An application of the Littlewood restriction formula to the Kostant-Rallis Theorem
HTML articles powered by AMS MathViewer
- by Jeb F. Willenbring PDF
- Trans. Amer. Math. Soc. 354 (2002), 4393-4419 Request permission
Abstract:
Consider a symmetric pair $(G,K)$ of linear algebraic groups with $\mathfrak {g} \cong \mathfrak {k} \oplus \mathfrak {p}$, where $\mathfrak {k}$ and $\mathfrak {p}$ are defined as the +1 and -1 eigenspaces of the involution defining $K$. We view the ring of polynomial functions on $\mathfrak {p}$ as a representation of $K$. Moreover, set $\mathcal {P}(\mathfrak {p}) = \bigoplus _{d=0}^\infty \mathcal {P}^d(\mathfrak {p})$, where $\mathcal {P}^d(\mathfrak {p})$ is the space of homogeneous polynomial functions on $\mathfrak {p}$ of degree $d$. This decomposition provides a graded $K$-module structure on $\mathcal {P}(\mathfrak {p})$. A decomposition of $\mathcal {P}^d(\mathfrak {p})$ is provided for some classical families $(G,K)$ when $d$ is within a certain stable range. The stable range is defined so that the spaces $\mathcal {P}^d(\mathfrak {p})$ 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 $\mathcal {P}^d(\mathfrak {p})$ is interpreted as a $q$–analog of the Kostant-Rallis theorem.References
- T. Enright and J. Willenbring, Hilbert series, Howe duality and branching rules, Preprint.
- 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
- Wim H. Hesselink, Characters of the nullcone, Math. Ann. 252 (1980), no. 3, 179–182. MR 593631, DOI 10.1007/BF01420081
- B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809. MR 311837, DOI 10.2307/2373470
- Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 158024, DOI 10.2307/2373130
- Leonard Eugene Dickson, New First Course in the Theory of Equations, John Wiley & Sons, Inc., New York, 1939. MR 0000002
- P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
- 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
- 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, DOI 10.4153/CJM-2000-020-0
- J. Willenbring, Stability properties for $q$-multiplicities and branching formulas for representations of the classical groups, Ph.D. thesis, University of California at San Diego, 2000.
Additional Information
- Jeb F. Willenbring
- Affiliation: Department of Mathematics, Yale University, 10 Hillhouse Avenue, PO Box 208283, New Haven, Connecticut 06520
- MR Author ID: 662347
- Email: jeb.willenbring@math.yale.edu
- 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.
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 4393-4419
- MSC (2000): Primary 22E47, 20G05, 05E05
- DOI: https://doi.org/10.1090/S0002-9947-02-03065-9
- MathSciNet review: 1926881