Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A generating function for Blattner's formula

Authors: Jeb F. Willenbring and Gregg J. Zuckerman
Journal: Proc. Amer. Math. Soc. 136 (2008), 2261-2270
MSC (2000): Primary 22E46; Secondary 17B10
Published electronically: January 28, 2008
MathSciNet review: 2383533
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a connected, semisimple Lie group with finite center and let $ K$ be a maximal compact subgroup. We investigate a method to compute multiplicities of $ K$-types in the discrete series using a rational expression for a generating function obtained from Blattner's formula. This expression involves a product with a character of an irreducible finite-dimensional representation of $ K$ and is valid for any discrete series system. Other results include a new proof of a symmetry of Blattner's formula, and a positivity result for certain low rank examples. We consider in detail the situation for $ G$ of type split $ \rm G_2$. The motivation for this work came from an attempt to understand pictures coming from Blattner's formula, some of which we include in the paper.

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

  • [1] Thomas J. Enright, On the fundamental series of a real semisimple Lie algebra: Their irreducibility, resolutions and multiplicity formulae, Ann. of Math. (2) 110 (1979), no. 1, 1-82. MR 541329 (81a:17003)
  • [2] Thomas J. Enright and V. S. Varadarajan, On an infinitesimal characterization of the discrete series, Ann. of Math. (2) 102 (1975), no. 1, 1-15. MR 0476921 (57:16472)
  • [3] Thomas J. Enright and Nolan R. Wallach, The fundamental series of representations of a real semisimple Lie algebra, Acta Math. 140 (1978), no. 1-2, 1-32. MR 0476814 (57:16368)
  • [4] R. Goodman and N.R. Wallach, Representations and invariants of the classical groups, Cambridge University Press, Cambridge, 1998. MR 1606831 (99b:20073)
  • [5] B. Gross and N. Wallach, Restriction of small discrete series representations to symmetric subgroups, The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998) , Proc. Sympos. Pure Math., vol. 68, Amer. Math. Soc., Providence, RI, 2000, pp. 255-272. MR 1767899 (2001f:22042)
  • [6] Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1-111. MR 0219666 (36:2745)
  • [7] Henryk Hecht and Wilfried Schmid, A proof of Blattner's conjecture, Invent. Math. 31 (1975), no. 2, 129-154. MR 0396855 (53:715)
  • [8] Anthony W. Knapp, Lie groups beyond an introduction, Second edition, Progress in Mathematics, vol. 140, Birkhäuser Boston Inc., Boston, MA, 2002. MR 1920389 (2003c:22001)
  • [9] Ivan Penkov and Gregg Zuckerman, Generalized Harish-Chandra modules with generic minimal $ \germ k$-type, Asian J. Math. 8 (2004), no. 4, 795-811. MR 2127949 (2005k:17007)
  • [10] Wilfried Schmid, $ L\sp{2}$-cohomology and the discrete series, Ann. of Math. (2) 103 (1976), no. 2, 375-394. MR 0396856 (53:716)
  • [11] David A. Vogan Jr., Representations of real reductive Lie groups, Progress in Mathematics, vol. 15, Birkhäuser, Boston, MA, 1981. MR 632407 (83c:22022)
  • [12] Nolan R. Wallach, Real reductive groups. I, Pure and Applied Mathematics, vol. 132, Academic Press Inc., Boston, MA, 1988. MR 929683 (89i:22029)
  • [13] Nolan R. Wallach, Real reductive groups. II, Pure and Applied Mathematics, vol. 132, Academic Press Inc., Boston, MA, 1992. MR 1170566 (93m:22018)
  • [14] Gregg J. Zuckerman, Coherent translation of characters of semisimple Lie groups, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Acad. Sci. Fennica, Helsinki, 1980, pp. 721-724. MR 562678 (81f:22025)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 22E46, 17B10

Retrieve articles in all journals with MSC (2000): 22E46, 17B10

Additional Information

Jeb F. Willenbring
Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 0413, Milwaukee, Wisconsin 53201-0413

Gregg J. Zuckerman
Affiliation: Department of Mathematics, Yale University, P.O. Box 208283; New Haven, Connecticut 06520-8283

Keywords: Blattner's formula, coherent continuation, discrete series
Received by editor(s): April 26, 2007
Published electronically: January 28, 2008
Additional Notes: The first author was supported in part by NSA Grant # H98230-05-1-0078.
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society