groups on elements related to regular unipotents

When n is odd, consider the finite general linear and unitary groups of rank n, extended by the inverse transpose automorphism. There are elements in the extended groups which square to a regular unipotent element, and we evaluate the values of irreducible characters of the extended groups on these elements. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.

Counting conjugacy classes in the unipotent radical of parabolic subgroups of $\GL_n(q)$

Let $q$ be a power of a prime $p$. Let $P$ be a parabolic subgroup of the general linear group $\GL_n(q)$ that is the stabilizer of a flag in $\FF_q^n$ of length at most 5, and let $U = O_p(P)$. In this note we prove that, as a function of $q$, the number $k(U)$ of conjugacy classes of $U$ is a polynomial in $q$ with integer coefficients.

The character table of the finite Chevalley group $F_4(q)$ for $q$ a power of~$2$

Let $q$ be a prime power and $F_4(q)$ be the Chevalley group of type $F_4$ over a finite field with $q$ elements. Marcelo--Shinoda (1995) determined the values of the unipotent characters of $F_4(q)$ on all unipotent elements, extending earlier work by Kawanaka and Lusztig to small characteristics. Assuming that $q$ is a power of~$2$, we explain how to construct the complete character table of~$F_4(q)$.

On the cohomology of integral $p$-adic unipotent radicals

Let $\mathrm G / F$ be a reductive split $p$-adic group and let $\mathrm U$ be the unipotent radical of a Borel subgroup. We study the cohomology with trivial $\mathbb Z_p$-coefficients of the profinite nilpotent group $N = \mathrm U(\mathcal O_F)$ and its Lie algebra $\mathfrak n$, by extending a classical result of Kostant to our integral $p$-adic setup. The techniques used are a combination of results from group theory, algebraic groups and homological algebra.

Geck's Conjecture and the Generalized Gelfand-Graev Representations in Bad Characteristic

For a connected reductive algebraic group $G$ defined over a finite field $\mathbb F_q$, Kawanaka introduced the generalized Gelfand-Graev representations (GGGRs for short) of the finite group $G(\mathbb F_q)$ in the case where $q$ is a power of a good prime for $G$. This representation has been widely studied and used in various contexts. Recently, Geck proposed a conjecture, characterizing Lusztig's special unipotent classes in terms of weighted Dynkin diagrams. Based on this conjecture, he gave a guideline for extending the definition of GGGRs to the case where $q$ is a power of a bad prime…

$Sp_{4}(F)$主系列表示的扭曲Jacquet模的结构

Structure of Twisted Jacquet Modules of principal series representations of $Sp_{4}(F)$

本文研究非阿基米德局部域上辛群$Sp_4(F)$的主系列表示，确定了由秩1二次型特征标给出的扭曲Jacquet模作为稳定子群模的结构，并推广到有限域情形。