On Harish-Chandra’s 𝜇-function for 𝑝-adic groups

Silberger, Allan J.

doi:10.1090/S0002-9947-1980-0570781-7

On Harish-Chandra’s $\mu$-function for $p$-adic groups
HTML articles powered by AMS MathViewer

by Allan J. Silberger

Trans. Amer. Math. Soc. 260 (1980), 113-121

DOI: https://doi.org/10.1090/S0002-9947-1980-0570781-7

PDF | Request permission

Abstract:

The Harish-Chandra $\mu$-function is, up to known constant factors, the Plancherel’s measure associated to an induced series of representations. In this paper we show that, when the series is induced from special representations lifted to a parabolic subgroup, the $\mu$-function is a quotient of translated $\mu$-functions associated to series induced from supercuspidal representations. It is now known, in both the real and p-adic cases, that the $\mu$-function is always an Euler factor.

References

Harish-Chandra, Harmonic analysis on reductive $p$-adic groups, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 167–192. MR 0340486
Harish-Chandra, Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1976), no. 1, 117–201. MR 439994, DOI 10.2307/1971058
A. W. Knapp and E. M. Stein, Singular integrals and the principal series. III, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 4622–4624. MR 367116, DOI 10.1073/pnas.71.11.4622
Allan J. Silberger, Introduction to harmonic analysis on reductive $p$-adic groups, Mathematical Notes, vol. 23, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1979. Based on lectures by Harish-Chandra at the Institute for Advanced Study, 1971–1973. MR 544991

Special representations of reductive p-adic groups are not integrable

Nolan R. Wallach, On Harish-Chandra’s generalized $C$-functions, Amer. J. Math. 97 (1975), 386–403. MR 399357, DOI 10.2307/2373718