Related papers: Kirillov's conjecture and $\CaD$-modules

The Hilbert-Smith Conjecture states that if G is a locally compact group which acts effectively on a connected manifold as a topological transformation group, then G is a Lie group. A rather straightforward proof of this conjecture is…

In the representation theory of split reductive algebraic groups, it is well known that every Weyl module with minuscule highest weight is irreducible over every field. Also, the adjoint representation of $E_8$ is also irreducible over…

Let $1 \to N \to G \to G/N \to 1$ be a short exact sequence of countable discrete groups and let $B$ be any $G$-$C^*$-algebra. In this paper, we show that the strong Novikov conjecture with coefficients in $B$ holds for such a group $G$…

We prove a version of Blattner's conjecture, for irreducible subquotients of principal series representations with integral infinitesimal character of a real reductive Lie group whose Beilinson-Bernstein D-module is supported on a K-orbit…

The fermionic formula conjectured by Kirillov and Reshetikhin describes the decomposition (as a module for $U_q(\frak g)$) of a tensor product of multiples of of fundamental representations $W(m\lambda_i)$ of the corresponding quantum…

Let $V$ be a $GL_n(\mathbb{R})$-distinguished, irreducible, admissible representation of $GL_n(\mathbb{C})$. We prove that any continuous linear functional on $V$, which is invariant under the action of the real mirabolic subgroup, is…

We prove the Kirillov-Reshetikhin (KR) conjecture in the general case : for all twisted quantum affine algebras we prove that the characters of KR modules solve the twisted Q-system and we get explicit formulas for the character of their…

We establish a general CCR (liminarity) property for uniformly bounded irreducible representations of nilpotent Lie groups on reflexive Banach spaces, extending the well known property of unitary irreducible representations of these groups…

We consider two algebraic invariants in the representation theory of quantized enveloping algebras: the dimension growth of simple modules for the De Concini-Kac quantum group at roots of unity, and the Gelfand-Kirillov dimension of simple…

Given a positive integer d, the Kaplansky-Lvov conjecture states that the set of values of a multilinear noncommutative polynomial f on the matrix algebra M_d(C) is a vector subspace. In this article the technique of using one-wiggle…

The main theorem of Galois theory states that there are no finite group-subgroup pairs with the same invariants. On the other hand, if we consider complex linear reductive groups instead of finite groups, the analogous statement is no…

Let $p$ be a prime number and $K$ a finite extension of $\mathbb{Q}_p$. We state conjectures on the smooth representations of $\mathrm{GL}_n(K)$ that occur in spaces of mod $p$ automorphic forms (for compact unitary groups). In particular,…

Let F be a non-Archimedean locally compact field of residue characteristic p, let D be a finite dimensional central division F-algebra and let R be an algebraically closed field of characteristic different from p. We classify all smooth…

We describe our conjecture about the irreducible unitary representations of reductive Lie groups, in the special case of $\mathrm{SL}(2,\mathbb{R})$.

In the local, characteristic 0, non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies that an…

A permutation group is {\it binary} if its orbits on $k$-tuples, for any integer $k\geq 2$, can be deduced from its orbits on $2$-tuples. Cherlin conjectured that a finite primitive binary permutation group $G$ must lie in one of three…

This article discusses the modular representation theory of finite groups of Lie type from the viewpoint of Broue's abelian defect group conjecture. We discuss both the defining characteristic case, the inspiration for Alperin's weight…

We prove one direction of a recently posed conjecture by Gan-Gross-Prasad, which predicts the branching laws that govern restriction from p-adic $GL_n$ to $GL_{n-1}$ of irreducible smooth representations within the Arthur-type class. We…

A group is irreducibly represented if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gasch\"utz…

Consider a standard representation $\pi_{st}$ of a quasi-split reductive p-adic group G. The generalized injectivity conjecture, posed by Casselman and Shahidi, asserts that any generic irreducible subquotient $\pi$ of $\pi_{st}$ is…