Small $K$-タイプに付随したRiemann対称空間上のベクトル束における球変換 (表現論とその周辺分野の広がり)

Abstract

For a connected semisimple real Lie group G of non-compact type, Wallach introduced a class of K-types called small. We classify all small K-types for all simple Lie groups and prove cxcept just one case that each elementary spherical function for each small K-type ($pi$, V) can be expressed as a product of hyperbolic cosines and a Heckman-Opdam hypergeometric function. As an application, the inversion formula for the spherical transform on Gtimes KV is obtained from Opdam's theory on hypergeometric Fourier transforms.