Description of unitary representations of the group of infinite $p$-adic integer matrices

In this paper we give a simple (local) proof of two principal results about irreducible tempered representations of general linear groups over a non-archimedean local division algebra. We give a proof of the parameterization of the irreducible square integrable representations of these groups by segments of cuspidal representations, and a proof of the irreducibility of the tempered parabolic induction. Our proofs are based on Jacquet modules (and the Geometric Lemma, incorpor...

We study a problem concerning parabolic induction in certain $p$-adic unitary groups. More precisely, for $E/F$ a quadratic extension of $p$-adic fields the associated unitary group $G=\mathrm{U}(n,n+1)$ contains a parabolic subgroup $P$ with Levi component $L$ isomorphic to $\mathrm{GL}_n(E) \times \mathrm{U}_1(E)$. Let $\pi$ be an irreducible supercuspidal representation of $L$ of depth zero. We use Hecke algebra methods to determine when the parabolically induced represent...

It is shown that the finite dimensional irreducible representaions of the quantum matrix algebra $ M_{ q,p}(2) $ ( the coordinate ring of $ GL_{q,p}(2) $) exist only when both q and p are roots of unity. In this case th e space of states has either the topology of a torus or a cylinder which may be thought of as generalizations of cyclic representations.

In the following article, we give a description of the distingushed irreducible principal series representations of the general linear group over a p-adic field in terms of inducing datum. This provides a counter-example to a conjecture of Jacquet about distinction (Conjecture 1 in U.K Anandavardhanan, "Distinguished non-Archimedean representations ", Proc. Hyderabad Conference on Algebra and Number Theory, 2005, 183-192).

Let $\F$ be a non-Archimedean locally compact field, $q$ be the cardinality of its residue field, and $\R$ be an algebraically closed field of characteristic $\ell$ not dividing $q$.We classify all irredu\-cible smooth $\R$-representations of $\GL\_n(\F)$ having a nonzero $\GL\_{n-1}(\F)$-inva\-riant linear form, when $q$ is not congruent to $1$ mod $\ell$.Partial results in the case when $q$ is $1$ mod $\ell$ show that, unlike the complex case, the space of $\GL\_{n-1}(\F)$-...

In this paper we give an explicit description of the universal unitary completion of certain locally Q_p-analytic representations of GL_2(F), where F is a finite extension of Q_p (this generalizes some results of Berger-Breuil for F=Q_p). To this aim, we make use of certain Banach spaces of C^r functions on O_F (for r a positive real number) introduced by the author.

The groups mentioned in the title are certain matrix groups of infinite size over a finite field $\mathbb F_q$. They are built from finite classical groups and at the same time they are similar to reductive $p$-adic Lie groups. In the present paper, we initiate the study of invariant measures for the coadjoint action of these infinite-dimensional groups. We examine first the group $\mathbb{GLB}$, a topological completion of the inductive limit group $\varinjlim GL(n, \mathb...

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 example, Diarra showed that the abelian group $G=\dZ$ has an enormous wealth of infinite dimensional, topologically irreducible Banach space representations. We therefore address the problem of finding an additional ''finiteness'' conditi...

I will survey some results in the theory of modular representations of a reductive $p$-adic group, in positive characteristic $\ell \neq p$ and $\ell=p$.

Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$. Let ${\mathcal U}_n(A)$ be the subgroup of $\mathrm{GL}_n(A)$ of all upper triangular matrices with 1's along the main diagonal. Let $P=H_n(A)\rtimes {\mathcal U}_n(A)$, where ${\mathcal U}_n(A)$ acts on $H_n(A)$ by $*$-congruence transformat...