Joseph Bernstein

• PhD
• Professor (Full) at Tel Aviv University

We examine from an algebraic point of view some families of unitary group representations that arise in mathematical physics and are associated to contraction families of Lie groups. The contraction families of groups relate different real forms of a reductive group and are continuously parametrized, but the unitary representations are defined over...

We examine from an algebraic point of view some families of unitary group representations that arise in mathematical physics and are associated to contraction families of Lie groups. The contraction families of groups relate different real forms of a reductive group and are continuously parametrized, but the unitary representations are defined over...

Mathematical physicists have studied degenerations of Lie groups and their representations, which they call contractions. In this paper we study these contractions, and also other families, within the framework of algebraic families of Harish-Chandra modules. We construct a family that incorporates both a real reductive group and its compact form,...

Mathematical physicists have studied degenerations of Lie groups and their representations, which they call contractions. In this paper we study these contractions, and also other families, within the framework of algebraic families of Harish-Chandra modules. We construct a family that incorporates both a real reductive group and its compact form,...

In this note I introduce a new approach to (or rather a new language for) representation theory of groups. Namely, I propose to consider a (complex) representation of a group $G$ as a sheaf on some geometric object (a stack). This point of view necessarily leads to a conclusion that the standard approach to (continuous) representations of algebraic...

We are interested in invariant functionals defined on automorphic representations via period integrals. We consider the action of an adelic subgroup on such an invariant functional. We show that in certain cases this action gives rise to another period integral, and this corresponds to a known relation between an automorphic period and a special va...

This is a lecture course for beginners on representation theory of semisimple finite dimensional Lie algebras. It is shown how to use infinite dimensional representations (Verma modules) to derive the Weyl character formula. We also provide a proof for Harish-Chandra's theorem on the center of the universal enveloping algebra and for Kostant's mult...

An elementary and self-contained approach to smooth Frechet globalizations of Harish-Chandra modules is presented. We provide applications to the theory of Eisenstein series and Automatic Continuity.

We describe a new method to estimate the trilinear period on automorphic representations of PGL(2,R). Such a period gives rise to a special value of the triple L-function. We prove a bound for the triple period which amounts to a subconvexity bound for the corresponding special value of the triple L-function. Our method is based on the study of the...

We prove that, for the irreducible complex crystallographic Coxeter group W , the following conditions are equivalent: • a) W is generated by reflections; • b) the analytic variety X/W is isomorphic to a weighted projective space. The result is of interest, for example, in application to topological conformal field theory. We also discuss the st...

We classify (up to an isomorphism in the category of affine groups) the complex crystallographic groups generated by reflections and such that d, its linear part, is a Coxeter group, i.e., d is generated by "real" reflections of order 2.

In tikya[GK] we developed a framework to study representations of groups of the form G((t)), where G is an algebraic group over a local field K. The main feature of this theory is that natural representations of groups of this kind are not on vector spaces, but rather on pro-vector spaces. In this paper we present some further constructions relate...

Dedicated to Anthony Joseph, this volume contains surveys and invited articles by leading specialists in representation theory. The focus here is on semisimple Lie algebras and quantum groups, where the impact of Joseph's work has been seminal and has changed the face of the subject. Two introductory biographical overviews of Joseph's contribution...

We describe a new method to estimate the trilinear period on automorphic representations of PGL(2,R). Such a period gives rise to a special value of the triple L-function. We prove a bound for the triple period which amounts to a subconvexity bound for the corresponding special value. Our method is based on the study of the analytic structure of th...

We characterize the range of the cosine transform on real Grassmannians in terms of the decomposition under the action of the special orthogonal group SO(n). We also give a geometric interpretation of this image in terms of valuations. In addition, we discuss the non-Archimedean analogues.

We present a new method of estimating trilinear period for automorphic representations of SL(2,R). The method is based on the uniqueness principle in representation theory. We show how to separate the exponentially decaying factor in the triple period from the essential automorphic factor which behaves polynomially. We also describe a general metho...

We present a new method of estimating trilinear period for automorphic representations of SL(2,R). The method is based on the uniqueness principle in representation theory. We show how to separate the exponentially decaying factor in the triple period from the essential automorphic factor which behaves polynomially. We also describe a general metho...

We prove a quantitative version of Frobenius reciprocity in the theory of automorphic functions. Namely, given an automorphic representation we show how to estimate the norm of the corresponding Γ-invariant functional with respect to some norms on the representation space. The answer is given by relative traces.

Properties of analytic vectors in representations of SL(2,R) are used to give new bounds for the triple products recently considered by P. Sarnak. A conjecture of Sarnak about such products is proved. The results of this paper generalize results of A. Good and M. Jutila about special cases, but the techniques are entirely different. One consequence...

We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with the n-th tensor power of the fundamental (two-dimensional) sl(2)-module. The action of U(sl(2)) is given by projective functors and the commuting action of the Temperley-Lieb algebra by Zuckerman functors. Indecomposable project...

In an earlier paper [P1], we studied self-dual complex representations of a finite group of Lie type. In this paper, we make an analogous study in the p-adic case. We begin by recalling the main result of that paper. Let G(F) be the group of F rational points of a connected reductive algebraic group G over a finite field F. Fix a Borel subgroup B o...

We propose a new approach to the study of eigenfunctions of the Laplace-Beltrami operator on a Riemann surface of curvature 1. It is based on Frobenius reciprocity from the theory of automorphic functions. We determine Sobolev class of arising automorphic functionals and discuss some applications.RésumeNous proposons une nouvelle approche pour étud...

We show that every admissible representation of a real re-ductive group has a canonical system of Sobolev norms parametrized by positive characters of a minimal parabolic subgroup. These norms are compatible with morphisms of representations. Similar statement also holds for representations of reductive p-adic groups. Let us fix a real reductive al...

Let (g, K) be a Harish-Chandra pair. In this paper we prove that if P and P′ are two projective (g, K)-modules, then Hom(P, P′) is a Cohen-Macaulay module over the algebra Z(g, K) of K-invariant elements in the center of U(g). This fact implies that the category of (g, K)-modules is locally equivalent to the category of modules over a Cohen-Macaula...

We consider a special category of Hopf algebras, depending on parameters $\Sigma$ which possess properties similar to the category of representations of simple Lie group with highest weight $\lambda$. We connect quantum groups to minimal objects in this categories---they correspond to irreducible representations in the category of representations w...

In this paper we present a simple proof of the fundamental result by B. Kostant which claims that the universal enveloping algebra of a reductive Lie algebra $\germ{g}$ is free over its center. We also indicate how this result allows to simplify the proof of another important result of B. Kostant-the description of the algebra of functions on the n...

We start with the observation that the quantum group SL_q(2), described in terms of its algebra of functions has a quantum subgroup, which is just a usual Cartan group. Based on this observation we develop a general method of constructing quantum groups with similar property. We also describe this method in the language of quantized universal envel...

The series of three lectures given at Tel-Aviv University in 1992: 1. Tensor categories. 2. Quantum groups. 3. Topological (quantum) field theories. Published as the preprint IAS 897-92 of Tel-Aviv University and The Mortimer and Raymond Sacler Institute of Advanced Studies.

Let Gℝ be a real reductive Lie group, g;ℝ its Lie algebra. Let M be an irreducible Harish-Chandra module. Using some fine analytic arguments, based on the study of asymptotic behavior of matrix coefficients, Casselman has proved that M can be imbedded into a principal series representation [2,3].