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December 2021 Lowest weight modules of Sp4(R) and nearly holomorphic Siegel modular forms
Ameya Pitale, Abhishek Saha, Ralf Schmidt
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Kyoto J. Math. 61(4): 745-814 (December 2021). DOI: 10.1215/21562261-2021-0012


We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), where G=Sp4(R) and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of degree 2.

Further, by explicating the algebraic structure of the relevant space of n-finite automorphic forms, we are able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight detsymm with respect to an arbitrary congruence subgroup of Sp4(Q). We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight detsymm with (,m) varying over a certain set. The structure theorem for the space of all modular forms is similar, except that we may now have an additional component coming from certain nearly holomorphic forms of weight det3symm that cannot be obtained from holomorphic forms.

As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction.


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Ameya Pitale. Abhishek Saha. Ralf Schmidt. "Lowest weight modules of Sp4(R) and nearly holomorphic Siegel modular forms." Kyoto J. Math. 61 (4) 745 - 814, December 2021.


Received: 25 May 2015; Revised: 27 March 2019; Accepted: 9 May 2019; Published: December 2021
First available in Project Euclid: 28 July 2021

Digital Object Identifier: 10.1215/21562261-2021-0012

Primary: 11F46
Secondary: 11F67 , 22E47

Keywords: lowest weight modules , nearly holomorphic , representation theory , Siegel modular forms

Rights: Copyright © 2021 by Kyoto University


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Vol.61 • No. 4 • December 2021
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