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

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).