arXiv subjects

Shai Shechter

Publications and source records attributed to Shai Shechter.

4 recordsLinked to original sources

arXiv2019

Bounds on multiplicities of spherical spaces over finite fields -- the general case

Let $G$ be a connected reductive group scheme acting on a spherical scheme $X$. In the case where $G$ is of type $A_n$, Aizenbud and Avni proved the existence of a number $C$ such that the multiplicity $\dim\hom(ρ,\mathbb{C}[X(F)])$ is bounded by $C$, for any finite field $F$ and any irreducible representation $ρ$ of $G(F)$. In this paper, we generalize this result to the case where $G$ is a connected reductive group scheme over $\mathbb{Z}$, and prove Conjecture A of [1].

math.RT↗

arXiv2018

Regular characters of classical groups over complete discrete valuation rings

Let $\mathfrak{o}$ be a complete discrete valuation ring with finide residue field $\mathsf{k}$ of odd characteristic, and let $\mathbf{G}$ be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $\ell\in\mathbb{N}$ let $G^\ell$ denote the $\ell$-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{\ell+1}$ for some $\ell$, and if its restriction to $G^\ell/G^{\ell+1}\simeq \mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}(\mathsf{k}^{\rm alg})$ stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.

math.RT↗

arXiv2017

Enumerating traceless matrices over compact discrete valuation rings

We enumerate traceless square matrices over finite quotients of compact discrete valuation rings by their image sizes. We express the associated rational generating functions in terms of statistics on symmetric and hyperoctahedral groups, viz. Coxeter groups of types A and B, respectively. These rational functions may also be interpreted as local representation zeta functions associated to the members of an infinite family of finitely generated class-2-nilpotent groups. As a byproduct of our work, we obtain descriptions of the numbers of traceless square matrices over a finite field of fixed rank in terms of statistics on the hyperoctahedral groups.

math.GR↗

arXiv2017

Characters of the Norm-One Units of Local Division Algebras of Prime Degree

We give an explicit construction of all complex continuous irreducible characters of the group ${\rm SL}_1(D)$, where $D$ is a division algebra of prime degree $\ell$ over a local field of odd residual characteristic different than $\ell$. For $\ell$ odd, we show that all such characters of ${\rm SL}_1(D)$ are induced from linear characters of compact-open subgroups of ${\rm SL}_1(D)$. We also compute an explicit formula for the representation zeta function of ${\rm SL}_1(D)$.

math.RT↗