A Realization of Semisimple Symmetric Spaces and Construction of Boundary Value Maps

Abstract

Contents § 0. Introduction § 1. Construction of a compact imbedding § 2. Invariant differential operators § 3. Boundary value maps § 4. Principal series § 0. Introduction

The Most Continuous Part of the Plancherel Decomposition for a Reductive Symmetric Space

“…In particular, in the above-mentioned case of split rank one, where the discrete series is complementary to L 2(G/H), one can now determine the full decomposition of L2(G/H) by combination of these results with the present work. In [35], top of p. 604, T. Oshima announced the determination of the Plancherel formula for G/H. In [35], top of p. 604, T. Oshima announced the determination of the Plancherel formula for G/H.…”

Some results on Semisimple Symmetric Spaces and Invariant Differential Operators

“…Moreover, by a similar way with the proof of Lemma 2.1 in [9], we have: Denote ( / ) for the algebra of invariant differential operators on / and ( / 0 ) for the algebra of invariant differential operators on / 0 . Then, we see that ( / 0 ) is naturally isomorphic to the algebra ( ) /( ( ) ∩ ( ) ), and it follows from Lemma 4.1 that ( / ) is also isomorphic to this algebra.…”

“…In this section, we shall show that the system of invariant differential operators on = / extends analytically on the compact -space ̂. First, we recall after [9] on the structure of the algebra of invariant differential operators on / .…”

Lie Theory

“…There is also a more analytic proof for Casselman's embedding theorem without using n-homologies: This proof consists of three steps: (i) to compactify G ( [79]), (ii) to realize π ∞ in C ∞ (G) via matrix coefficients, (iii) to take the boundary values into principal series representations (see [80], [104] and references therein).…”