On invariant eigendistributions on \(U(p,q)/(U(r)\times{} U(p-r,q))\) (Q1184880)

Let \(X=G/H\) be a semisimple symmetric space, \(D(X)\) be the ring of invariant differential operators on \(X\), and \(\chi\) a character of \(D(X)\). Take \(X=U(p,q)/(U(r) \times U(p-r,q))\). The authors' aim is to determine invariant eigendistributions (IED) on \(X\) as explicitly as possible. For this end, they study the following problem, to which a corresponding problem for semisimple Lie groups was investigated in detail by Hirai: Problem. Find a necessary and sufficient condition for an IED on \(X'\) to be extendible to an IED on \(X\). In this article, they give a necessary condition in the case where the infinitesimal character is regular. It will be shown that their condition is also sufficient, when the infinitesimal character \(\chi\) is ``generic''. They conjecture that this will hold even in the case where \(\chi\) is not generic. Results in the case of singular (i.e. nonregular) infinitesimal character will appear in a forthcoming paper.