Related papers: A generalization of Duflo's conjecture

We present some results about the irreducible representations appearing in the exterior algebra $\Lambda \mathfrak{g}$, where $ \mathfrak{g}$ is a simple Lie algebra over $\mathbb{C}$. For Lie algebras of type $B$, $C$ or $D$ we prove that…

In this mostly expository article, we consider certain homological aspects of branching laws for representations of a group restricted to its subgroups in the context of $p$-adic groups. We follow our earlier paper, ICM 2018 proceedings,…

All indecomposable finite-dimensional representations of the homogeneous Galilei group which when restricted to the rotation subgroup are decomposed to spin 0, 1/2 and 1 representations are constructed and classified. These representations…

For a central division algebra $D$ of dimension $d^2$ over a finite extension $F$ of $\mathbb Q_p$ or of $\mathbb F_p((t))$, a field $R$ of characteristic prime to $p$, and an irreducible smooth $R$-representation $\pi$ of $G=GL_n(D)$, we…

Let $\mathfrak g$ be a finite dimensional Lie algebra over a field $\mathbf k$, $U\mathfrak g$ be its enveloping algebra and $S\mathfrak g$ be the symmetric algebra on $\mathfrak g$. Extending the work of Braverman and Gaitsgory on the…

The two-parametric quantum superalgebra $U_{pq}[gl(2/2)]$ and its representations are considered. All finite-dimensional irreducible representations of this quantum superalgebra can be constructed and classified into typical and nontypical…

Let $\pi$ be a set of primes such that $|\pi|\geqslant 2$ and $\pi$ differs from the set of all primes. Denote by $r$ the smallest prime which does not belong to $\pi$ and set $m=r$ if $r=2,3$ and $m=r-1$ if $r\geqslant 5$. We study the…

We study the relationship between $n$-cluster tilting modules over $n$ representation finite algebras and the Euler forms. We show that the dimension vectors of cluster-indecomposable modules give the roots of the Euler form. Moreover, we…

In this paper, we consider the branching law of the Speh representation $\mathrm{Sp}(\pi,n+l)$ of $\mathrm{GL}_{2n+2l}$ with respect to the block diagonal subgroup $\mathrm{GL}_n\times\mathrm{GL}_{n+2l}$ for any irreducible generic…

We show that Caratheodory's conjecture, on umbilical points of closed convex surfaces, may be reformulated in terms of the existence of at least one umbilic in the graphs of functions f: R^2-->R whose gradient decays uniformly faster than…

Let $F$ be a non-archimedean local field. The classification of the irreducible representations of $GL_n(F)$, $n\ge0$ in terms of supercuspidal representations is one of the highlights of the Bernstein--Zelevinsky theory. We give an…

We provide some experimental results on the decomposition of the parabolic induction of $\pi\otimes\pi$ in the Grothendieck group where $\pi$ is an irreducible representation of $GL_n$ over a local non-archimedean field.

In the case of p-adic general linear groups, each irreducible representation is parabolically induced by a tensor product of irreducible representations supported by cuspidal lines. One gets in this way a parameterization of the irreducible…

Let $F$ be a non-archimedean local field of characteristic different from 2 and residual characteristic $p$. This paper concerns the $\ell$-modular representations of a connected reductive group $G$ distinguished by a Galois involution,…

Suppose $G$ is a real reductive group. The determination of the irreducible unitary representations of $G$ is one of the major unsolved problem in representation theory. There is evidence to suggest that every irreducible unitary…

The unitary dual of $GL(n, \mathbb{R})$ was classified by Vogan in the 1980s. Focusing on the irreducible unitary representations of $GL(n, \mathbb{R})$ with half-integral infinitesimal characters, we find that Speh representations and the…

The generalized Mordell-Lang conjecture (GML) is the statement that the irreducible components of the Zariski closure of a subset of a group of finite rank inside a semi-abelian variety are translates of closed algebraic subgroups. M.…

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…

We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the `Irreducibility theorem' and `Duflo theorem', to much wider classes of algebras. Our general version of Irreducibility theorem says…

In this paper, we prove the Generalized P\'{o}lya conjecture for the Dirichlet eigenvalues. In other words, we show that $\lambda_k(\alpha) \ge \frac{(2\pi)^{\alpha} k^{\alpha/n}}{\big(\omega_n \cdot {vol}(\Omega)\big)^{\alpha/n}}, \quad\,…