A Plancherel formula for parabolic subgroups

Author:
Mie Nakata

Journal:
Trans. Amer. Math. Soc. **319** (1990), 243-256

MSC:
Primary 22E35; Secondary 22E50, 43A32

DOI:
https://doi.org/10.1090/S0002-9947-1990-1019522-6

MathSciNet review:
1019522

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Abstract | References | Similar Articles | Additional Information

Abstract: We obtain explicit Plancherel formulas for the parabolic subgroups of -adic unitary groups which fix one dimensional isotropic subspaces. By means of certain limits of difference operators (called strong derivatives), we construct a Dixmier-Pukanszky operator which compensates for the nonunimodularity of the group .

Then, we compute the Plancherel formula of , where is the nilradical of and , the multiplicative group of nonzero -adic numbers, by formulating a -adic change of variable formula and using the strong derivative.

**[1]**J. Dixmier,*Algèbres quasi-unitaires*, Comment. Math. Helv.**26**(1952), 275–322 (French). MR**0052697**, https://doi.org/10.1007/BF02564306**[2]**M. Duflo and Calvin C. Moore,*On the regular representation of a nonunimodular locally compact group*, J. Functional Analysis**21**(1976), no. 2, 209–243. MR**0393335****[3]**I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro,*Representation theory and automorphic functions*, Translated from the Russian by K. A. Hirsch, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. MR**0233772****[4]**Frederick W. Keene,*Square integrable representations and a Plancherel theorem for parabolic subgroups*, Trans. Amer. Math. Soc.**243**(1978), 61–73. MR**0498983**, https://doi.org/10.1090/S0002-9947-1978-0498983-X**[5]**Frederick W. Keene, Ronald L. Lipsman, and Joseph A. Wolf,*The Plancherel formula for parabolic subgroups*, Israel J. Math.**28**(1977), no. 1-2, 68–90. MR**0507242**, https://doi.org/10.1007/BF02759782**[6]**A. A. Kirillov,*Unitary representations of nilpotent Lie groups*, Uspehi Mat. Nauk**17**(1962), no. 4 (106), 57–110 (Russian). MR**0142001****[7]**Adam Kleppner and Ronald L. Lipsman,*The Plancherel formula for group extensions. I, II*, Ann. Sci. École Norm. Sup. (4)**5**(1972), 459–516; ibid. (4) 6 (1973), 103–132. MR**0342641****[8]**-,*The Plancherel formula for group extensions*. II, Ann. Sci. Ecole Norm. Sup.**6**(1973), 103-132.**[9]**T. Y. Lam,*The algebraic theory of quadratic forms*, W. A. Benjamin, Inc., Reading, Mass., 1973. Mathematics Lecture Note Series. MR**0396410****[10]**R. L. Lipsman and J. A. Wolf,*The Plancherel formula for parabolic subgroups of the classical groups*, Trans. Amer. Math. Soc.**260**(1980), 607-622.**[11]**Ronald L. Lipsman and Joseph A. Wolf,*Canonical semi-invariants and the Plancherel formula for parabolic groups*, Trans. Amer. Math. Soc.**269**(1982), no. 1, 111–131. MR**637031**, https://doi.org/10.1090/S0002-9947-1982-0637031-6**[12]**Calvin C. Moore,*Decomposition of unitary representations defined by discrete subgroups of nilpotent groups*, Ann. of Math. (2)**82**(1965), 146–182. MR**0181701**, https://doi.org/10.2307/1970567**[13]**-,*A Plancherel formula for non-unimodular groups*, Address presented to the Internat. Conf. on Harmonic Analysis, Univ. of Maryland, 1971.**[14]**Calvin C. Moore,*Representations of solvable and nilpotent groups and harmonic analysis on nil and solvmanifolds*, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Amer. Math. Soc., Providence, R.I., 1973, pp. 3–44. MR**0385001****[15]**M. Nakata,*Harmonic analysis on local fields*, Doctoral Dissertation, Univ. of California, Berkeley, 1983.**[16]**C. W. Onneweer,*On the definition of dyadic differentiation*, Applicable Anal.**9**(1979), no. 4, 267–278. MR**553959**, https://doi.org/10.1080/00036817908839275**[17]**L. Pukánszky,*On the theory of quasi-unitary algebras*, Acta Sci. Math. Szeged**16**(1955), 103–121. MR**0073961****[18]**I. Satake,*Classification theory of semi-simple algebraic groups*, Marcel Dekker, Inc., New York, 1971. With an appendix by M. Sugiura; Notes prepared by Doris Schattschneider; Lecture Notes in Pure and Applied Mathematics, 3. MR**0316588****[19]**M. H. Taibleson,*Fourier analysis on local fields*, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1975. MR**0487295****[20]**Nobuhiko Tatsuuma,*Plancherel formula for non-unimodular locally compact groups*, J. Math. Kyoto Univ.**12**(1972), 179–261. MR**0299729**, https://doi.org/10.1215/kjm/1250523567**[21]**G. van Dijk,*Smooth and admissible representations of 𝑝-adic unipotent groups*, Compositio Math.**37**(1978), no. 1, 77–101. MR**0492085****[22]**J. A. Wolf,*Fourier inversion problems on Lie groups and a class of pseudo-differential operators*, Conf. Partial Differential Equations and Geometry, Park City, Utah, 1977.**[23]**Joseph A. Wolf,*Unitary representations of maximal parabolic subgroups of the classical groups*, Mem. Amer. Math. Soc.**8**(1976), no. 180, iii+193. MR**0444847**, https://doi.org/10.1090/memo/0180**[24]**Edwin Weiss,*Algebraic number theory*, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. MR**0159805**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1990-1019522-6

Keywords:
Parabolic subgroup,
-adic number field,
Plancherel formula,
Dixmier-Pukanszky operator,
strong derivative

Article copyright:
© Copyright 1990
American Mathematical Society