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Joseph Bernstein

Publications and source records attributed to Joseph Bernstein.

At least 19 recordsLinked to original sources
arXiv2026

A new model for the quantum mechanics of the Hydrogen atom

To every Lorentzian quadratic space $(V,q)$ of even dimension $n$ such that $n\geq 4$ we attach a canonical algebraic quantum mechanical model for a corresponding generalized hydrogen atom system. In our model the configuration space is the regular null cone $C$ of the quadratic space. The Hilbert space $H$ is a canonical $L^2$ space on the cone $C$, and observables are realized in the algebra $D(C)$ of algebraic differential operators on $C$. We also construct a distinguished Schwartz space $S(H)\subset H$, which carries a self-adjoint action of $D(C)$ and encodes the boundary conditions of the standard theory. The role of the Schr\"odinger operator is played by a one-parameter Schr\"odinger family of operators in $D(C)$. We explain how the model relates to the realization of the minimal representation of $O(n,2)$ on $H$. For $n=4$, which corresponds to the physical hydrogen atom system, we prove that the spectrum of the Schr\"odinger family on the upper-half component $S(H)_+$ coincides with the usual spectrum of the hydrogen atom and that the corresponding solution spaces recover the standard physical solutions. The spectrum of the Schr\"odinger family on the lower-half component $S(H)_-$ gives additional positive-energy solution spaces not present in the usual formulation.

math-ph↗
arXiv2025

On norms on Harish-Chandra modules

The Casselman-Wallach theorem is a foundational result in the theory of representations of real reductive groups connecting algebraic representations to topological representations. We provide a quantitative version of this theorem. For that we introduce the notion of {\it Sobolev gap} for a Harish-Chandra module. This is a new invariant whose finiteness is highly non-trivial. We determine the Sobolev gap for representations in the unitary dual of the group $\SL(2,\R)$ and establish uniform finiteness results in general for representations of the discrete series and the minimal principal series. We use these notions to reformulate and extend classical results of Bernstein and Reznikov concerning automorphic functionals with respect to cocompact lattices. In particular, we prove an abstract convexity bound which applies to automorphic functionals with respect to general lattices in $\SL(2,\R)$ and is independent of the type of unitarizable irreducible Harish-Chandra module. Finally, we offer an extensive list of open problems.

math.RT↗
arXiv2024

On the classification of hypergeometric families of orthogonal polynomials on the real line

Several important families of orthogonal polynomials on the real line are called ``hypergeometric'' since they can be explicitly described in terms of some hypergeometric series $_pF_q$ that uses the degree $n$ of the polynomial as a parameter. It is natural to ask if one can classify all such families. Indeed many classification results have been obtained in this direction, but only under the additional assumption that the polynomials are eigenfunctions of some second order operator. In this paper we initiate a new approach to this classification. We propose a definition of an HG family that makes precise, but also generalizes, the notion of a ``hypergeometric'' family. Our main result is that there are exactly 10 types of orthogonal HG families, 8 from the well-known Askey scheme and 2 additional types of families that can be expressed in terms of Lommel polynomials. Our methods in this paper are algebraic. In particular, we classify a wider class of quasi-orthogonal HG families, and this classification is valid over an arbitrary field of characteristic zero. We also define a more general class of rational HG families and prove a structure theorem for quasi-orthogonal families in this class. We provide examples for such families, that are in particular new families of orthogonal polynomials of potential interest.

math.CA↗
arXiv2022

$P$-strict promotion and $Q$-partition rowmotion: the graded case

Promotion and rowmotion are intriguing actions in dynamical algebraic combinatorics which have inspired much work in recent years. In this paper, we study $P$-strict labelings of a finite, graded poset $P$ of rank $n$ and labels at most $q$, which generalize semistandard Young tableaux with $n$ rows and entries at most $q$, under promotion. These $P$-strict labelings are in equivariant bijection with $Q$-partitions under rowmotion, where $Q$ equals the product of $P$ and a chain of $q-n-1$ elements. We study the case where $P$ equals the product of chains in detail, yielding new homomesy and order results in the realm of tableaux and beyond. Furthermore, we apply the bijection to the cases in which $P$ is a minuscule poset and when $P$ is the three element $V$ poset. Finally, we give resonance results for promotion on $P$-strict labelings and rowmotion on $Q$-partitions.

math.CO↗
arXiv2022

Strong density of spherical characters attached to unipotent subgroups

We prove the following result in relative representation theory of a reductive p-adic group $G$: Let $U$ be the unipotent radical of a minimal parabolic subgroup of $G$, and let $\psi$ be an arbitrary smooth character of $U$. Let $S \subset Irr(G)$ be a Zariski dense collection of irreducible representations of $G$. Then the span of the Bessel distributions $B_{\pi}$ attached to representations $\pi$ from $S$ is dense in the space $\mathcal S^*(G)^{U\times U,\psi \times \psi}$ of all $(U\times U,\psi \times \psi)$-equivariant distributions on $G.$ We base our proof on the following results: 1. The category of smooth representations $\mathcal M(G)$ is Cohen-Macaulay. 2. The module $ind_U^G(\psi)$ is a projective module.

math.RT↗
arXiv2020

$P$-strict promotion and $B$-bounded rowmotion, with applications to tableaux of many flavors

We define P-strict labelings for a finite poset P as a generalization of semistandard Young tableaux and show that promotion on these objects is in equivariant bijection with a toggle action on B-bounded Q-partitions of an associated poset Q. In many nice cases, this toggle action is conjugate to rowmotion. We apply this result to flagged tableaux, Gelfand-Tsetlin patterns, and symplectic tableaux, obtaining new cyclic sieving and homomesy conjectures. We also show P-strict promotion can be equivalently defined using Bender-Knuth and jeu de taquin perspectives.

math.CO↗
arXiv2020

Hidden sign in Langlands' correspondence

In this note I describe some modification of the Langlands correspondence and explain why it is more natural. I will also discuss its significance to the theory of automorphic L-functions.

math.RT↗
arXiv2019

On the meromorphic continuation of Eisenstein series

Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a "soft" proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg's later proofs.

math.NT↗
arXiv2017

Algebraic Families of Harish-Chandra Pairs

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, separate parts of which have been studied individually as contractions. We give a complete classification of generically irreducible families of Harish-Chandra modules in the case of the family associated to SL(2, R).

math.RT↗
arXiv2017

Contractions of Representations and Algebraic Families of Harish-Chandra Modules

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 a parameter subspace that includes both discrete and continuous parts. Both finite- and infinite-dimensional representations can occur, even within the same family. We shall study the simplest nontrivial examples, and use the concepts of algebraic families of Harish-Chandra pairs and Harish-Chandra modules, introduced in a previous paper, together with the Jantzen filtration, to construct these families of unitary representations algebraically.

math.RT↗
arXiv2017

Deligne-Lusztig duality and wonderful compactification

We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne-Lusztig (or Alvis-Curtis) duality for $p$-adic groups and the homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for $G=GL(n)$ by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct we obtain a description of the Serre functor for representations of a p-adic group.

math.RT↗
arXiv2016

Stacks in Representation Theory. What is a continuous representation of an algebraic group ?

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 groups is ideologically inconsistent. I propose a way to modify this approach using sheaves on stacks. In new version I corrected some misprints and added an explanation how stack's approach is related to Vogan's picture of representations.

math.RT↗
arXiv2013

Periods and global invariants of automorphic representations

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 $L$-functions. This leads to an introduction of a global invariant of an automorphic representation arising from two such periods. We compute this invariant in some cases.

math.NT↗
arXiv2010

Subconvexity bounds for triple L-functions and representation theory

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 analytic structure of the corresponding unique trilinear functional on unitary representations of PGL(2,R).

math.NT↗
arXiv2005

Periods, subconvexity of L-functions and representation theory

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 the corresponding unique trilinear functional on unitary representations of PGL(2,R).

math.RT↗
arXiv2003

Estimates of automorphic functions

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 method which gives an estimate on the average of the automorphic factor and thus prove a convexity bound for the triple period.

math.RT↗