Paul Sally

Admissible Invariant Distributions on Reductive -adic Groups

Introduction Fourier transforms on the Lie algebra An extension and proof of Howes Theorem Theory on the group Bibliography List of symbols Index.

The fourier transform on semisimple lie groups of real rank one

SO~E RESULTS OF H~RIsH-CItA~DRA . . . . . . . . . . . . . . . . . . . . . . 3 The structure of (~ and G . . . . . . . . . . . . . . . . . . . . . . . . 3 The invariant integral on G . . . . . . . . . . . . . . . . . . . . . . . . 4 The characters of the discrete series . . . . . . . . . . . . . . . . . . . . 6 The characters of the principal series . . . . . . . . . . . . . . . . . . . 7 FOURIER T R A N S I ~ O R ~ OF A R E G U L A R O R B I T . . . . . . . . . . . . . . . . . 8

Intertwining operators and the representations of SL(2, R)

Abstract The irreducible unitary representations of SL (2, R ) were originally determined by Bargmann. The infinitesimal methods of Bargmann were later applied by Pukanszky to determine all the irreducible unitary representations of the universal covering group of SL (2, R ). In the present note, it is shown that a study of the intertwining operators relating two unitarily equivalent members of the principal series of irreducible unitary representations of the covering group leads naturally to global realizations of all the irreducible unitary representations of SL (2, R ), in particular the discrete series. The irreducibility and square integrability of the discrete series, realized in this form, are easily proved, and the characters of the discrete series are computed by using the generating function for Jacobi polynomials. All the irreducible, finite dimensional representations of SL (2, R ) also appear naturally in the above context. The present construction gives an interesting geometric explanation of the classical picture in which pairs of members of the discrete series occur at so-called integral points.