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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Combinatorics and spherical functions on the Heisenberg group
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by Chal Benson and Gail Ratcliff
Represent. Theory 2 (1998), 79-105
Published electronically: April 1, 1998


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 $\mathbb {C}[V]$ is multiplicity free. One obtains a canonical basis $\{p_\alpha \}$ for the space $\mathbb {C}[V_{\mathbb {R}}]^K$ of $K$-invariant polynomials on $V_{\mathbb {R}}$ 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 $H=V\times \mathbb {R}$.
  • C. Benson, J. Jenkins, and G. Ratcliff, The spherical transform of a Schwartz function on the Heisenberg group. (to appear in J. Functional Analysis).
  • Chal Benson, Joe Jenkins, and Gail Ratcliff, Bounded $K$-spherical functions on Heisenberg groups, J. Funct. Anal. 105 (1992), no. 2, 409–443. MR 1160083, DOI 10.1016/0022-1236(92)90083-U
  • Chal Benson, Joe Jenkins, and Gail Ratcliff, $\textrm {O}(n)$-spherical functions on Heisenberg groups, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 181–197. MR 1216189, DOI 10.1090/conm/145/1216189
  • Chal Benson, Joe Jenkins, Gail Ratcliff, and Tefera Worku, Spectra for Gelfand pairs associated with the Heisenberg group, Colloq. Math. 71 (1996), no. 2, 305–328. MR 1414831, DOI 10.4064/cm-71-2-305-328
  • Chal Benson and Gail Ratcliff, A classification of multiplicity free actions, J. Algebra 181 (1996), no. 1, 152–186. MR 1382030, DOI 10.1006/jabr.1996.0113
  • Giovanna Carcano, A commutativity condition for algebras of invariant functions, Boll. Un. Mat. Ital. B (7) 1 (1987), no. 4, 1091–1105 (English, with Italian summary). MR 923441
  • Hacen Dib, Fonctions de Bessel sur une algèbre de Jordan, J. Math. Pures Appl. (9) 69 (1990), no. 4, 403–448 (French). MR 1076516
  • J. Faraut and A. Koranyi, Analysis on Symmetric Cones. Oxford University Press, New York, 1994.
  • Roger Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, The Schur lectures (1992) (Tel Aviv), Israel Math. Conf. Proc., vol. 8, Bar-Ilan Univ., Ramat Gan, 1995, pp. 1–182. MR 1321638
  • Roger Howe and T\B{o}ru Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), no. 3, 565–619. MR 1116239, DOI 10.1007/BF01459261
  • V. G. Kac, Some remarks on nilpotent orbits, J. Algebra 64 (1980), no. 1, 190–213. MR 575790, DOI 10.1016/0021-8693(80)90141-6
  • F. Knop and S. Sahi, Difference equations and symmetric polynomials defined by their zeroes. International Math. Research Notes 10, 473–486, 1996.
  • A. Leahy, Ph.D. Thesis, Rutgers University.
  • I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
  • G. Olshanski, Quasi-symmetric functions and factorial Schur functions. (preprint).
  • A. Okounkov and G. Olshanski, Shifted Jack polynomials, binomial formula, and applications. Math. Res. Letters 4, 69–78, 1997.
  • Siddhartha Sahi, The spectrum of certain invariant differential operators associated to a Hermitian symmetric space, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 569–576. MR 1327549, DOI 10.1007/978-1-4612-0261-5_{2}1
  • Z. Yan, Special functions associated with multiplicity-free representations. (preprint).
  • Zhi Min Yan, Generalized hypergeometric functions and Laguerre polynomials in two variables, Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991) Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 239–259. MR 1199131, DOI 10.1090/conm/138/1199131
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Bibliographic Information
  • Chal Benson
  • Affiliation: Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, Missouri 63121
  • Email:
  • Gail Ratcliff
  • Affiliation: Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, Missouri 63121
  • Email:
  • Received by editor(s): October 22, 1997
  • Received by editor(s) in revised form: February 17, 1998
  • Published electronically: April 1, 1998
  • © Copyright 1998 American Mathematical Society
  • Journal: Represent. Theory 2 (1998), 79-105
  • MSC (1991): Primary 22E30, 43A55
  • DOI:
  • MathSciNet review: 1613063