Generic differential operators on Siegel modular forms and special polynomials
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Generic differential operators on Siegel modular forms and special polynomials Tomoyoshi Ibukiyama1 Accepted: 23 August 2020 © The Author(s) 2020
Abstract Holomorphic vector valued differential operators acting on Siegel modular forms and preserving automorphy under the restriction to diagonal blocks are important in many respects, including application to critical values of L functions. Such differential operators are associated with vectors of new special polynomials of several variables defined by certain harmonic conditions. They include the classical Gegenbauer polynomial as a prototype, and are interesting as themselves independently of Siegel modular forms. We will give formulas for all such polynomials in two different ways. One is to describe them using polynomials characterized by monomials in off-diagonal block variables. We will give an explicit and practical algorithm to give the vectors of polynomials through these. The other one is rather theoretical but seems much deeper. We construct an explicit generating series of polynomials mutually related under certain mixed Laplacians. Here substituting the variables of the polynomials to partial derivatives, we obtain the generic differential operator from which any other differential operators of this sort are obtained by certain projections. This process exhausts all the differential operators in question. This is also generic in the sense that for any number of variables and block partitions, it is given by a recursive unified expression. As an application, we prove that the Taylor coefficients of Siegel modular forms with respect to off-diagonal block variables, or of corresponding expansion of Jacobi forms, are essentially vector valued Siegel modular forms of lower degrees, which are obtained as images of the differential operators given above. We also show that the original forms are recovered by the images of our operators. This is an ultimate generalization of Eichler–Zagier’s results on Jacobi forms of degree one. Several more explicit results and practical construction are also given. Keywords Siegel modular forms · Differential operators · Jacobi forms · Special polynomials
This work was supported by JSPS KAKENHI Grant Nos. 25247001 and 19K03424. The author would like to thank Max Planck Institute for Mathematics in Bonn for their kind hospitality during the preparation of this paper. He also thanks D. Zagier for inspiring discussion. Extended author information available on the last page of the article 0123456789().: V,-vol
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T. Ibukiyama
Mathematics Subject Classification Primary 11F46; Secondary 11F50 · 33C47
1 Introduction First we explain a general problem setting. Assume that Di are bounded symmetric domains for i = 1, 2 such that D2 ⊂ D1 . For i = 1, 2, we denote by Aut(Di ) the group of biholomorphic automorphisms of Di and fix subgroups G i ⊂ Aut(Di ). We assume that there is an embedding ι : G 2 → G 1 acting equivariantly on Di for the embedding D2 ⊂ D1 . Let Vi (i = 1, 2) be finite
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