A proof of Casselman's comparison theorem for standard minimal parabolic subalgebra

Publication date: 5 February 2021

Abstract: Let

G

be a real linear reductive group and

K

be a maximal compact subgroup. Let

P

be a minimal parabolic subgroup of

G

with complexified Lie algebra

m a t h f r a k p

, and

m a t h f r a k n

be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fr'echet representation

V

of

G

, the inclusion

V_{K} s u b s e t V

induces isomorphisms

H_{i} (m a t h f r a k n, V_{K}) c o n g H_{i} (m a t h f r a k n, V)

(

i g e q 0

), where

V_{K}

denotes the

(m a t h f r a k g, K)

module of

K

finite vectors in

V

. This is called Casselman's comparison theorem. As a consequence, we show that: for any

k g e q 1

,

m a t h f r a k n^{k} V

is a closed subspace of

V

and the inclusion

V_{K} s u b s e t V

induces an isomorphism

V_{K} / m a t h f r a k n^{k} V_{K} = V / m a t h f r a k n^{k} V

. This strengthens Casselman's automatic continuity theorem.

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