Avraham Aizenbud

Schwartz Functions on Nash Manifolds

The goal of this paper we extend the notions of Schwartz functions, tempered functions, and generalized Schwartz functions to Nash (i.e. smooth semi-algebraic) manifolds. We reprove for this case the classically known properties of Schwartz functions on and build some additional tools that are important in representation theory.

Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem

In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field F of characteristic zero. Our main tool is the Luna Slice Theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pairs (GL(n+k,F), GL(n,F) x GL(k,F)) and (GL(n,E), GL(n,F)) are Gelfand pairs for any local field F and its quadratic extension E. In the non-Archimedean case, the first result was proven earlier by Jacquet and Rallis and the second by Flicker. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-Archimedean F the latter is a classical theorem of Gelfand and Kazhdan.

Multiplicity one theorem for \(({\rm GL}_{n+1}({\mathbb{R}}), {\rm GL} _ {n} ({ \mathbb{R}}))\)

Let F be either R or C. Consider the standard embedding GLn(F ) ↪→ GLn+1(F ) and the action of GLn(F ) on GLn+1(F ) by conjugation. In this paper we show that any GLn(F )-invariant distribution on GLn+1(F ) is invariant with respect to transposition. We show that this implies that for any irreducible admissible smooth Frechet representations π of GLn+1(F ) and τ of GLn(F ), dim HomGLn(F )(π, τ) ≤ 1. For p-adic fields those results were proven in [AGRS]. Mathematics Subject Classification (2000). 20G05, 22E45, 20C99, 46F10.Abstract.Let F be either

Representation growth and rational singularities of the moduli space of local systems

{\mathbb{R}}

Smooth transfer of Kloosterman integrals (the Archimedean case)

or

Derivatives for smooth representations of GL(n, ℝ) and GL(n, ℂ)

{\mathbb{C}}