TL;DR
This paper provides an overview of matrix beta-integrals, including their historical development, connections to symmetric spaces, interpolation methods, adelic analogs, and applications to flag spaces.
Contribution
It synthesizes existing knowledge on matrix beta-integrals, highlighting their extensions, interpolations, and relations to various geometric and algebraic structures.
Findings
Historical origins of matrix beta-integrals from 1930s-50s
Development of multi-parametric series by Gindikin in the 1960s
Connections to symmetric spaces and flag spaces
Abstract
First examples of matrix beta-integrals were discovered on 1930-50s by Siegel and Hua, in 60s Gindikin obtained multi-parametric series of such integrals. We discuss beta-integrals related to 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.
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