ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

7. Lie Groups and Lie Algebras

1991 MS Classification: 22, 33, 42, 46 Aoki, Shigeru, Takushoku University, Japan Invariant Eigendistributions on U(1, 2)/(U(1, 1) × U(1, 1)) Let X = G/H be a reductive summetric space, X′ the set of all regular semisimple elements in X. An H-invariant Schwartz distribution on X is said to be an invariant eigendistribution (=IED) or a spherical distribution, if it is an eigendistribution of every invariant differential operator. An IED is necessarily real analytic on X′. In the case of X = R = (R×R)/R, IED’s are exactly exp(xz)’s, so IED’s should be fundamental constituents in the harmonic analysis on a reductive summetric space. Our present aim is to determine IED’s on X = U(2, 2)/(U(1, 1)×U(1, 1)) as explicitly as possible. We will first establish matching conditions for an IED, between adjacent Cartan subspaces. Then, on X′, we will express IED’s explicitly in terms of solutions of a hypergeometric differential equation. (Joint work with Suehiro Kato)