Distributions sphériques invariantes sur les espaces symétriques semi-simples \(G_ c/G\). (Invariant spherical distributions over the semi-simple symmetric spaces \(G_ c/G\))
The results of the paper under review were announced [in Lect. Notes Math. 1243, 283-309 (1985; Zbl 0627.43007)] and proved [in RIMS Kôkyûroku 598, 117-180 (1986; Zbl 0675.22006)]. Let \(G\) be a connected semi- simple Lie group, \(H\) the subgroup consisting of the fixed points of an involutive automorphism \(\sigma\) of \(G\). With this involutive automorphism one can associate a map \(\varphi: G\to G\); \(\varphi(g):=g\sigma(g)^{-1}\) and set \(X\) its image in \(G\). Thus \(G/H\) can be included in \(G\) as symmetric \(G\)-space. The problem is to study the invariant spherical distributions on \(X\). There is a natural Weyl formula of integration over \(X\) (Proposition 2.1). The restriction of an invariant spherical distribution on the subset \(X'\) of regular elements of \(X\) is always an analytic function. Conversely, given an analytic function over the subset \(X'\), the author finds in Theorem 11.1 a necessary and sufficient condition, with which this analytic function defines an invariant spherical distribution on \(X\).
scientific article; zbMATH DE number 27578
scientific article