A description of discrete series for semisimple symmetric spaces.

*(English)*Zbl 0577.22012
Group representations and systems of differential equations, Proc. Symp., Tokyo 1982, Adv. Stud. Pure Math. 4, 331-390 (1984).

[For the entire collection see Zbl 0566.00014.]

A fundamental problem in harmonic analysis on semisimple symmetric spaces is the determination of the discrete series. The paper under review gives a complete solution of this problem. The problem is difficult and the solution deep and long, using for example the theory of hyperfunction in an essential way, lengthy step by step reductions to special cases, and involved combinatorial arguments. Even a precise statement of the result takes more than a page, in addition to several pages of explanation and notation.

All of this is hardly surprising if one considers that Harish-Chandra’s solution of the discrete series problem for semisimple groups is a very special case. But it should be emphasized that the proof does not rely on an extension of Harish-Chandra’s methods in the group case but is fundamentally different. The proof builds rather on the basic work of M. Flensted-Jensen [Ann. Math., II. Ser. 111, 253-311 (1980; Zbl 0462.22006)]. The K-finite eigenfunctions which are analytic and \(L^ 2\) are realized by a Poisson transform using Flensted-Jensen’s duality principle. This gives as well the reduction of these eigenspaces into irreducible subspaces and, of course, criteria for their non-triviality, completing the description of the discrete series.

A fundamental problem in harmonic analysis on semisimple symmetric spaces is the determination of the discrete series. The paper under review gives a complete solution of this problem. The problem is difficult and the solution deep and long, using for example the theory of hyperfunction in an essential way, lengthy step by step reductions to special cases, and involved combinatorial arguments. Even a precise statement of the result takes more than a page, in addition to several pages of explanation and notation.

All of this is hardly surprising if one considers that Harish-Chandra’s solution of the discrete series problem for semisimple groups is a very special case. But it should be emphasized that the proof does not rely on an extension of Harish-Chandra’s methods in the group case but is fundamentally different. The proof builds rather on the basic work of M. Flensted-Jensen [Ann. Math., II. Ser. 111, 253-311 (1980; Zbl 0462.22006)]. The K-finite eigenfunctions which are analytic and \(L^ 2\) are realized by a Poisson transform using Flensted-Jensen’s duality principle. This gives as well the reduction of these eigenspaces into irreducible subspaces and, of course, criteria for their non-triviality, completing the description of the discrete series.

Reviewer: W.Rossmann