Schwartz space of parabolic basic affine space and asymptotic Hecke algebras

Publication date: 27 November 2020

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

m a t h c a l J (G)

of functions on

G

which contains the Hecke algebra

m a t h c a l H (G)

and is contained in the Harish-Chandra Schwartz algebra

m a t h c a l C (G)

. We consider

m a t h c a l J (G)

as an algebraic analog the algebra

m a t h c a l 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

m a t h c a l S (X_{P}) : = m a t h c a l J ({m a t h c a l S}_{c} (X_{P}))

and the 2nd is the space spanned by functions of the form

P h i_{Q, P} (p h i)

where

Q

is another parabolic with the same Levi subgroup,

p h i i n m a t h c a l S_{c} (X_{Q})

and

P h i_{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

m a t h c a l S^{'} (X_{P}) s u b s e t m a t h c a l 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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