Periods and harmonic analysis on spherical varieties

Details Periods and harmonic analysis on spherical varieties Periods and harmonic analysis on spherical varieties

2012-02-29

arxiv.org/abs/1203.0039v1

Mathematics

Representation Theory

Scientific paper

This paper develops a "relative Langlands program" in a local context: Given a spherical variety X for a reductive group G over a non-archimedean local field, the Plancherel decomposition for L^2(X) should be related to "distinguished" Arthur parameters into a dual group closely related to that defined by Gaitsgory and Nadler. We discuss this and give evidence for it. In particular we give (under some assumptions on the spherical variety) a Plancherel formula for L^2(X), modulo knowledge of the discrete spectra of other spherical G-varieties which model the geometry of X at infinity. Along the way, we discuss the theory of asymptotics for subrepresentations of C^{infty}(X), and establish conjectures of Ichino--Ikeda and Lapid--Mao. The full discussion of the conjectures, and their global counterparts about spherical period integrals of automorphic forms, has not been included in the current preprint.

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