On the asymptotics of Harish-Chandra modules By Henryk Hecht*) at Salt Lake City and WilfriedSchmid**} at Cambridge, Massachusetts. 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 repre- sentations 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. To describe the Statement, we fix an Iwasawa decomposition G = KAN of the group in question; then g = i 0 á ö n, on the level of the complexified Lie algebras. We recall the passage from an admissible representation ð of G to its Harish-Chandra module—the infinitesimal representation of g on the space of ^-fmite vectors. The Harish-Chandra module reflects all important properties of ð, except those which depend on the choice of a topology [11]. To each Harish-Chandra module F, Casselman associates the " Jacquet module" (1) F [n] = lhnF/n fe F [5]. It has a natural g-module structure, and its o-fmite part belongs to a category &' of g-modules closely related to the category (S of Bernstein-Gelfand-Gelfand [1]. The coefficients in the asymptotic expansion of the matrix coefficients along the minimal parabolic subgroup P=MAN constitute a g-module F P , the "asymptotic module along P", which also belongs to the category G'. In his article [5], and more concretely in [4], Casselman conjectured a canonical isomorphism (2) V P £ á-finite part of F [n] . Our theorem (26) establishes the conjecture, not only for the minimal parabolic sub- group P, but for arbitrary parabolic subgroups s well. This settles the question of Harish-Chandra and Casselman: the á-finite part of F [n] determines all embeddings of F into principal series representations for essentially formal reasons [5], whereas Y P is an *) Sloan Fellow. **) Partially supported by NSF grant MCS 79-13190. Brought to you by | University of Iowa Libraries Authenticated Download Date | 6/4/15 12:56 AM