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Harish-Chandra’s Plancherel theorem for \(\mathfrak{p}\)-adic groups. (English) Zbl 0869.22007
Trans. Am. Math. Soc. 348, No. 11, 4673-4686 (1996); correction ibid. 352, No. 4, 1947-1949 (2000).
In Harish-Chandra’s study of the Plancherel formula for reductive \({\mathfrak p}\)-adic groups a crucial point is the finite-dimensionality of the space of \(Z\)-finite and Hecke-finite cusp forms in the Schwartz space [see: Harish-Chandra, Collected papers, Vol. IV (New York etc., 1984; Zbl 0546.01013); 353-370]. The proof of this fact depends on the finiteness of a certain integral, stated as Theorem 11 and not proved in loc. cit. In the present paper the author proves a special case of Theorem 11, which suffices to prove the Plancherel theorem, and he derives Theorem 11 from the Plancherel theorem.

22E35 Analysis on \(p\)-adic Lie groups
22E50 Representations of Lie and linear algebraic groups over local fields
Zbl 0546.01013
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