Schwartz space of parabolic basic affine space and asymptotic Hecke algebras

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 X_P:= G/U_P . Let S_c (X_P ) be the space of locally constant functions on X_P 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 (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 ∈ 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 S^′ (X_P ) ⊂ S(X_P ) 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).