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Invariant harmonic analysis on the Schwartz space of a reductive $$p$$-adic group. (English) Zbl 0760.22023
Harmonic analysis on reductive groups, Proc. Conf., Brunswick/ME (USA) 1989, Prog. Math. 101, 101-121 (1991).
[For the entire collection see Zbl 0742.00061.]
At the 1972 Williamstown Conference on Harmonic Analysis, R. Howe announced his fundamental conjecture concerning the orbital integrals of functions on a reductive $$p$$-adic group and on its Lie algebra. Although Howe proved the Lie algebra version himself, it was only recently that the author succeeded in proving “Howe’s conjecture” on the group [cf. Ann. Math., II. Ser. 129, 237-251 (1989; Zbl 0675.22007) and Duke Math. J. 61, 255-302 (1990; Zbl 0731.22011)].
In the present paper, the author gives proofs of several related important results, including the following: (1) temperedness of regular orbital integrals; (2) orthogonality relations for discrete series representations; and (3) existence of pseudo-coefficients for discrete series and limits of discrete series representations. Granted Howe’s conjectures, the author observes that most of the results surveyed in this paper were known to Harish-Chandra; moreover, alternate approaches to some (notably the orthogonality relations and existence of pseudo- coefficients for discrete series) were found by others, such as Bernstein, Deligne and Kazhdan.

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 22E35 Analysis on $$p$$-adic Lie groups 43A80 Analysis on other specific Lie groups