Related papers: Classification of $\mathrm{GL}_{n}(\mathbb{C})$-Re…

Let $\mathrm{G} = \mathrm{Gl}_{n}(K)$, and $\mathrm{H} = \mathrm{G}^{\sigma}$ for $\sigma$ an involution of the form $g\rightarrow aga^{-1}$, It is known that for $K =\mathbb{Q}_q$ any irreducible representation of $\mathrm{G}$ with an…

We continue our study of Hilbert space representations of the Reflection Equation Algebra, again focusing on the algebra constructed from the $R$-matrix associated to the $q$-deformation of $GL(N,\mathbb{C})$ for $0<q<1$. We develop a form…

This is a successive paper of arXiv:1909.11492. We prove an equivalence between the category of finite-dimensional representations of degenerate supergroup $\underline{GL}(M|N)$ and the category of $(GL_M(O) \ltimes U_{M, N}(F), \chi_{M,…

On restriction to the maximal compact subgroup $\mathrm{GL}(3,\mathscr{R})$, an unramified principal series representation of the $p$-adic group $\mathrm{GL}(3,F)$ decomposes into a direct sum of finite-dimensional irreducibles each…

Selfdual representations of any group fall into two classes when they are irreducible: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the…

In this paper we study irreducible unitary representations of GL(n,R) and prove a number of results. Our first result establishes a precise connection between the annihilator of a representation and the existence of degenerate Whittaker…

Let $X$ be a smooth, separated, geometrically connected scheme defined over a number field $K$ and $\{\rho_\lambda\}_\lambda$ a system of n-dimensional semisimple $\lambda$-adic representations of the \'etale fundamental group of $X$ such…

Let S be a principally embedded sl_2 subalgebra in sl_n for n > 2. A special case of results of the third author and Gregg Zuckerman implies that there exists a positive integer b(n) such that for any finite-dimensional irreducible sl_n…

Let G be a finite subgroup of SL(2,C). Let S_N#G^N be the wreath product of G by the symmetric group of degree N, acting symplectically on a complex vector space V of dimension 2N, with symplectic basis {x_i, y_i} i=1,...,N. In this paper…

The representation theory for categorical groups is constructed. Each categorical group determines a monoidal bicategory of representations. Typically, these categories contain representations which are indecomposable but not irreducible. A…

We study the complex irreducible representations of special linear, symplectic, orthogonal and unitary groups over principal ideal local rings of length two. We construct a canonical correspondence between the irreducible representations of…

We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman-Wallach representation of $\mathrm{GL}_n$ that quantifies the extent to which this representation may be ramified. We also determine a distinguished…

Let $k$ be a finite field of characteristic $p$. In the 1960s, Kondo attached non-abelian Gauss sums to irreducible $\mathbb{C}$-representations of $\mathrm{GL}_n(k)$, and computed them in terms of Green parameters. On the other hand, the…

We give a method to produce representations of the braid group $B_n$ of $n-1$ generators ($n\leq \infty$). Moreover, we give sufficient conditions over a non unitary representation for being of this type. This method produces examples of…

Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F. Let C an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible…

Explicit expressions for the generators of the quantum superalgebra $U_q[gl(n/m)]$ acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a…

Let $(\mathcal{G},\nu)$ be a $t$-discrete ergodic groupoid. Consider a finite Von Neumann algebra $\mathcal{M}$ with separable predual. We prove that every uniformly bounded measurable representation $\rho:\mathcal{G} \rightarrow…

We classify irreducible unitary representations of the group of all infinite matrices over a $p$-adic field ($p\ne 2$) with integer elements equipped with a natural topology. Any irreducible representation passes through a group $GL$ of…

We introduce a class of finite dimensional nonlinear superalgebras $L = L_{\bar{0}} + L_{\bar{1}}$ providing gradings of $L_{\bar{0}} = gl(n) \simeq sl(n) + gl(1)$. Odd generators close by anticommutation on polynomials (of degree $>1$) in…

Let $F$ be a non-Archimedean local field. This paper studies homological properties of irreducible smooth representations restricted from $\mathrm{GL}_{n+1}(F)$ to $\mathrm{GL}_n(F)$. A main result shows that each Bernstein component of an…