Nagoya Mathematical Journal

Singular invariant hyperfunctions on the square matrix space and the alternating matrix space

Masakazu Muro

Full-text: Open access

Abstract

Fundamental calculations on singular invariant hyperfunctions on the $n \times n$ square matrix space and on the $2n \times 2n$ alternating matrix space are considered in this paper. By expanding the complex powers of the determinant function or the Pfaffian function into the Laurent series with respect to the complex parameter, we can construct singular invariant hyperfunctions as their Laurent expansion coefficients. The author presents here the exact orders of the poles of the complex powers and determines the exact supports of the Laurent expansion coefficients. By applying these results, we prove that every quasi-relatively invariant hyperfunction can be expressed as a linear combination of the Laurent expansion coefficients of the complex powers and that every singular quasi-relatively invariant hyperfunction is in fact relatively invariant on the generic points of its support. In the last section, we give the formula of the Fourier transforms of singular invariant tempered distributions.

Article information

Source
Nagoya Math. J., Volume 169 (2003), 19-75.

Dates
First available in Project Euclid: 27 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1114631809

Mathematical Reviews number (MathSciNet)
MR1962523

Zentralblatt MATH identifier
1068.11077

Subjects
Primary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}
Secondary: 11E39: Bilinear and Hermitian forms 11S90: Prehomogeneous vector spaces 32A45: Hyperfunctions [See also 46F15]

Citation

Muro, Masakazu. Singular invariant hyperfunctions on the square matrix space and the alternating matrix space. Nagoya Math. J. 169 (2003), 19--75. https://projecteuclid.org/euclid.nmj/1114631809


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References

  • I. N. Bernstein, The analytic continuation of generalized functions with respect to a parameter , Functional Anal. Appl., 6 (1972), 26--40.
  • I. M. Gelfand and G. E. Shilov, Generalized functions -- properties and operations, Generalized Functions, vol. 1, Academic Press, New York and London (1964).
  • A. Gyoja, Bernstein-Sato's polynomial for several analytic functions , J. Math. Kyoto Univ., 33 (1993, no. 2), 399--411.
  • --------, Local $b$-functions of prehomogeneous Lagrangians , J. Math. Kyoto Univ., 33 (1993, no. 2), 413--436.
  • J. Igusa, An introduction to the theory of local zeta functions, Studies in Advanced Mathematics, vol. 14, American Mathematical Society (2000).
  • M. Kashiwara, T. Kawai, and T. Kimura, Daisuukaisekigaku no Kiso (Foundations of Algebraic Analysis), Kinokuniya, Tokyo, 1980 (Japanese) ; The English translation was published by Princeton UP in 1985.
  • sho Kôkyûroku, 283 (1974), 60--147.
  • M. Muro, Microlocal analysis and calculations on some relatively invariant hyperfunctions related to zeta functions associated with the vector spaces of quadratic forms , Publ. Res. Inst. Math. Sci. Kyoto Univ., (1986, no. 3), 395--463.
  • --------, Singular invariant tempered distributions on regular prehomogeneous vector spaces , J. Funct. Anal., 76 (1988, no. 2), 317--345.
  • --------, Invariant hyperfunctions on regular prehomogeneous vector spaces of commutative parabolic type , Tôhoku Math. J. (2), 42 (1990, no. 2), 163--193.
  • --------, Singular invariant hyperfunctions on the space of real symmetric matrices , Tôhoku Math. J. (2), 51 (1999), 329--364.
  • --------, Singular invariant hyperfunctions on the space of complex and quaternion hermitian matrices , J. Math. Soc. Japan, 53 (2001, no. 3), 589--602.
  • --------, Invariant hyperfunction solutions to invariant differential equations on the space of real symmetric matrices , J. Funct. Anal., 193 (2002, no. 2), 346--384.
  • S. Rallis and G. Schiffmann, Distributions invariantes par le groupe orthogonal , Lecture Note in Math. (Springer), 497 (1975), 494--642.
  • H. Rubenthaler, Distributions bi-invariantes par $SL_n(k)$ , Lecture Note in Math. (Springer), 497 (1975), 383--493.
  • H. Saito, Explicit form of the zeta functions of prehomogeneous vector spaces , Math. Ann., 315 (1998), 587--615.
  • I. Satake, On zeta functions associated with self-dual homogeneous cone , Number theory and Related Topics (Bombay) (S. Raghavan, ed.), Tata Institute of Fundamental Research, Tata Institute of Fundamental Research and Oxford UP (1989), 177--193.
  • M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces , Ann. of Math. (2), 100 (1974), 131--170.
  • T. Shintani, On zeta functions associated with the vector spaces of quadratic forms , J. Fac. Sci. Univ. Tokyo Sect. IA Math., 22 (1975), 25--65.
  • H. Weyl, The classical groups, Princeton University Press (1946).