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.

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.

Reviewer: S.Gelbart (Rehovot)

##### 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 |