Holonomic systems of Gegenbauer type polynomials of matrix arguments related with Siegel modular forms. (English) Zbl 1272.11066

Summary: Differential operators on Siegel modular forms which behave well under the restriction of the domain are essentially intertwining operators of the tensor product of holomorphic discrete series to its irreducible components. These are characterized by polynomials in the tensor of pluriharmonic polynomials with some invariance properties. We give a concrete study of such polynomials in the case of the restriction from Siegel upper half space of degree \(2n\) to the product of degree \(n\). These generalize the Gegenbauer polynomials which appear for \(n = 1\). We also describe their radial parts parametrization and differential equations which they satisfy, and show that these differential equations give holonomic systems of rank \(2^{n}\).


11F60 Hecke-Petersson operators, differential operators (several variables)
32C38 Sheaves of differential operators and their modules, \(D\)-modules
11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
33C67 Hypergeometric functions associated with root systems
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[1] C. Bachoc, R. Coulangeon and G. Nebe, Designs in Grassmannian spaces and lattices, J. Algebraic Combin., 16 (2002), 5-19. · Zbl 1035.05027 · doi:10.1023/A:1020826329555
[2] R. J. Beerends and E. M. Opdam, Certain hypergeometric series related to the root system \(BC\), Trans. Amer. Math. Soc., 339 (1993), 581-609. · Zbl 0794.33009 · doi:10.2307/2154288
[3] S. Böcherer, Über die Fourier Jacobi-Entwicklung Siegelscher Eisensteinreihen, II (German), Math. Z., 189 (1985), 81-110. · Zbl 0558.10022 · doi:10.1007/BF01246946
[4] S. Böcherer, T. Satoh and T. Yamazaki, On the pullback of a differential operator and its application to vector valued Eisenstein series, Comment. Math. Univ. St. Paul., 41 (1992), 1-22. · Zbl 0759.11015
[5] A. G. Constantine, Some non-central distribution problems in multivariate analysis, Ann. Math. Statist., 34 (1963), 1270-1285. · Zbl 0123.36801 · doi:10.1214/aoms/1177703863
[6] W. Eholzer and T. Ibukiyama, Rankin-Cohen type differential operators for Siegel modular forms, Internat. J. Math., 9 (1998), 443-463. · Zbl 0919.11037 · doi:10.1142/S0129167X98000191
[7] M. Eichler and D. Zagier, The theory of Jacobi forms, Progr. Math., 55 , Birkhäuser, Boston, Inc., Boston MA, 1985, v+148 pp. · Zbl 0554.10018
[8] R. Goodman and N. R. Wallach, Representations and invariants of the classical groups, Encyclopedia of Mathematics and its Application, 68 , Cambridge University Press, Cambridge, 1998, xvi+685 pp. · Zbl 0901.22001
[9] G. J. Heckman, Root systems and hypergeometric functions, II, Compositio Math., 64 (1987), 353-374. · Zbl 0656.17007
[10] G. J. Heckman and E. M. Opdam, Root systems and hypergeometric functions, I, Compositio Math., 64 (1987), 329-352. · Zbl 0656.17006
[11] G. Heckman and H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces, (ed. S. Helgason), Perspectives in Mathematics, 16 , Academic Press Inc., San Diego, CA, 1994, xii+225 pp. · Zbl 0836.43001
[12] R. Hotta, K. Takeuchi and T. Tanisaki, \(\mathscr{D}\)-Modules, Perverse Sheaves, and Representation Theory, Progr. Math., 236 , Birkhäuser Boston Inc., Boston, MA, 2008, xi+ 407 pp. · Zbl 1136.14009
[13] T. Ibukiyama, On differential operators on automorphic forms and invariant pluri-harmonic polynomials, Comment. Math. Univ. St. Paul., 48 (1999), 103-118. · Zbl 1007.11023
[14] T. Ibukiyama and D. Zagier, Higher spherical polynomials, in preparation.
[15] T. Ibukiyama and D. Zagier, Higher spherical functions, in preparation.
[16] A. T. James, Zonal polynomials of the real positive definite symmetric matrices, Ann. of Math. (2), 74 (1961), 456-469. · Zbl 0104.02803 · doi:10.2307/1970291
[17] A. T. James and A. G. Constantine, Generalized Jacobi polynomials as spherical functions of the Grassmann manifold, Proc. London Math. Soc. (3), 29 (1974), 174-192. · Zbl 0289.33031 · doi:10.1112/plms/s3-29.1.174
[18] M. Kashiwara, Algebraic Analysis, Iwanami Shoten, Tokyo, 2000, 276 pp (Japanese).
[19] M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math., 44 (1978), 1-47. · Zbl 0375.22009 · doi:10.1007/BF01389900
[20] H. Katsurada, Exact standard zeta values of Siegel modular forms, Experiment. Math., 19 (2010), 65-77. · Zbl 1206.11064 · doi:10.1080/10586458.2010.10129062
[21] R. Muirhead, Systems of partial differential equations for hypergeometric functions of matrix argument, Ann. Math. Statist., 41 (1970), 991-1001. · Zbl 0225.62078 · doi:10.1214/aoms/1177696975
[22] M. Sato, M. Kashiwara, T. Kimura and T. Oshima, Micro-local analysis of prehomogeneous vector spaces, Invent. Math., 62 (1980), 117-179. · Zbl 0456.58034 · doi:10.1007/BF01391666
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