Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem

Abstract

Abstract. In the first part of the paper we generalize a descent technique due to Harish-Chandra to

On a new functional equation for local integrals

“…Recall that such L, if exists, is unique up to scalar. (This is clear in the split case; in the inert case see [Fli91,AG09]. )…”

Some regular symmetric pairs

“…In this paper we develop further the tools from [AG2] for proving regularity of symmetric pairs. We also introduce a systematic way to compute descendants of classical symmetric pairs.…”

“…In subsection 2.1 we discuss the notion of a Gelfand pair and review a classical technique for proving the Gelfand property due to Gelfand and Kazhdan. In subsection 2.2 we review the results of [AG2], introduce the notions of symmetric pair, descendants of a symmetric pair, good symmetric pair and regular symmetric pair mentioned above and discuss their relations to the Gelfand property.…”

“…[AG1]). It does not matter here which notion to choose since in the cases of consideration of this paper, if there are no nonzero equivariant Schwartz distributions, then there are no nonzero equivariant distributions at all (see Theorem 4.0.2 in [AG2]). …”

“…In [AG2] we explored the question what symmetric pairs are Gelfand pairs. We introduced the notion of regular symmetric pair and conjectured that all symmetric pairs are regular.…”