An application of the Littlewood restriction formula to the Kostant-Rallis Theorem

Author:
Jeb F. Willenbring

Journal:
Trans. Amer. Math. Soc. **354** (2002), 4393-4419

MSC (2000):
Primary 22E47, 20G05, 05E05

Published electronically:
June 24, 2002

MathSciNet review:
1926881

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider a symmetric pair of linear algebraic groups with , where and are defined as the +1 and -1 eigenspaces of the involution defining . We view the ring of polynomial functions on as a representation of . Moreover, set , where is the space of homogeneous polynomial functions on of degree . This decomposition provides a graded -module structure on . A decomposition of is provided for some classical families when is within a certain stable range.

The stable range is defined so that the spaces are within the hypothesis of the classical Littlewood restriction formula. The Littlewood restriction formula provides a branching rule from the general linear group to the standard embedding of the symplectic or orthogonal subgroup. Inside the stable range the decomposition of is interpreted as a -analog of the Kostant-Rallis theorem.

**1.**T. Enright and J. Willenbring,*Hilbert series, Howe duality and branching rules*, Preprint.**2.**Roe Goodman and Nolan R. Wallach,*Representations and invariants of the classical groups*, Encyclopedia of Mathematics and its Applications, vol. 68, Cambridge University Press, Cambridge, 1998. MR**1606831****3.**Wim H. Hesselink,*Characters of the nullcone*, Math. Ann.**252**(1980), no. 3, 179–182. MR**593631**, 10.1007/BF01420081**4.**B. Kostant and S. Rallis,*Orbits and representations associated with symmetric spaces*, Amer. J. Math.**93**(1971), 753–809. MR**0311837****5.**Bertram Kostant,*Lie group representations on polynomial rings*, Amer. J. Math.**85**(1963), 327–404. MR**0158024****6.**Dudley E. Littlewood,*The Theory of Group Characters and Matrix Representations of Groups*, Oxford University Press, New York, 1940. MR**0002127****7.**D. E. Littlewood,*On invariant theory under restricted groups*, Philos. Trans. Roy. Soc. London. Ser. A.**239**(1944), 387–417. MR**0012299****8.**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****9.**N. R. Wallach and J. Willenbring,*On some 𝑞-analogs of a theorem of Kostant-Rallis*, Canad. J. Math.**52**(2000), no. 2, 438–448. MR**1755786**, 10.4153/CJM-2000-020-0**10.**J. Willenbring,*Stability properties for -multiplicities and branching formulas for representations of the classical groups*, Ph.D. thesis, University of California at San Diego, 2000.

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Additional Information

**Jeb F. Willenbring**

Affiliation:
Department of Mathematics, Yale University, 10 Hillhouse Avenue, PO Box 208283, New Haven, Connecticut 06520

Email:
jeb.willenbring@math.yale.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03065-9

Keywords:
Kostant-Rallis theorem,
Littlewood restriction formula,
skew Schur polynomial

Received by editor(s):
October 22, 2001

Published electronically:
June 24, 2002

Additional Notes:
This research was funded by the Yale Gibbs Instructorship as well as the NSF VIGRE postdoctoral fellowship.

Article copyright:
© Copyright 2002
American Mathematical Society