We prove a strong induction theorem and classify the tempered and square integrable representations of graded Hecke algebras.

and Applied Analysis 3 Let L2 Λ be a square integrable function space with inner product and norm as follows:

We consider the classic problem of minimizing a quadratic cost functional for well-posed linear systems over the class of inputs that are square integrable and that produce a square integrable output. As is well-known, the minimum cost can be expressed in terms of a bounded nonnegative selfadjoint operator X that in the finite-dimensional case satisfies a Riccati equation. Unfortunately, the in...

The goal of this paper is to generalize Demailly’s asymptotic holomorphic Morse inequalities to the case of a covering manifold of a compact manifold. We shall obtain estimates which involve Atiyah’s “normalized dimension” of the square integrable harmonic spaces. The techniques used are those of Shubin who gave a proof for the usual Morse inequalities in the presence of a group action relying ...

Let G denote an infinite-dimensional Heisenberg-like group, which is a class of infinite-dimensional step 2 stratified Lie groups. We consider holomorphic functions on G that are square integrable with respect to a heat kernel measure which is formally subelliptic, in the sense that all appropriate finite dimensional projections are smooth measures. We prove a unitary equivalence between a subc...

Together with Cogdell, Piatetski-Shapiro and Shahidi, we proved earlier the existence of a weak functorial lift of a generic cuspidal representation of SO2n+1 to GL2n. Recently, Ginzburg, Rallis and Soudry obtained a more precise form of the lift using their integral representation technique, namely, the lift is an isobaric sum of cuspidal representations of GLni (more precisely, cuspidal repre...

We consider the long time limit theorems for the solutions of a discrete wave equation with a weak stochastic forcing. The multiplicative noise conserves the energy, and in the unpinned case also conserves the momentum. We obtain a time-inhomogeneous OrnsteinUhlenbeck equation for the limit wave function that holds both for square integrable and statistically homogeneous initial data. The limit...

Abstract. The Calogero model is a one-dimensional quantum integrable system with inverse-square long-range interactions confined in an external harmonic well. It shares the same algebraic structure with the Sutherland model, which is also a one-dimensional quantum integrable system with inverse-sine-square interactions. Inspired by the Rodrigues formula for the Jack polynomials, which form the ...

Let G be a reductive group over a non-Archimedean local field. Then the canonical functor from the derived category of smooth tempered representations of G to the derived category of all smooth representations of G is fully faithful. Here we consider representations on bornological vector spaces. As a consequence, if G is semi-simple, V and W are tempered irreducible representations of G, and V...

The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this note we consider such a problem for the hyperbolic KleinGordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An infinite family of square integrable solutions for the Klein-Gordon equation on...