Related papers: Cuspidal representations which are not strongly cu…

Word maps have been studied for matrix groups over a field. We initiate the study of problems related to word maps in the context of the group $\mathrm{GL}_n(\mathscr O_2)$, where $\mathscr O_2$ is a finite local principal ideal ring of…

Let $D$ be the quatenion division algebra over a non-Archimedean local field $F$ of characteristic zero and odd residual characterisitc. We show that an irreducible discrete series representation of $\mathrm{GL}_n(D)$ is…

For local non-archimedean fields $k$, Piatetski-Shapiro has defined local spinor $L$-factors for irreducible representations $\Pi$ of $\mathrm{GSp}(4,k)$ of dimension $>1$, attached to a choice of a Bessel model $\Lambda$. We classify…

We give a complete description of the finite-dimensional irreducible representations of the Yangian associated with the orthosymplectic Lie superalgebra $\frak{osp}_{1|2}$. The representations are parameterized by monic polynomials in one…

Let $p$ be a prime number, $F $ a non-archimedean local field with residue characteristic $p$, and $R$ an algebraically closed field of characteristic different from $ p$. We thoroughly investigate the irreducible smooth $R$-representations…

Let G be a simple, simply-connected algebraic group over the complex numbers with Lie algebra $\mathfrak g$. The main result of this article is a proof that each irreducible representation of the fundamental group of the orbit O through a…

We construct the finite-dimensional continuous complex representations of $\mathrm{SL}_2$ over compact discrete valuation rings of even residual characteristic. We also prove that the complex group algebras of $\mathrm{SL}_2$ over finite…

Let F be a non-Archimedean locally compact field with residual characteristic p, let G be an inner form of GL(n,F) for a positive integer n and let R be an algebraically closed field of characteristic different from p. When R has…

The general linear group GL(n, K) over a field K contains a particularly prominent subgroup U(n, K), consisting of all the upper triangular unipotent elements. In this paper we are interested in the case when K is the finite field F_q, and…

Recently, Hazeltine-Liu, and independently Haan-Kim-Kwon, proved a local converse theorem for $\mathrm{SO}_{2n}(F)$ over a $p$-adic field $F$, which says that, up to an outer automorphism of $\mathrm{SO}_{2n}(F)$, an irreducible generic…

In this paper, we will point out a gap in the proof of a theorem of G.Hill (J. Algebra, 174 (1995), 610-635) and will give new arguments to give a remedy in the non-dyadic case modulo a conjecture on the triviality of certain Schur…

The orthogonal groups are a series of simple Lie groups associated to symmetric bilinear forms. There is no analogous series associated to symmetric trilinear forms. We introduce an infinite dimensional group-like object that can be viewed…

Let $\mathbb{F}_q$ be the finite field with $q = p^f$ elements. We study the restriction of two classes of mod $p$ representations of $G_q = \text{GL}_2({\mathbb{F}_q})$ to $G_p = \text{GL}_2(\mathbb{F}_p)$. We first study the restrictions…

This paper examines 8-dimensional Riemannian manifolds whose structure group reduces to ${SO(4)}_{ir}\subset GL(8,\mathbb R)$, the image of an irreducible representation of $SO(4)$ on $\mathbb R^8$. We demonstrate that such a reduction can…

Let $F$ be a local non-archimedean field and $\mathcal{O}_F$ its ring of integers. Let $\Omega$ be a Bernstein component of the category of smooth representations of $GL_n(F)$, let $(J, \lambda)$ be a Bushnell-Kutzko $\Omega$-type, and let…

We investigate the mod-$p$ supersingular representations of $GL_2(D)$, where $D$ is a division algebra over a $p$-adic field with characteristic 0, by computing a basis for the vector space of the pro-$p$ Iwahori subgroup invariants of a…

Let $p\geq 7$ be a prime and $n>1$ be a natural number. We show that there exist infinitely many Galois representations $\varrho:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow GL_{n}(\mathbb{Z}_p)$ which are unramified outside $\{p, \infty\}$…

Let $F/F_{\mathsf{o}}$ be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and $\sigma$ be its non-trivial automorphism. We show that any $\sigma$-self-dual cuspidal representation of ${\rm…

Let F be a non-archimedean local field of characteristic zero. In this paper we construct examples of supercuspidal representations showing that the bound $[N/2]$ for the local converse theorem of $GL_N(F)$ is sharp, N general, when the…

Let $p$ be an odd prime and $\mathbb{F}_p$ be the prime field of order $p$. Consider a $2$-dimensional orthogonal group $G$ over $\mathbb{F}_p$ acting on the standard representation $V$ and the dual space $V^*$. We compute the invariant…