×

The Capelli identity for Grassmann manifolds. (English) Zbl 1271.22011

Generalizations and eigenvalue problems of Capelli-type identities have received a lot of attention. In [Colloq. Math. 118, No. 1, 349–364 (2010; Zbl 1194.22015)], R. Howe and S.-T. Lee considered a certain \(O_n\)-invariant Capelli-type differential operator, a product of the determinants of matrices of variables and corresponding partial derivatives, in the context of Grassmannians of \(k\) planes in \(\mathbb{C}^n\). In the article, they raised an eigenvalue problem for the differential operator. When \(k=1\), the problem is reduced to the classical theory of harmonic polynomials, and they solved the problem when \(k=2\).
The article under review answers the question for general \(k\). More precisely, the author solves the eigenvalue problem for a certain family of \(O_n\)-invariant Cappelli-type differential operators, to which the differential operator that Howe and Lee considered belongs. This is done by methods, which are completely different from those Howe and Lee used. The author achieved it by imbedding the problem into a more general setting of the symmetric space \(\mathrm{SO}_n(\mathbb{R})/(\mathrm{SO}_k(\mathbb{R})\times \mathrm{SO}_l(\mathbb{R}))\). In the appendix, by combining the results of the article with those of M. Alexander and M. Nazarov [Math. Ann. 313, No. 2, 315–357 (1999; Zbl 0989.17006)] and M. Itoh [J. Lie Theory 10, No. 2, 463–489 (2000; Zbl 0981.17005)], the author also gives new Capelli-type identities for invariant differential operators for orthogonal Lie algebras.

MSC:

22E46 Semisimple Lie groups and their representations
43A90 Harmonic analysis and spherical functions
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M. Atiyah, R. Bott, and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279 – 330. · Zbl 0257.58008
[2] Armand Borel, Essays in the history of Lie groups and algebraic groups, History of Mathematics, vol. 21, American Mathematical Society, Providence, RI; London Mathematical Society, Cambridge, 2001. · Zbl 1087.01011
[3] Alfredo Capelli, Ueber die Zurückführung der Cayley’schen Operation \Omega auf gewöhnliche Polar-Operationen, Math. Ann. 29 (1887), no. 3, 331 – 338 (German). · JFM 19.0151.01
[4] Sergio Caracciolo, Alan D. Sokal, and Andrea Sportiello, Noncommutative determinants, Cauchy-Binet formulae, and Capelli-type identities. I. Generalizations of the Capelli and Turnbull identities, Electron. J. Combin. 16 (2009), no. 1, Research Paper 103, 43. · Zbl 1192.15001
[5] William Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University Press, Cambridge, 1997. With applications to representation theory and geometry. · Zbl 0878.14034
[6] Sigurdur Helgason, Groups and geometric analysis, Mathematical Surveys and Monographs, vol. 83, American Mathematical Society, Providence, RI, 2000. Integral geometry, invariant differential operators, and spherical functions; Corrected reprint of the 1984 original. · Zbl 0965.43007
[7] Roger Howe, Remarks on classical invariant theory, Trans. Amer. Math. Soc. 313 (1989), no. 2, 539 – 570. , https://doi.org/10.1090/S0002-9947-1989-0986027-X Roger Howe, Erratum to: ”Remarks on classical invariant theory”, Trans. Amer. Math. Soc. 318 (1990), no. 2, 823. · Zbl 0674.15021
[8] 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. · Zbl 0194.53802
[9] Roger Howe and Soo Teck Lee, Spherical harmonics on Grassmannians, Colloq. Math. 118 (2010), no. 1, 349 – 364. · Zbl 1194.22015
[10] Roger Howe and Tōru Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), no. 3, 565 – 619. · Zbl 0733.20019
[11] Minoru Itoh, Capelli elements for the orthogonal Lie algebras, J. Lie Theory 10 (2000), no. 2, 463 – 489. · Zbl 0981.17005
[12] Minoru Itoh, Capelli identities for reductive dual pairs, Adv. Math. 194 (2005), no. 2, 345 – 397. · Zbl 1154.17301
[13] Bertram Kostant and Siddhartha Sahi, The Capelli identity, tube domains, and the generalized Laplace transform, Adv. Math. 87 (1991), no. 1, 71 – 92. · Zbl 0748.22008
[14] Bertram Kostant and Siddhartha Sahi, Jordan algebras and Capelli identities, Invent. Math. 112 (1993), no. 3, 657 – 664. · Zbl 0999.17043
[15] Alexander Molev and Maxim Nazarov, Capelli identities for classical Lie algebras, Math. Ann. 313 (1999), no. 2, 315 – 357. · Zbl 0989.17006
[16] Maxim Nazarov, Yangians and Capelli identities, Kirillov’s seminar on representation theory, Amer. Math. Soc. Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 139 – 163. · Zbl 0927.17007
[17] Andrei Okounkov, Quantum immanants and higher Capelli identities, Transform. Groups 1 (1996), no. 1-2, 99 – 126. · Zbl 0864.17014
[18] Andrei Okounkov, Young basis, Wick formula, and higher Capelli identities, Internat. Math. Res. Notices 17 (1996), 817 – 839. · Zbl 0878.17008
[19] 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. · Zbl 0851.22010
[20] Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. · Zbl 1024.20501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.