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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?

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

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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... –  user1832 Jan 28 '10 at 21:40
    
Any element can be written as p'k', you don't actually have to do it. –  MBN Feb 1 '10 at 20:14
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