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Symmetries for spherical functions of type $χ$ for quantum symmetric pairs (2405.15401v2)
Published 24 May 2024 in math.RT and math.QA
Abstract: Let $\mathfrak{g}$ be a complex semisimple Lie algebra and let $\mathbf{U}_q(\mathfrak{g})$ denote the associated Drinfel'd Jimbo quantized enveloping algebra. In this paper we study spherical functions of $\mathbf{U}_q(\mathfrak{g})$ related to characters. We show invariance under the Wang-Zhang braid group operators and show relative Weyl group invariance, when restricted to the quantum torus.
- Branching Rules for Finite-Dimensional 𝐔q(𝔰𝔲(3))subscript𝐔𝑞𝔰𝔲3\mathbf{U}_{q}(\mathfrak{su}(3))bold_U start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT ( fraktur_s fraktur_u ( 3 ) )-Representations with Respect to a Right Coideal Subalgebra. Algebras and Representation Theory, 20(4):821–842, Mar 2017.
- The bar involution for quantum symmetric pairs. Representation Theory of the American Mathematical Society, 19(8):186–210, Oct 2015.
- Universal K-matrix for quantum symmetric pairs. Journal für die reine und angewandte Mathematik (Crelles Journal), 2019(747):299–353, Jul 2016.
- Construction of Irreducible Representations over Khovanov–Lauda–Rouquier Algebras of Finite Classical Type. International Mathematics Research Notices, 2012(5):1312–1366, Nov 2014.
- Canonical bases arising from quantum symmetric pairs. Inventiones mathematicae, 213(3):1099–1177, Apr 2018.
- Factorisation of quasi K𝐾Kitalic_K-matrices for quantum symmetric pairs. Selecta Mathematica, 25(4):Article: 63, Oct 2019.
- Sigurdur Helgason. Differential geometry, Lie groups, and symmetric spaces. Number 80 in Pure and applied mathematics. Academic Press, San Diego, 1998.
- Harmonic analysis and special functions on symmetric spaces. Number 16 in Perspectives in mathematics. Academic Press, San Diego, 1995.
- Jens Carsten Jantzen. Lectures on quantum groups. Number volume 6 in Graduate studies in mathematics. American Mathematical Society, Providence, Rhode Island, 1996.
- Masaki Kashiwara. Global crystal bases of quantum groups. Duke Mathematical Journal, 69(2):455–485, Feb 1993.
- Stefan Kolb. Quantum symmetric Kac–Moody pairs. Advances in Mathematics, 267:395–469, Dec 2014.
- Stefan Kolb. A brief introduction to quantum symmetric pairs. OCAMI Reports 2023, 005:5–21, Sep 2023.
- Tom H. Koornwinder. Askey–Wilson Polynomials as Zonal Spherical Functions on the SU(2) Quantum Group. SIAM Journal on Mathematical Analysis, 24(3):795–813, May 1993.
- Gail Letzter. Symmetric Pairs for Quantized Enveloping Algebras. Journal of Algebra, 220(2):729–767, Oct 1999.
- Gail Letzter. Quantum Symmetric Pairs and Their Zonal Spherical Functions. Transformation Groups, 8(3):261–292, Sep 2003.
- Gail Letzter. Quantum zonal spherical functions and Macdonald polynomials. Advances in Mathematics, 189(1):88–147, Dec 2004.
- George Lusztig. Introduction to Quantum Groups. Birkhäuser Boston, 3rd edition, 2010.
- I. G. Macdonald. Affine Hecke Algebras and Orthogonal Polynomials. Cambridge University Press, Mar 2003.
- Quantum homogeneous spaces with faithfully flat module structures. Israel Journal of Mathematics, 111(1):157–190, Dec 1999.
- Hideya Watanabe. Crystal Basis Theory for a Quantum Symmetric Pair (𝐔,𝐔ȷ)𝐔superscript𝐔italic-ȷ(\mathbf{U},\mathbf{U}^{\jmath})( bold_U , bold_U start_POSTSUPERSCRIPT italic_ȷ end_POSTSUPERSCRIPT ). International Mathematics Research Notices, 22:1099–1177, Oct 2018.
- Hideya Watanabe. Stability of ıitalic-ı\imathitalic_ı-canonical bases of irreducible finite type of real rank one. Representation Theory of the American Mathematical Society, 27(1):1–29, Mar 2023.
- Hideya Watanabe. Stability of ıitalic-ı\imathitalic_ıcanonical bases of locally finite type arXiv:2306.12199. Jun 2023.
- An intrinsic approach to relative braid group symmetries on ıitalic-ı\imathitalic_ıquantum groups. Proceedings of the London Mathematical Society, 127:1338–1423, Sep 2023.