Let G be a real reductive group, and P any parabolic subgroup. In the paper 'Canonical extensions of Harish-Chandra modules to representations of $G$' by Casselman, a result says that if we begin with a smooth representation of moderate growth of P, then the smooth induced representation of G induced from P is also of moderate growth.(Proposition 4.1 in that paper)

Casselman didn't give details of the proof. Does anybody know any proof of this result in other places?


1 Answer 1


This follows from the definitions and the remark he has written. I don't think anywhere else the proof would be any different.

  • $\begingroup$ Suppose f is in the induced space, X in the Lie algebra of G, g any element of G and k in K(a max compact subgroup), write down (R_XR_gf)(k),here $R$ denotes the right regular action, then I have f(kexp(tX)g) in the derivative, and I don't know how to use his remark to rewrite kg as p'k'. Maybe I'm missing or confused by something... $\endgroup$
    – user1832
    Jan 28, 2010 at 21:40
  • $\begingroup$ Any element can be written as p'k', you don't actually have to do it. $\endgroup$
    – MBN
    Feb 1, 2010 at 20:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.