Related papers: $\mathbb{P}\mathfrak{gl}_{2}$ is Multiplicity-Free…

Let $\mathbb{F}$ be a non-archimedean local field of positive characteristic different from 2. We consider distributions on $\mathrm{GL}(n+1,\mathbb{F})$ which are invariant under the adjoint action of $\mathrm{GL}(n,\mathbb{F})$. We prove…

We define a generalization of Shalika models for $GL_{n+m}(F)$ and prove that they are multiplicity-free, where $F$ is either a non-Archimedean local field or a finite field and $n,m$ are any natural numbers. In particular, we give new…

Let $\mathfrak{o}$ be the ring of integers of a non-archimedean local field with the maximal ideal $\wp$ and the finite residue field of characteristic $p.$ Let $\mathbf{G}$ be the General Linear or Special Linear group with entries from…

We classify the irredible representations of $\mathrm{GL}_{2}(q)$ for which the induction to the product group $\mathrm{GL}_{2}(q)\times\mathrm{GL}_{2}(q)$, under the diagonal embedding, decomposes multiplicity free. It turns out that only…

Let F be a non-archimedean local field of characteristic zero. We consider distributions on GL(n+1,F) which are invariant under the adjoint action of GL(n,F). We prove that any such distribution is invariant with respect to transposition.…

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,…

Let G be a connected reductive group acting on a finite dimensional vector space V. Assume that V is equipped with a G-invariant symplectic form. Then the ring C[V] of polynomial functions becomes a Poisson algebra. The ring C[V]^G of…

Let F be a non-Archimedean local field or a finite field. Let n be a natural number and k be 1 or 2. Consider G:=GL(n+k,F) and let M:=GL(n,F) x GL(k,F)<G be a maximal Levi subgroup. Let U< G be the corresponding unipotent subgroup and let…

In [AGRS] a multiplicity one theorem is proven for general linear groups, orthogonal groups and unitary groups ($GL, O,$ and $U$) over $p$-adic local fields. That is to say that when we have a pair of such groups $G_n\subseteq G_{n+1}$, any…

For every genuine irreducible admissible smooth representation $\pi$ of the metaplectic group $\widetilde{\Sp}(2n)$ over a p-adic field, and every smooth oscillator representation $\omega_\psi$ of $\widetilde{\Sp}(2n)$, we prove that the…

Let $F$ be a totally real field unramified at all places above $p$ and $D$ be a quaternion algebra which splits at either none, or exactly one, of the infinite places. Let $\bar{r}:\mathrm{Gal}(\bar{F}/F)\to…

Let $F$ be a non-archimedean local field. Let $\Pi$ be a principal series representation of $\mathrm{GL}_n(F)$ induced from an irreducible cuspidal representation of a Levi subgroup. When $\pi$ is an essentially square integrable…

We are considering the commuting variety of the Lie algebra $\mathfrak{pgl}_n$ over an algebraically closed field of characteristic $p >0$, namely the set of pairs $ \{ (A,B) \in \mathfrak{pgl}_n \times \mathfrak{pgl}_n \mid [A,B]=0 \} $.…

We show that every sequence of torsion-free arithmetic congruence lattices in $\mathrm{PGL}(2,\mathbb R)$ or $\mathrm{PGL}(2,\mathbb C)$ satisfies a strong quantitative version of the Limit Multiplicity property. We deduce that for $R>0$ in…

Let $F$ be a finite field of $char F > 3$ and $sl_{2}(F)$ be the Lie algebra of traceless $2\times 2$ matrices over $F$. In this paper, we find a basis for the $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$.

We study homological multiplicities of spherical varieties of reductive group $G$ over a $p$-adic field $F$. Based on Bernstein's decomposition of the category of smooth representations of a $p$-adic group, we introduce a sheaf that…

Let $\pi$ be a cuspidal, cohomological automorphic representation of an inner form $G$ of $\mathrm{PGL}_2$ over a number field $F$ of arbitrary signature. Further, let $\mathfrak{p}$ be a prime of $F$ such that $G$ is split at…

Branching of symplectic groups is not multiplicity-free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra $B$. The algebra $B$ is a graded algebra whose components…

Let $E/F$ be a quadratic extension of non-archimedean local fields of characteristic different from $2$. Let $A$ be an $F$-central simple algebra of even dimension so that it contains $E$ as a subfield, set $G=A^\times$ and $H$ for the…

Let $K$ be an algebraically closed field of characteristic zero, and let $G$ be a connected reductive algebraic group over $K$. We address the problem of classifying triples $(G,H,V)$, where $H$ is a proper connected subgroup of $G$, and…