Invariant spherical hyperfunctions on the tangent space of a symmetric space. (English) Zbl 0578.22011

Algebraic groups and related topics, Proc. Symp., Kyoto and Nagoya/Jap. 1983, Adv. Stud. Pure Math. 6, 83-126 (1985).
[For the entire collection see Zbl 0561.00006.]
Harish-Chandra [Bull. Am. Math. Soc. 69, 117–123 (1963; Zbl 0115.10801)] showed that every invariant eigendistribution on a real semisimple Lie algebra is a locally integrable function. This paper deals with the generalization of this result to invariant spherical hyperfunctions. Suppose \(\sigma\) is an involution on the real semisimple Lie algebra \(\mathfrak g\), and let \(\mathfrak g=\mathfrak h+\mathfrak q\) with \(\sigma =1,-1\) on \(\mathfrak h\), \(\mathfrak q\), respectively.
The author defines the concept of an invariant spherical hyperfunction on \(\mathfrak q\) and derives conditions on when such an object is determined by its action on \(\mathfrak q_{rs}\), the set of \(\mathfrak q\)-regular semisimple elements of \(\mathfrak q\). He points out that these conditions are satisfied for Riemannian symmetric spaces from already known results. A detailed structural analysis is then carried out to determine the pairs \((\mathfrak g,\mathfrak h)\) to which the theorem applies.


22E20 General properties and structure of other Lie groups
58J15 Relations of PDEs on manifolds with hyperfunctions
43A85 Harmonic analysis on homogeneous spaces
53C35 Differential geometry of symmetric spaces