Remote Access Representation Theory
Green Open Access

Representation Theory

ISSN 1088-4165

 
 

 

Combinatorics and spherical functions
on the Heisenberg group


Authors: Chal Benson and Gail Ratcliff
Journal: Represent. Theory 2 (1998), 79-105
MSC (1991): Primary 22E30, 43A55
DOI: https://doi.org/10.1090/S1088-4165-98-00040-5
Published electronically: April 1, 1998
MathSciNet review: 1613063
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $V$ be a finite dimensional Hermitian vector space and $K$ be a compact Lie subgroup of $U(V)$ for which the representation of $K$ on $\hbox{${\mathbb C}$}[V]$ is multiplicity free. One obtains a canonical basis $\{p_\alpha\}$ for the space $\hbox{${\mathbb C}$}[\hbox{$V_{\mathbb R}$}]^K$ of $K$-invariant polynomials on $\hbox{$V_{\mathbb R}$}$ and also a basis $\{q_\alpha\}$ via orthogonalization of the $p_\alpha$'s. The polynomial $p_\alpha$ yields the homogeneous component of highest degree in $q_\alpha$. The coefficients that express the $q_\alpha$'s in terms of the $p_\beta$'s are the generalized binomial coefficients of Z. Yan. We present some new combinatorial identities that involve these coefficients. These have applications to analysis on Heisenberg groups. Indeed, the polynomials $q_\alpha$ completely determine the generic bounded spherical functions for a Gelfand pair obtained from the action of $K$ on a Heisenberg group $H=V\times \hbox{${\mathbb R}$}$.


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

  • [BJR] C. Benson, J. Jenkins, and G. Ratcliff, The spherical transform of a Schwartz function on the Heisenberg group. (to appear in J. Functional Analysis).
  • [BJR92] C. Benson, J. Jenkins, and G. Ratcliff, Bounded $K$-spherical functions on Heisenberg groups. J. Functional Analysis 105, 409-443, 1992. MR 93e:22017
  • [BJR93] C. Benson, J. Jenkins, and G. Ratcliff, $O(n)$-spherical functions on Heisenberg groups. Contemporary Math. 145, 181-197, 1993. MR 94h:22002
  • [BJRW96] C. Benson, J. Jenkins, G. Ratcliff, and T. Worku, Spectra for Gelfand pairs associated with the Heisenberg group. Colloq. Math. 71, 305-328, 1996. MR 98b:22016
  • [BR96] C. Benson and G. Ratcliff, A classification for multiplicity free actions. Journal of Algebra 181, 152-186, 1996. MR 97c:14046
  • [Car87] G. Carcano, A commutativity condition for algebras of invariant functions. Boll. Un. Mat. Italiano 7, 1091-1105, 1987. MR 89h:22011
  • [Dib90] H. Dib, Fonctions de Bessel sur une algèbre de Jordan. J. Math Pures et Appl. 69, 403-448, 1990. MR 92e:33017
  • [FK94] J. Faraut and A. Koranyi, Analysis on Symmetric Cones. Oxford University Press, New York, 1994. CMP 97:12
  • [How95] R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond. Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995. MR 96e:13006
  • [HU91] R. Howe and T. Umeda, The Capelli identity, the double commutant theorem and multiplicity-free actions. Math. Annalen 290, 565-619, 1991. MR 92j:17004
  • [Kac80] V. Kac, Some remarks on nilpotent orbits. Journal of Algebra 64, 190-213, 1980. MR 81i:17005
  • [KS96] F. Knop and S. Sahi, Difference equations and symmetric polynomials defined by their zeroes. International Math. Research Notes 10, 473-486, 1996. CMP 96:15
  • [Lea] A. Leahy, Ph.D. Thesis, Rutgers University.
  • [Mac95] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Second Edition. Clarendon Press, Oxford, 1995. MR 96h:05207
  • [Ols] G. Olshanski, Quasi-symmetric functions and factorial Schur functions. (preprint).
  • [OO97] A. Okounkov and G. Olshanski, Shifted Jack polynomials, binomial formula, and applications. Math. Res. Letters 4, 69-78, 1997. CMP 97:08
  • [Sah94] S. Sahi, The spectrum of certain invariant differential operators associated to Hermitian symmetric spaces. Lie Theory and Geometry, vol. 123, Progress in Math.
    (J. L. Brylinski, editor), Birkhäuser, Boston, pp. 569-576, 1994. MR 96d:43013
  • [Yan] Z. Yan, Special functions associated with multiplicity-free representations. (preprint).
  • [Yan92] Z. Yan, Generalized hypergeometric functions and Laguerre polynomials in two variables. Contemporary Math. 138, 239-259, 1992. MR 94j:33019

Similar Articles

Retrieve articles in Representation Theory of the American Mathematical Society with MSC (1991): 22E30, 43A55

Retrieve articles in all journals with MSC (1991): 22E30, 43A55


Additional Information

Chal Benson
Affiliation: Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, Missouri 63121
Email: benson@arch.umsl.edu

Gail Ratcliff
Affiliation: Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, Missouri 63121
Email: ratcliff@arch.umsl.edu

DOI: https://doi.org/10.1090/S1088-4165-98-00040-5
Received by editor(s): October 22, 1997
Received by editor(s) in revised form: February 17, 1998
Published electronically: April 1, 1998
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society