Schwartz space of parabolic basic affine space and asymptotic Hecke algebras

TL;DR

This paper explores Schwartz spaces related to parabolic affine spaces over non-archimedean fields, proposing conjectures about their structure and establishing partial proofs, thus advancing understanding of harmonic analysis on reductive groups.

Contribution

It introduces two versions of Schwartz spaces on parabolic affine spaces and formulates conjectures about their properties, providing partial proofs for some conjectures.

Findings

Defined algebraic analogs of Schwartz spaces on $X_P$

Formulated conjectures on the structure and embeddings of these spaces

Proved some conjectures related to the $M$-cuspidal parts

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 $\Phi_{Q,P}(\phi)$ where $Q$ is another parabolic with the same Levi subgroup, $\phi\in \mathcal S_c(X_Q)$ and $\Phi_{Q,P}$ 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 $\mathcal S'(X_P)\subset \mathcal S(X_P)$ and that this embedding is an isomorphism on $M$-cuspidal part. We give a proof of some of our conjectures.