Joseph Bernstein

Analytic continuation of representations and estimates of automorphic forms

0.1. Analytic vectors and their analytic continuation. Let G be a Lie group and (�,G,V ) a continuous representation of G in a topological vector space V. A vector v ∈ V is called analytic if the functionv : g 7→ �(g)v is a real analytic function on G with values in V. This means that there exists a neighborhood U of G in its complexification GC such thatv extends to a holomorphic function on U. In other words, for each element g ∈ U we can unambiguously define the vector �(g)v asv(g), i.e., we can extend the action of G to a somewhat larger set. In this paper we will show that the possibility of such an extension sometimes allows one to prove some highly nontrivial estimates. Unless otherwise stated, G = SL(2, R), so GC = SL(2, C). We consider a typical representation ofG, i.e., a representation of the principal series. Namely, fix� ∈ C and consider the space Dof smooth homogeneous functions of degree � − 1 on R 2 \ 0, i.e., D� = {� ∈ C ∞ (R 2 \ 0) : �(ax,ay) = |a| �−1 �(x,y)}; we

On the support of Plancherel measure

Abstract Let G be a real reductive group. As follows from Plancherel formula for G, proved by Harish-Chandra, only tempered representations of G contribute to the decomposition of the regular representation in L 2 (G). We give a simple direct proof of this result, based on Gelfand-Kostyuchenko method. We also prove similar results for representations, which appear in the decomposition of L 2 (X), where X is a homogeneous G-space of polynomial growth. (See precise definition in 3.5). Important examples of such space X are semisimple symmetric spaces and quotient of G by arithmetic subgroups.

Range characterization of the cosine transform on higher Grassmannians

Abstract 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.

A Generalization of Casselman’s Submodule Theorem

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].

Sobolev norms of automorphic functionals

It is well known that Frobenius reciprocity is one of the central tools in the representation theory. In this paper, we discuss Frobenius reciprocity in the theory of automorphic functions. This Frobenius reciprocity was discovered by Gel’fand, Fomin, and PiatetskiShapiro in the 1960s as the basis of their interpretation of the classical theory of automorphic functions in terms of the representation theory (eventually, of adelic groups, see [7, 8, 9]). Later, Ol’shanski gave a more transparent proof of it (see [14]). However, in the subsequent rapid development of the theory of automorphic functions, Frobenius reciprocity was barely noticeable. We believe that this is due to the incompleteness of the above-mentioned results. In this paper, we prove a general theorem (see Theorem 1.1), which we view as a quantitative version of Frobenius reciprocity. We then illustrate it by looking into the example of SL(2,R). We think that these methods will play a more prominent role in the theory of automorphic functions.

Complex crystallographic Coxeter groups and affine root systems

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

Sobolev norms of automorphic functionals and Fourier coefficients of cusp forms

Abstract 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.

Chevalley's theorem for the complex crystallographic groups

Abstract 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 status of the above statement for other types of complex crystallographic group W and certain generalizations of the statement.

Lectures on Lie Algebras

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 multiplicity formula.

On the quantum groupSLq(2)

We start with the observation that the quantum groupSLq(2), described in terms of the 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 develop this method in the language of quantized universal enveloping algebras, which is another common method of studying quantum groups. We carry out our method in detail for root systems of typeSL(2); as a byproduct, we find a new series of quantum groups-metaplectic groups ofSL(2)-type. Representations of these groups can provide interesting examples of bimodule categories over monoidal category of representations ofSLq(2).