Related papers: Generic representations, open parameters and ABV-p…

Let $F$ be a non-archimedean local field. Let $\overline{F}$ be an algebraic closure of $F$. Let $G$ be a connected reductive group over $F$. Let $\varphi$ be an elliptic $L$-parameter. For every irreducible representation $\pi$ of $G(F)$…

Mixed-parity module emerges for instance when a de Rham Galois representation is being tensored with a square root of cyclotomic character, which produces half odd integers as the corresponding Hodge-Tate weights. We build the whole…

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,…

If $G$ is a Polish group and $\Gamma$ is a countable group, denote by $\Hom(\Gamma, G)$ the space of all homomorphisms $\Gamma \to G$. We study properties of the group $\cl{\pi(\Gamma)}$ for the generic $\pi \in \Hom(\Gamma, G)$, when…

In this note, we aim to establish a number of embeddings between various function spaces that are frequently considered in the theory of Fourier series. More specifically, we give sufficient conditions for the embeddings $\Phi V[h]\subseteq…

Each infinitesimally faithful representation of a reductive complex connected algebraic group $G$ induces a dominant morphism $\Phi$ from the group to its Lie algebra $\g$ by orthogonal projection in the endomorphism ring of the…

Let $\pi$ be an irreducible, complex, smooth representation of $GL_n$ over a local non-archimedean (skew) field. Assuming $\pi$ has regular Zelevinsky parameters, we give a geometric necessary and sufficient criterion for the irreducibility…

Let $\pi$ be a simple supercuspidal representation of the symplectic group $Sp_{2l}(F)$, over a $p$-adic field $F$. In this work, we explicitly compute the Rankin-Selberg $\gamma$-factor of rank-$1$ twists of $\pi$. We then completely…

Let $F$ be an arbitrary $p$-adic field and let $G$ be an arbitrary reductive group over $F$ with Langlands dual group $^LG$. We show that the change-of-group morphism of Emerton-Gee stacks $\mathcal{X}_{^LG}\to\mathcal{X}_{GL_d}$ is…

Let $k$ be a field, let $G$ be a reductive group, and let $V$ be a linear representation of $G$. Let $V//G = Spec(Sym(V^*))^G$ denote the geometric quotient and let $\pi: V \to V//G$ denote the quotient map. Arithmetic invariant theory…

The lack of a $p$-adic Haar measure causes many methods of traditional representation theory to break down when applied to continuous representations of a compact $p$-adic Lie group $G$ in Banach spaces over a given $p$-adic field $K$. For…

Let K be a complete discretely valued field of mixed characteristics (0, p) with perfect residue field. One of the central objects of study in p-adic Hodge theory is the category of continuous representations of the absolute Galois group of…

We consider the following class of unitary representations $\pi $ of some (real) Lie group $G$ which has a matched pair of symmetries described as follows: (i) Suppose $G$ has a period-2 automorphism $\tau $, and that the Hilbert space…

Let $G$ be a finite group, and let $V$ be a completely reducible faithful $G$-module. It has been known for a long time that if $G$ is abelian, then $G$ has a regular orbit on $V$. In this paper we show that $G$ has an orbit of size at…

Consider a unitary representation $\pi$ of a discrete group $G$, which, when restricted to an almost normal subgroup $\Gamma\subseteq G$, is of type II. We analyze the associated unitary representation $\overline{\pi}^{\rm{p}}$ of $G$ on…

We can associate an admissible unitary representation $\Pi(\rho_p)$ of $\GL_2(\Q_p)$ with every local Galois representation $\rho_p$ by the $p$-adic local Langlands correspondence. If $\rho_p$ is ordinary, we prove local and global…

Let $G$ be a connected reductive $p$-adic group. As verified for unipotent representations, it is expected that there is a close relation between the (Harish-Chandra-Howe) wavefronts sets of irreducible smooth representations and their…

For a smooth affine algebraic group $G$ over an algebraically closed field, we consider several two-variables generalizations of the affine Grassmannian $G(\!(t)\!)/G[\![t]\!]$, given by quotients of the double loop group…

In an earlier paper, we considered several restriction problems in the representation theory of classical groups over local and global fields. Assuming the Langlands-Vogan parameterization of irreducible representations, we formulated…

Let $\pi$ be an irreducible cuspidal representation of $\mathrm{GL}_{kn}\left(\mathbb{F}_q\right)$. Assume that $\pi = \pi_{\theta}$, corresponds to a regular character $\theta$ of $\mathbb{F}_{q^{kn}}^{*}$. We consider the twisted Jacquet…