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I'm currently reading Jacquet's paper 'smooth transfer of Kloosterman integrals', in which Jacquet proved there is a smooth transfer for compactly supported smooth functions on GL(n,F) to smooth functions on Hermitian matrices for E/F, where F is a p-adic field and E/F is a quadratic extension.

The method is linearization, which is suggested by J.Waldspurger, that is, to prove the existence of such a transfer for smooth functions on $M(n\times n,F)$, the Lie algebra of GL(n,F). It seems that this method is a useful tool (maybe most powerful?) for similar problems.

But I'm not quite understand why smooth transfer on Lie algebra level implies that on group level, and this is not explained in Jacquet's paper. Maybe it is not too difficult, but I couldn't figure it out. I appreciate it a lot if someone could explain it in some details or suggest some references on it.

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