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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On a relation between $\widetilde {\mathrm {SL}}_{2}$ cusp forms and cusp forms on tube domains associated to orthogonal groups
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by S. Rallis and G. Schiffmann
Trans. Amer. Math. Soc. 263 (1981), 1-58
DOI: https://doi.org/10.1090/S0002-9947-1981-0590410-7

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).
References
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Bibliographic Information
  • © Copyright 1981 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 263 (1981), 1-58
  • MSC: Primary 10D40; Secondary 22E50, 32N10
  • DOI: https://doi.org/10.1090/S0002-9947-1981-0590410-7
  • MathSciNet review: 590410