Matrix beta-integrals: An overview

Abstract

First examples of matrix beta-integrals were discovered in 1930-50s by Siegel and Hua and in the 60s Gindikin obtained multi-parametric series of such integrals. We discuss beta-integrals related to the symmetric spaces, their interpolation with respect to the dimension of a ground field, and adelic analogs; also we discuss beta-integrals related to flag spaces.