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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

On a relation between $ \widetilde {\rm SL}\sb{2}$ cusp forms and cusp forms on tube domains associated to orthogonal groups


Authors: S. Rallis and G. Schiffmann
Journal: Trans. Amer. Math. Soc. 263 (1981), 1-58
MSC: Primary 10D40; Secondary 22E50, 32N10
MathSciNet review: 590410
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Abstract: We use the decomposition of the discrete spectrum of the Weil representation of the dual reductive pair $ ({\tilde{SL}_2},\;O(Q))$ to construct a generalized Shimura correspondence between automorphic forms on $ O(Q)$ and $ \widetilde{S{L_2}}$. We prove a generalized Zagier identity which gives the relation between Fourier coefficients of modular forms on $ \widetilde{S{L_2}}$ and $ O(Q)$. We give an explicit form of the lifting between $ \widetilde{S{L_2}}$ and $ O(n,2)$ in terms of Dirichlet series associated to modular forms. For the special case $ n = 3$, we construct certain Euler products associated to the lifting between $ S{L_2}$ and $ {\text{S}}{{\text{p}}_2} \cong O(3,2)$ (locally).


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DOI: https://doi.org/10.1090/S0002-9947-1981-0590410-7
Article copyright: © Copyright 1981 American Mathematical Society