Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-25T08:55:29.007Z Has data issue: false hasContentIssue false
  • English
  • Français

Correspondences of the Gelfand invariants in reductive dual pairs

Part of: Lie algebras and Lie superalgebras

Published online by Cambridge University Press:  09 April 2009

Minoru Itoh
Minoru Itoh
Affiliation:
Department of Mathematics Faculty of Science Kyoto UniversityKyoto 606-8502Japan e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For each complex reductive dual pair introduced by R. Howe, this paper presents a formula for the central elements of the universal enveloping algebras given by I. M. Gelfand. This formula provides an explicit description of the correspondence between the ‘centers’ of the two universal enveloping algebras.

Keywords

center of universal enveloping algebraGelfand invariantsdual pair

MSC classification

Secondary: 17B35: Universal enveloping (super)algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 75 , Issue 2 , October 2003 , pp. 263 - 278
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Capelli, A., ‘Über die Zurückführung der Cayley'schen Operation Ω auf gewöhnliche Polar-Operationen’, Math. Ann. 29 (1887), 331–338.CrossRefGoogle Scholar
[2]Capelli, A., ‘Sur les opérations dans la théorie des formes algébriques’, Math. Ann. 37 (1890), 1–37.CrossRefGoogle Scholar
[3]Gelfand, I. M., ‘Center of the infinitesimal groups’, Mat. Sbornik N. S. 26 (1950), 103–112;Google Scholar
English translation: Collected papers Vol. II, pp. 22–30.Google Scholar
[4]Howe, R., ‘θ-series and invariant theory’, in: Automorphic forms, representations, and L-functions. Part I (eds. Borel, A. and Casselman, W.), Proc. Sympos. Pure Math. XXXIII (Amer. Math. Soc., Providence, 1979) pp. 275–285.Google Scholar
[5]Howe, R., ‘Remarks on classical invariant theory’, Trans. Amer. Math. Soc. 313 (1989), 539–570;Google Scholar
Erratum in: Trans. Amer. Math. Soc. 318 (1990) 823.Google Scholar
[6]Howe, R., ‘Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond’, in: The Schur lectures (1992), Israel Math. Conf. Proc. 8 (Bar-Ilan Univ., Ramat Gan, 1995) pp. 1–182.Google Scholar
[7]Howe, R. and Umeda, T., ‘The Capelli identity, the double commutant theorem, and multiplicity-free actions’, Math. Ann. 290 (1991), 565–619.CrossRefGoogle Scholar
[8]Itoh, M., ‘Explicit Newton's formulas for gln’, J. Algebra 208 (1998), 687–697.CrossRefGoogle Scholar
[9]Kashiwara, M. and Vergne, M., ‘On the Segal-Shale-Weil representation and harmonic polynomial’, Invent. Math. 44 (1978), 1–47.CrossRefGoogle Scholar
[10]Klink, W. H. and Ton-That, T., ‘On resolving the multiplicity of arbitrary tensor products of the U(N) groups’, J. Phys. A 21 (1988), 3877–3892.CrossRefGoogle Scholar
[11]Leung, E. Y., ‘On resolving the multiplicity of tensor products of irreducible representations of symplectic groups’, J. Phys. A 26 (1993), 5851–5866.Google Scholar
[12]Leung, E. Y. and Ton-That, T., ‘Invariant theory of the dual pairs (SO*(2n), Sp(2k, C)) and (Sp(2n, k), O(N))’, Proc. Amer. Math. Soc. 120 (1994), 53–65.Google Scholar
[13]Louck, J. D. and Biedenharn, L. C., ‘Canonical unit adjoint tensor operators in U(n)’, J. Math. Phys. 11 (1970), 2368–2414.CrossRefGoogle Scholar
[14]Mæ, C., Vignéras, M.-F. and Waldspurger, J.-L., Correspondances de Howe sur un corps padique, Lecture Notes in Math. 1291 (Springer, Berlin, 1987).Google Scholar
[15]Molev, A., ‘Sklyanin determinant, Laplace operators, and characteristic identities for classical Lie algebras’, J. Math. Phys. 36 (1995), 923–943.Google Scholar
[16]Molev, A. and Nazarov, M., ‘Capelli identities for classical Lie algebras’, Math. Ann. 313 (1999), 315–357.CrossRefGoogle Scholar
[17]Nazarov, M., ‘Quantum Berezinian and the classical Capelli identity’, Lett. Math. Phys. 21 (1991), 123–131.CrossRefGoogle Scholar
[18]Perelomov, A. M. and Popov, V. S., ‘Casimir operators for U(n) and SU(n)’, Soviet J. Nuclear Phys. 3 (1966), 676–680.Google Scholar
[19]Perelomov, A. M. and Popov, V. S., ‘Casimir operators for the orthogonal and symplectic groups’, Soviet J. Nuclear Phys. 3 (1966), 819–824.Google Scholar
[20]Schmidt, M., ‘Classification and partial ordering of reductive Howe dual pairs of classical Lie groups’, J. Geom. Phys. 29 (1999), 283–318.CrossRefGoogle Scholar
[21]Umeda, T., ‘The Capelli identities, a century after’, Sugaku 46 (1994), 206–227 (in Japanese);Google Scholar
English translation: Selected papers on harmonic analysis, groups, and invariants (ed. Nomizu, K.) Amer. Math. Soc. Transl. ser. 2, 183 (Amer. Math. Soc., Providence, RI, 1998) pp. 51–78.Google Scholar
[22]Umeda, T., ‘Newton's formula for ’, Proc. Amer. Math. Soc. 126 (1998), 3169–3175.CrossRefGoogle Scholar
[23]Želobenko, D. P., Compact Lie groups and their representations, Transl. Math. Monographs 40 (Amer. Math. Soc., Providence, 1973).Google Scholar