The Capelli eigenvalue problem for quantum groups

• Gail Letzter ^[1] ; Siddhartha Sahi ^[2] ; Hadi Salmasian ^[3]
  1. [1] Mathematics Research Group, USA
  2. [2] Department of Mathematics, USA
  3. [3] Department of Mathematics and Statistics, Canada

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• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 31, Nº. 1, 2025
• Idioma: inglés
• DOI: 10.1007/s00029-024-01003-8
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• Resumen

  • We introduce and study quantum Capelli operators inside newly constructed quantum Weyl algebras associated to three families of symmetric pairs (Letzter et al. in J Algebra, 2024). Both the center of a particular quantized enveloping algebra and the Capelli operators act semisimply on the polynomial part of these quantum Weyl algebras. We show how to transfer well-known properties of the center arising from the theory of quantum symmetric pairs to the Capelli operators. Using this information, we provide a natural realization of Knop-Sahi interpolation polynomials as functions that produce eigenvalues for quantum Capelli operators.

• Referencias bibliográficas
  • 1. Alldridge, A., Sahi, S., Salmasian, H.: Schur q-functions and the Capelli eigenvalue problem for the Lie superalgebra q(n). In: Representation...
  • 2. Bershtein, O.: Regular functions on the Shilov boundary. J. Algebra Appl. 4(6), 613–629 (2005)
  • 3. Bershtein, O.: On a q-analog of a Sahi result. J. Math. Phys. 48, 043510 (2007)
  • 4. Dikhhuizen, M.S., Stokman, J.V.: Some limit transitions between BC type orthogonal polynomials interpreted on quantum complex Grassmannians....
  • 5. Heckenberger, I., Schmüdgen, K.: Classification of bicovariant differential calculi on the quantum groups SLq (n + 1) and Spq (2n)....
  • 6. Joseph, A.: Quantum Groups and Their Primitive Ideals, A Series of Modern Surveys in Mathematics, vol. 29. Springer, Berlin (2012)
  • 7. Joseph, A., Letzter, G.: Local finiteness of the adjoint action for quantized enveloping algebras. J. Algebra 153, 289–318 (1992)
  • 8. Joseph, A., Letzter, G.: Separation of variables for quantized enveloping algebras. Am. J. Math. 116(1), 127–177 (1994)
  • 9. Kolb, S.: Quantum symmetric Kac-Moody pairs. Adv. Math. 267, 395–469 (2014)
  • 10. Knop, F.: Symmetric and non-symmetric quantum Capelli polynomials. Comment. Math. Helv. 72, 84–100 (1997)
  • 11. Klimyk, A., Schmüdgen, K.: Quantum Groups and Their Representations. Texts and Monographs in Physics. Springer, Berlin (1997)
  • 12. Kostant, B., Sahi, S.: The Capelli Identity, tube domains, and the generalized Laplace transform. Adv. Math. 87(1), 71–92 (1991)
  • 13. Kostant, B., Sahi, S.: Jordan algebras and Capelli identities. Invent. Math. 112(3), 657–664 (1993)
  • 14. Letzter, G.: Symmetric pairs for quantized enveloping algebras. J. Algebra 220, 729–767 (1999)
  • 15. Letzter, G.: Coideal subalgebras and quantum symmetric Pairs. In: New Directions in Hopf Algebras. MSRI Publications 43, 117–165 (2002)
  • 16. Letzter, G.: Quantum symmetric Pairs and their zonal spherical functions. Transform. Groups 8, 261– 292 (2003)
  • 17. Letzter, G.: Quantum zonal spherical functions and Macdonald polynomials. Adv. Math. 189, 88–147 (2004)
  • 18. Letzter, G.: Invariant differential operators for quantum symmetric spaces, Memoirs of the American Mathematical Society, Volume 193,...
  • 19. Letzter, G., Sahi, S., Salmasian, H.: Quantized Weyl Algebras, the Double Centralizer Property, and a new first Fundamental Theorem, preprint...
  • 20. Letzter, G., Sahi, S., Salmasian, H.: Weyl algebras for quantum homogeneous spaces. J. Algebra 655, 651–721 (2024)
  • 21. Macdonald, I.G.: Orthogonal polynomials associated with root systems. Sém. Lotharingien Combin. 45, Art B45a, 40 pp
  • 22. Noumi, M.: Macdonald’s symmetric polynomials as zonal spherical functions on quantum homogeneous spaces. Adv. Math. 123, 16–77 (1996)
  • 23. Noumi, M., Dijkhuizen, M.S., Sugitani, T.: Multivariable Askey-Wilson polynomials and quantum complex Grassmannians. Fields Inst. Commun....
  • 24. Noumi, M., Sugitani, T.: Quantum symmetric spaces and related q-orthogonal polynomials. In: Group Theoretical Methods in Physics (ICGTMP),...
  • 25. Okounkov, A.: Shifted Macdonald polynomials: q-Integral representation and combinatorial formula. Compos. Math. 112, 147–182 (1998)
  • 26. Olshanski, G.: Interpolation Macdonald polynomials and Cauchy-type identities. J. Comb. Theory Ser. A 162, 65–117 (2019)
  • 27. Sahi, S.: Interpolation, integrality, and a generalization of Macdonald’s polynomials. Int. Math. Res. Not. 10, 457–471 (1996)
  • 28. Sahi, S.: The spectrum of certain invariant differential operators associated to a Hermitian symmetric space In: Lie theory and geometry,...
  • 29. Sahi, S., Salmasian, H.: The Capelli problem for gl(m|n) and the spectrum of invariant differential operators. Adv. Math. 303, 1–38 (2016)
  • 30. Sahi, S., Salmasian, H., Serganova, V.: The Capelli eigenvalue problem for Lie superalgebras. Math. Z. 294, 359–395 (2020)
  • 31. Sahi, S., Salmasian, H., Serganova, V.: Capelli operators for spherical superharmonics and the DougallRamanujan identity. arXiv:1912:06301,...
  • 32. Shklyarov, D., Sinel’shchikov, S., Vaksman, L.: Fock representations and quantum matrices. Int. J. Math. 15(9), 855–894 (2004)