Journal of the Mathematical Society of Japan

Holonomic systems of Gegenbauer type polynomials of matrix arguments related with Siegel modular forms

Tomoyoshi IBUKIYAMA, Takako KUZUMAKI, and Hiroyuki OCHIAI

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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 2n.

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J. Math. Soc. Japan, Volume 64, Number 1 (2012), 273-316.

First available in Project Euclid: 26 January 2012

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Primary: 11F60: Hecke-Petersson operators, differential operators (several variables) 32C38: Sheaves of differential operators and their modules, D-modules [See also 14F10, 16S32, 35A27, 58J15]
Secondary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 33C67: Hypergeometric functions associated with root systems

Siegel modular forms differential operators holonomic system


IBUKIYAMA, Tomoyoshi; KUZUMAKI, Takako; OCHIAI, Hiroyuki. Holonomic systems of Gegenbauer type polynomials of matrix arguments related with Siegel modular forms. J. Math. Soc. Japan 64 (2012), no. 1, 273--316. doi:10.2969/jmsj/06410273.

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