Matrix Spherical Functions for $(\mathrm{SU}(n+m),\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(m)))$: Two Specific Classes

Jie Liu

Symmetry, integrability and geometry: methods and applications, Tome 19 (2023)

Résumé

We consider the matrix spherical function related to the compact symmetric pair $(G,K)=(\mathrm{SU}(n+m),\mathrm{S}(\mathrm{U}(n)\times\mathrm{U}(m)))$. The irreducible $K$ representations $(\pi,V)$ in the ${\rm U}(n)$ part are considered and the induced representation $\mathrm{Ind}_K^G\pi$ splits multiplicity free. In this case, the irreducible $K$ representations in the ${\rm U}(n)$ part are studied. The corresponding spherical functions can be approximated in terms of the simpler matrix-valued functions. We can determine the explicit spherical functions using the action of a differential operator. We consider several cases of irreducible $K$ representations and the orthogonality relations are also described.

Détail

Keywords: representation theory, special functions.
Mots-clés : Lie group

Bibliographie

[1] Bourbaki N., Éléments de mathématique. Fasc. XXXIV Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: Systèmes de racines, Actual. Sci. Ind., 1337, Hermann, Paris, 1968 | MR

[2] Casselman W., Miličić D., “Asymptotic behavior of matrix coefficients of admissible representations”, Duke Math. J., 49 (1982), 869–930 | DOI | MR | Zbl

[3] Deitmar A., “Invariant operators on higher $K$-types”, J. Reine Angew. Math., 412 (1990), 97–107 | DOI | MR | Zbl

[4] Gangolli R., Varadarajan V.S., Harmonic analysis of spherical functions on real reductive groups, Ergeb. Math. Grenzgeb. (3), 101, Springer, Berlin, 1988 | DOI | MR | Zbl

[5] Gantmacher F.R., The theory of matrices, v. 1, AMS Chelsea Publishing, Providence, RI, 1998 | MR

[6] Grünbaum F.A., Pacharoni I., Tirao J., “Matrix valued spherical functions associated to the complex projective plane”, J. Funct. Anal., 188 (2002), 350–441, arXiv: math.RT/0108042 | DOI | MR | Zbl

[7] Heckman G., Schlichtkrull H., Harmonic analysis and special functions on symmetric spaces, Perspect. Math., 16, Academic Press, Inc., San Diego, CA, 1994 | MR | Zbl

[8] Helgason S., Differential geometry and symmetric spaces, Pure Appl. Math., 12, Academic Press, New York, 1962 | MR | Zbl

[9] Kass S.N., “Explicit decompositions of some tensor products of modules for simple complex Lie algebras”, Comm. Algebra, 15 (1987), 2251–2261 | DOI | MR | Zbl

[10] Kobayashi T., “Multiplicity-free representations and visible actions on complex manifolds”, Publ. Res. Inst. Math. Sci., 41 (2005), 497–549 | DOI | MR | Zbl

[11] Koelink E., Liu J., “$BC_{2}$ type multivariable matrix functions and matrix spherical functions”, Publ. Res. Inst. Math. Sci. (to appear) , arXiv: 2110.02287

[12] Koelink E., van Pruijssen M., Román P., “Matrix-valued orthogonal polynomials related to $({\rm SU}(2)\times{\rm SU}(2),{\rm diag})$”, Int. Math. Res. Not., 2012 (2012), 5673–5730, arXiv: 1012.2719 | DOI | MR | Zbl

[13] Koelink E., van Pruijssen M., Román P., “Matrix-valued orthogonal polynomials related to $({\rm SU}(2)\times{\rm SU}(2),{\rm diag})$, II”, Publ. Res. Inst. Math. Sci., 49 (2013), 271–312, arXiv: 1203.0041 | DOI | MR | Zbl

[14] Koelink E., van Pruijssen M., Román P., “Matrix elements of irreducible representations of ${\rm SU}(n+1)\times {\rm SU}(n+1)$ and multivariable matrix-valued orthogonal polynomials”, J. Funct. Anal., 278 (2020), 108411, 48 pp., arXiv: 1706.01927 | DOI | MR | Zbl

[15] Koornwinder T.H., “Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. I”, Indag. Math., 77 (1974), 48–58 | DOI | MR | Zbl

[16] Koornwinder T.H., “Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. II”, Indag. Math., 77 (1974), 59–66 | DOI | MR | Zbl

[17] Koornwinder T.H., “Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. III”, Indag. Math., 77 (1974), 357–369 | DOI | MR | Zbl

[18] Koornwinder T.H., “Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators. IV”, Indag. Math., 77 (1974), 370–381 | DOI | MR | Zbl

[19] Koornwinder T.H., “Matrix elements of irreducible representations of ${\rm SU}(2)\times{\rm SU}(2)$ and vector-valued orthogonal polynomials”, SIAM J. Math. Anal., 16 (1985), 602–613 | DOI | MR | Zbl

[20] Krämer M., “Sphärische {U}ntergruppen in kompakten zusammenhängenden Liegruppen”, Compositio Math., 38 (1979), 129–153 | MR | Zbl

[21] Kumar S., “Tensor product decomposition”, Proceedings of the International Congress of Mathematicians, v. III, eds. R. Bhatia, A. Pal, G. Rangarajan, V. Srinivas, M. Vanninathan, Hindustan Book Agency, New Delhi, 2010, 1226–1261 | DOI | MR

[22] Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Science Publications, Clarendon Press, Oxford, 1998 | MR | Zbl

[23] Pacharoni I., Tirao J., “One-step spherical functions of the pair $({\rm SU}(n+1),{\rm U}(n))$”, Lie Groups: Structure, Actions, and Representations, Progr. Math., 306, Birkhäuser, New York, 2013, 309–354, arXiv: 1209.4500 | DOI | MR | Zbl

[24] Pezzini G., van Pruijssen M., “On the extended weight monoid of a spherical homogeneous space and its applications to spherical functions”, Represent. Theory (to appear) , arXiv: 2005.09490 | MR

[25] Stokman J.V., Reshetikhin N., “$N$-point spherical functions and asymptotic boundary KZB equations”, Invent. Math., 229 (2022), 1–86, arXiv: 2002.02251 | DOI | MR | Zbl

[26] van Pruijssen M., The branching rules for $(\mathrm{SL}(n + 1, \mathbb{C}), \mathrm{GL}(n, \mathbb{C}))$ revisited: a spherical approach and applications to orthogonal polynomials, Preprint, 2018 https://www.math.ru.nl/m̃pruijssen/MVP-SphericalProof2018.pdf

[27] van Pruijssen M., “Multiplicity free induced representations and orthogonal polynomials”, Int. Math. Res. Not., 2018 (2018), 2208–2239, arXiv: 1405.0796 | DOI | MR | Zbl

[28] Warner G., Harmonic analysis on semi-simple Lie groups, v. II, Grundlehren Math. Wiss., 189, Springer, New York, 1972 | DOI | MR | Zbl