Related papers: Automorphic $L$\snug-functions, intertwining opera…

Dunkl operators associated with finite reflection groups generate a commutative algebra of differential-difference operators. There exists a unique linear operator called intertwining operator which intertwines between this algebra and the…

In this expository paper we describe the study of certain non-self-adjoint operator algebras, the Hardy algebras, and their representation theory. We view these algebras as algebras of (operator valued) functions on their spaces of…

We give a microlocal description of the Aubert--Zelevinsky involution for all unipotent representations of all inner forms of simple adjoint unramified $p$-adic groups. Via the realization of enhanced $L$-parameters as perverse sheaves, we…

Let T be the Pascal-adic transformation. For any measurable function g, we consider the corrections to the ergodic theorem sum_{k=0}^{j-1} g(T^k x) - j/l sum_{k=0}^{l-1} g(T^k x). When seen as graphs of functions defined on {0,...,l-1}, we…

We discuss some basic problems and conjectures in a program to construct general orbifold conformal field theories using the representation theory of vertex operator algebras. We first review a program to construct conformal field theories.…

The authors study the spectral theory of self-adjoint operators that are subject to certain types of perturbations. An iterative introduction of infinitely many randomly coupled rank-one perturbations is one of our settings. Spectral…

We prove Lusztig's conjectures ${\bf P1}$-${\bf P15}$ for the affine Weyl group of type $\tilde{C}_2$ for all choices of positive weight function. Our approach to computing Lusztig's $\mathbf{a}$-function is based on the notion of a…

This paper studies Emerton's Jacquet module functor for locally analytic representations of p-adic reductive groups. When P is a parabolic subgroup whose Levi factor M is not commutative, we show that passing to an isotypical subspace for…

The article is devoted to the following question. Consider a periodic self-adjoint difference (differential) operator on a graph (quantum graph) G with a co-compact free action of the integer lattice Z^n. It is known that a local…

We consider periods of automorphic representations of adele groups defined by integrals along Gelfand subgroups. We define natural maps between local components of such periods and construct corresponding global maps using automorphic…

We present a method to calculate intertwining operators between the underlying Harish-Chandra modules of degenerate principal series representations of a semisimple Lie group $G$ and a semisimple subgroup $G'$, and between their composition…

In this article, we focus on how to embed a torus $\rT$ into a reductive group $\rG$ with respect to a given root datum $\Psi$ over a scheme $\rS$. This problem also relates to how to embed an \'etale algebra with involution into a central…

In the case of split $GSpin$ groups, we prove an equality of $L$-functions between automorphic local $L$-functions defined by the Langlands-Shahidi method and local Artin $L$-functions. Our method of proof is based on previous results of…

In the series of papers [FL,FL2] we approach quaternionic analysis from the point of view of representation theory of the conformal group SL(4,C) and its real forms. This approach has proven very fruitful and pushed further the parallel…

Let L be a finite field extension of Q_p and let G be the group of L-rational points of a split connected reductive group over L. We view G as a locally L-analytic group with Lie algebra g. We define a functor from admissible locally…

This is the third part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part III), we introduce and study…

Let $\tilde{W}$ be an extended affine Weyl group, $\mathbf{H}$ be the corresponding affine Hecke algebra over the ring $\mathbb{C}[\mathbf{q}^\frac{1}{2}, \mathbf{q}^{-\frac{1}{2}}]$, and $J$ be Lusztig's asymptotic Hecke algebra, viewed as…

This is a brief survey of recent results by the authors devoted to one of the most important operators of integral geometry. Basic facts about the analytic family of cosine transforms on the unit sphere and the corresponding Funk transform…

Fix an integral semisimple element $\lambda$ in the Lie algebra $\mathfrak{g}$ of a complex reductive algebraic group $G$. Let $L$ denote the centralizer of $\lambda$ in $G$ and let $\mathfrak{g}(-1)$ denote the $-1$ eigenspace of…

We study relationships between different formulations of the local principle. Also we establish a connection among the local principle and the non-commutative Fourier transform approach to the investigation of convolution operator algebras.…