Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We obtain explicit Plancherel formulas for the parabolic subgroups $ P$ of $ p$-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 $ P$.

Then, we compute the Plancherel formula of $ N \cdot A$, where $ N$ is the nilradical of $ P$ and $ A = {Q'_p}$, the multiplicative group of nonzero $ p$-adic numbers, by formulating a $ p$-adic change of variable formula and using the strong derivative.


References [Enhancements On Off] (What's this?)

  • [1] J. Dixmier, Algébres quasi-unitaries, Comment. Math. Helv. 26 (1952), 275-322. MR 0052697 (14:660b)
  • [2] M. Duflo and C. C. Moore, On the regular representation of a non-unimodular locally compact group, J. Funct. Anal. 21 (1976), 209-243. MR 0393335 (52:14145)
  • [3] I. M. Gel' fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, Saunders, Philadelphia, Pa., 1969. MR 0233772 (38:2093)
  • [4] F. W. Keene, Square integrable representations and Plancherel theorem for parabolic groups, Trans. Amer. Math. Soc. 243 (1978), 61-73. MR 0498983 (58:16967)
  • [5] F. W. Keene, R. L. Lipsman, and J. A. Wolf, The Plancherel formula for parabolic subgroups, Israel J. Math. 28 (1977), 68-90. MR 0507242 (58:22400)
  • [6] A. A. Kirillov, Unitary representations of nilpotent Lie groups, English transl., Russian Math. Surveys 17 (1962), no. 4, 53-104. MR 0142001 (25:5396)
  • [7] A. Kleppner and R. L. Lipsman, The Plancherel formula for group extensions, Ann. Sci. Ecole Norm. Sup. 5 (1972), 459-516. MR 0342641 (49:7387)
  • [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, Benjamin, 1973. MR 0396410 (53:277)
  • [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] -, Canonical semi-invariants and the Plancherel formula for parabolic groups, Trans. Amer. Math. Soc. 269 (1982), 111-131. MR 637031 (83k:22026)
  • [12] C. C. Moore, Decomposition of unitary representations defined by discrete subgroups of nilpotent groups, Ann. of Math. 82 (1965), 146-182. MR 0181701 (31:5928)
  • [13] -, A Plancherel formula for non-unimodular groups, Address presented to the Internat. Conf. on Harmonic Analysis, Univ. of Maryland, 1971.
  • [14] -, Representations of solvable and nilpotent groups and harmonic analysis on nil and solomanifolds, Proc. Sympos. Pure Math., vol. 26, Amer. Math. Soc., Providence, R. I., 1973, pp. 3-44. MR 0385001 (52:5871)
  • [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, Department of Math. and Stat., Univ. of New Mexico, Tech. Rep. No. 344 (1978). MR 553959 (80m:43011)
  • [17] L. Pukanszky, On the theory of quasi-unitary algebras, Acta Sci. Math. 16 (1955), 103-121. MR 0073961 (17:515a)
  • [18] I. Satake, Classification of semi-simple algebraic groups, Dekker, 1971. MR 0316588 (47:5135)
  • [19] M. H. Taibleson, Fourier analysis on local fields, Math. Notes, Princeton University Press, 1975. MR 0487295 (58:6943)
  • [20] N. Tatsuuma, Plancharel formula for non-unimodular locally compact groups, J. Math. Kyoto Univ. 12 (1972), 179-261. MR 0299729 (45:8777)
  • [21] G. van Dijk, Smooth and admissible representations of $ p$-adic unipotent groups, Compositio Math. 37 (1978), 77-101. MR 0492085 (58:11239)
  • [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] -, Unitary representations of maximal parabolic subgroup of the classical group, Mem. Amer. Math. Soc., vol. 8, no. 180 (1976). MR 0444847 (56:3194)
  • [24] E. Weiss, Algebraic number theory, McGraw-Hill, 1963. MR 0159805 (28:3021)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E35, 22E50, 43A32

Retrieve articles in all journals with MSC: 22E35, 22E50, 43A32


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1990-1019522-6
Keywords: Parabolic subgroup, $ p$-adic number field, Plancherel formula, Dixmier-Pukanszky operator, strong derivative
Article copyright: © Copyright 1990 American Mathematical Society

American Mathematical Society