Schwartz space of parabolic basic affine space and asymptotic Hecke algebras

2018/04/01 by Braverman, Alexander, Kazhdan, David · 1 citation
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1804.00336

Abstract

Let F be a local non-archimedian field and G be the group of F-points of a split connected reductive group over F. In a previous aricle we defined an algebra \mathcal J(G) of functions on G which contains the Hecke algebra \mathcal H(G) and is contained in the Harish-Chandra Schwartz algebra \mathcal C(G). We consider \mathcal J(G) as an algebraic analog the algebra \mathcal C(G). Given a parabolic subgroup P of G with a Levi subgroup M and the unipotent radical U_P we write X_P:=G/U_P. In this paper we study two versions of the Schwartz space of X_P. The first is \mathcal S(X_P):=\mathcal J(\mathcal S _c(X_P)) and the 2nd is the space spanned by functions of the form Φ_Q,P(ϕ) where Q is another parabolic with the same Levi subgroup, ϕ∈ \mathcal S_c(X_Q) and Φ_Q,P is a normalized intertwining operator from L²(X_Q) to L²(X_P). We formulate a series of conjectures about these spaces, for example, we conjecture that \mathcal S'(X_P)⊂ \mathcal S(X_P) and that this embedding is an isomorphism on M-cuspidal part. We give a proof of some of our conjectures.

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• Asymptotics of Schwartz functions
• Remarks on the asymptotic Hecke algebra
• Normalized intertwining operators and nilpotent elements in the Langlands dual group
• On the Schwartz space of the basic affine space
• Localization of the Parabolic Hecke Algebra at a Strictly Positive Element