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On the asymptotics of Harish-Chandra modules.

In his study of irreducible, admissible representations of semisimple Lie groups, Harish-Chandra makes extensive use of the asymptotic expansions of matrix coefficients [12], [13]. In particular, he constructs certain embeddings into principal series representations from the asymptotic expansions [14]; these ideas were then extended and refined by Casselman [5]. Both Harish-Chandra and Casselman raised the question whether the asymptotic expansions determine all embeddings into principal series representations. Our main result is a Statement, conjectured by Casselman, that answers the question affirmatively.