Matrix Valued Classical Pairs Related to Compact Gelfand Pairs of Rank One

Maarten Van Pruijssen ; Pablo Román

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

We present a method to obtain infinitely many examples of pairs $(W,D)$ consisting of a matrix weight $W$ in one variable and a symmetric second-order differential operator $D$. The method is based on a uniform construction of matrix valued polynomials starting from compact Gelfand pairs $(G,K)$ of rank one and a suitable irreducible $K$-representation. The heart of the construction is the existence of a suitable base change $\Psi_{0}$. We analyze the base change and derive several properties. The most important one is that $\Psi_{0}$ satisfies a first-order differential equation which enables us to compute the radial part of the Casimir operator of the group $G$ as soon as we have an explicit expression for $\Psi_{0}$. The weight $W$ is also determined by $\Psi_{0}$. We provide an algorithm to calculate $\Psi_{0}$ explicitly. For the pair $(\mathrm{USp}(2n),\mathrm{USp}(2n-2)\times\mathrm{USp}(2))$ we have implemented the algorithm in GAP so that individual pairs $(W,D)$ can be calculated explicitly. Finally we classify the Gelfand pairs $(G,K)$ and the $K$-representations that yield pairs $(W,D)$ of size $2\times2$ and we provide explicit expressions for most of these cases.

Détail

Keywords: matrix valued classical pairs; multiplicity free branching.

Bibliographie

[1] Baldoni Silva M. W., “Branching theorems for semisimple Lie groups of real rank one”, Rend. Sem. Mat. Univ. Padova, 61 (1979), 229–250 | MR | Zbl

[2] Berezans'kiĭ J. M., Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, 17, Amer. Math. Soc., Providence, R.I., 1968 | MR | Zbl

[3] Bochner S., “Über Sturm–Liouvillesche Polynomsysteme”, Math. Z., 29 (1929), 730–736 | DOI | MR | Zbl

[4] Camporesi R., “A generalization of the Cartan–Helgason theorem for Riemannian symmetric spaces of rank one”, Pacific J. Math., 222 (2005), 1–27 | DOI | MR | Zbl

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

[6] Damanik D., Pushnitski A., Simon B., “The analytic theory of matrix orthogonal polynomials”, Surv. Approx. Theory, 4 (2008), 1–85, arXiv: 0711.2703 | MR | Zbl

[7] Dixmier J., Algèbres envellopantes, Éditions Jacques Gabay, Paris, 1996

[8] Duistermaat J. J., Grünbaum F. A., “Differential equations in the spectral parameter”, Comm. Math. Phys., 103 (1986), 177–240 | DOI | MR | Zbl

[9] Durán A. J., “Matrix inner product having a matrix symmetric second order differential operator”, Rocky Mountain J. Math., 27 (1997), 585–600 | DOI | MR | Zbl

[10] Durán A. J., Grünbaum F. A., “Orthogonal matrix polynomials satisfying second-order differential equations”, Int. Math. Res. Not., 2004:10 (2004), 461–484 | DOI | MR | Zbl

[11] Fulton W., Harris J., Representation theory, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991 | DOI | MR | Zbl

[12] GAP — Groups, Algorithms, and Programming, Ver. 4.6.4, , 2013 http://www.gap-system.org

[13] Geronimo J. S., “Scattering theory and matrix orthogonal polynomials on the real line”, Circuits Systems Signal Process., 1 (1982), 471–495 | DOI | MR | Zbl

[14] Groenevelt W., Ismail M. E. H., Koelink E., “Spectral decomposition and matrix-valued orthogonal polynomials”, Adv. Math., 244 (2013), 91–105, arXiv: 1206.4785 | DOI | MR | Zbl

[15] Groenevelt W., Koelink E., “A hypergeometric function transform and matrix-valued orthogonal polynomials”, Constr. Approx., 38 (2013), 277–309, arXiv: 1210.3958 | DOI | MR | Zbl

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

[17] Grünbaum F. A., Tirao J., “The algebra of differential operators associated to a weight matrix”, Integral Equations Operator Theory, 58 (2007), 449–475 | DOI | MR | Zbl

[18] He X., Ochiai H., Nishiyama K., Oshima Y., “On orbits in double flag varieties for symmetric pairs”, Transform. Groups, 18 (2013), 1091–1136, arXiv: 1208.2084 | DOI | MR | Zbl

[19] Heckman G., van Pruijssen M., Matrix valued orthogonal polynomials for Gelfand pairs of rank one, arXiv: 1310.5134

[20] Knapp A. W., Lie groups beyond an introduction, Progress in Mathematics, 140, 2nd ed., Birkhäuser Boston, Inc., Boston, MA, 2002 | MR | Zbl

[21] 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

[22] 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

[23] 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

[24] Krein M. G., “Infinite $J$-matrices and a matrix-moment problem”, Doklady Akad. Nauk SSSR, 69 (1949), 125–128 | MR

[25] Krein M. G., “Fundamental aspects of the representation theory of Hermitian operators with deficiency index $(m,m)$”, Amer. Math. Soc. Translations Ser. 2, 97 (1971), 75–143 | MR

[26] Lepowsky J., “Multiplicity formulas for certain semisimple Lie groups”, Bull. Amer. Math. Soc., 77 (1971), 601–605 | DOI | MR | Zbl

[27] Lepowsky J., “Algebraic results on representations of semisimple Lie groups”, Trans. Amer. Math. Soc., 176 (1973), 1–44 | DOI | MR | Zbl

[28] Miranian L., “On classical orthogonal polynomials and differential operators”, J. Phys. A: Math. Gen., 38 (2005), 6379–6383 | DOI | MR | Zbl

[29] Pacharoni I., Román P., “A sequence of matrix valued orthogonal polynomials associated to spherical functions”, Constr. Approx., 28 (2008), 127–147, arXiv: math.RT/0702494 | DOI | MR | Zbl

[30] 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/Springer, New York, 2013, 309–354, arXiv: 1209.4500 | DOI | MR | Zbl

[31] Pacharoni I., Tirao J., Zurrián I., “Spherical functions associated to the three dimensional sphere”, Ann. Mat. Pura Appl., 146 (2014), 1727–1778, arXiv: 1203.4275 | DOI | MR

[32] Pacharoni I., Zurrián I., Matrix ultraspherical polynomials: the $2\times 2$ fundamental cases, arXiv: 1309.6902

[33] van Pruijssen M., Matrix valued orthogonal polynomials related to compact Gel'fand pairs of rank one, Ph.D. Thesis, Radboud University Nijmegen, 2012 http://repository.ubn.ru.nl/dspace31xmlui/handle/2066/100840

[34] van Pruijssen M., Román P., GAP codes SP.g and SP2.g, http://www.mvanpruijssen.nl

[35] Shapovalov N. N., “A certain bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra”, Funct. Anal. Appl., 6 (1972), 307–312 | DOI | MR

[36] Tirao J. A., “The matrix-valued hypergeometric equation”, Proc. Natl. Acad. Sci. USA, 100 (2003), 8138–8141 | DOI | MR | Zbl

[37] Tirao J. A., Zurrián I. N., “Spherical functions of fundamental $K$-types associated with the $n$-dimensional sphere”, SIGMA, 10 (2014), 071, 41 pp., arXiv: 1312.0909 | DOI | MR | Zbl

[38] Vretare L., “Elementary spherical functions on symmetric spaces”, Math. Scand., 39 (1976), 343–358 | MR

[39] Wang H. C., “Two-point homogeneous spaces”, Ann. of Math., 55 (1952), 177–191 | DOI | MR | Zbl

[40] Warner G., Harmonic analysis on semi-simple Lie groups, v. II, Die Grundlehren der mathematischen Wissenschaften, 189, Springer-Verlag, New York–Heidelberg, 1972 | MR | Zbl