The Plancherel decomposition for a reductive symmetric space. II. Representation theory

Abstract

We obtain the Plancherel decomposition for a reductive symmetric space in the sense of representation theory. Our starting point is the Plancherel formula for spherical Schwartz functions, obtained in part I. The formula for Schwartz functions involves Eisenstein integrals obtained by a residual calculus. In the present paper we identify these integrals as matrix coefficients of the generalized principal series.

The Plancherel decomposition for a reductive symmetric space. I. Spherical functions

“…The resulting proof of the Plancherel theorem in the present paper and [15] is independent of the one in [24]; moreover, it follows a completely different approach. Finally, we mention that T. Oshima has announced a Plancherel formula in [39], p. 604, but the details have not appeared.…”

“…The results of this paper and [15] were found and announced in the fall of 1995 when both authors were visitors of the Mittag-Leffler Institute in Djursholm, Sweden. At the same time P. Delorme announced his proof of the Plancherel theorem.…”

“…On the other hand, we also announced the proof of a Paley-Wiener theorem for reductive symmetric spaces, generalizing Arthur's theorem [1] for the case of the group. The proof of the Paley-Wiener theorem has now appeared in [16], which is independent of the present paper and [15]. The present paper as well as [15] and [16] rely on [12] and [14].…”

“…The proof of the Paley-Wiener theorem has now appeared in [16], which is independent of the present paper and [15]. The present paper as well as [15] and [16] rely on [12] and [14].…”

“…In this paper and its sequel [15] we determine the Plancherel decomposition for a reductive symmetric space X = G/H. Here G is a real reductive Lie group of Harish-Chandra's class and H is an open subgroup of the group G σ of fixed points for an involution σ of G. In the present paper we establish the Plancherel formula for K-finite (spherical) Schwartz functions on X, with K a σ-invariant maximal compact subgroup of G. In [15] we shall derive the Plancherel decomposition, in the sense of representation theory, from it.…”