Schwartz space of parabolic basic affine space and asymptotic Hecke algebras

摘要

Let F be a local non-archimedean field and G be the group of F-points of a split connected reductive group over F. In arXiv:1704.03019 we define an algebra J(G) of functions on G which contains the Hecke algebra H(G) and is contained in the Harish-Chandra Schwartz algebra C(G). We consider J(G) as an algebraic analog the algebra C(G). Given a parabolic subgroup P of G with a Levi subgroup M and the unipotent radical U_P we write Xp := G/Up. Let S_c(X_P) be the space of locally constant functions on Xp with compact support and S_(cusp,c)(X_P) ? S_c(X_P) be subspace of functions whose right shifts span a cuspidal representation of M. In this paper we study two versions of the Schwartz space of X_P. The first is S(X_P): = J(S_c(Xp)) 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 ∈ S_c(X_Q) and Φ_(Q,P) is a normalized intertwining operator from L~2(X_Q) to L~2(X_P). We formulate a series of conjectures about these spaces; for example, we conjecture that S'(Xp) C S(Xp) and that this embedding is an isomorphism on the M-cuspidal part. We give a proof of some of our conjectures (cf. Theorem 1.9).