### Abstract

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}.

Original language | English |
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Pages (from-to) | 273-316 |

Number of pages | 44 |

Journal | Journal of the Mathematical Society of Japan |

Volume | 64 |

Issue number | 1 |

DOIs | |

Publication status | Published - Oct 5 2012 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Journal of the Mathematical Society of Japan*,

*64*(1), 273-316. https://doi.org/10.2969/jmsj/06410273

**Holonomic systems of Gegenbauer type polynomials of matrix arguments related with Siegel modular forms.** / Ibukiyama, Tomoyoshi; Kuzumaki, Takako; Ochiai, Hiroyuki.

Research output: Contribution to journal › Article

*Journal of the Mathematical Society of Japan*, vol. 64, no. 1, pp. 273-316. https://doi.org/10.2969/jmsj/06410273

}

TY - JOUR

T1 - Holonomic systems of Gegenbauer type polynomials of matrix arguments related with Siegel modular forms

AU - Ibukiyama, Tomoyoshi

AU - Kuzumaki, Takako

AU - Ochiai, Hiroyuki

PY - 2012/10/5

Y1 - 2012/10/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84866910817&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866910817&partnerID=8YFLogxK

U2 - 10.2969/jmsj/06410273

DO - 10.2969/jmsj/06410273

M3 - Article

AN - SCOPUS:84866910817

VL - 64

SP - 273

EP - 316

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

SN - 0025-5645

IS - 1

ER -