Wave forms on \(O(1,q+1)\) and associated Dirichlet series
Let \(S_ 0\) be a symmetric positive definite matrix of size \(q>0\) with coefficients in \({\mathbb{Q}}\), the rational number field. The matrix \[ s= \begin{pmatrix} && 1\\ & S_ 0 \\ 1 \end{pmatrix} \] is a symmetric matrix of signature \((1,q+1)\). Let \(\tilde G\) be the reductive algebraic group over \({\mathbb{Q}}\) whose \({\mathbb{Q}}\) rational points are \(\tilde G_{{\mathbb{Q}}}=\{g\subset GL(q+2,{\mathbb{Q}})|^ tgSg=\nu(g)S\) for \(\nu (g)\in {\mathbb{Q}}^*\}\). The semi-simple part G of \(\tilde G\) is obtained by assigning \(\nu (g)=1\) for \(g\in G\). The adelization of \(\tilde G\) (resp. \({\mathbb{Q}}^*)\) are denoted by \(\tilde G_ A\) (resp. \({\mathbb{Q}}^*_ A)\). A wave form of type \((\omega,\rho)\) is a continuous \({\mathbb{C}}\)-valued function \(\Phi\) on \(\tilde G_ A\) satisfying certain conditions. By adopting \(S_ 0\) instead of S in the definition of \(\tilde G_{{\mathbb{Q}}}\), the groups \(\tilde G_{0,{\mathbb{Q}}}\) and \(\tilde G_ 0\) are defined. \(\tilde G_{0,f}\) is the finite part of \(\tilde G_{0,A}\), the adelization of \(\tilde G_ 0\). For a wave form \(\Phi\) of type \((\omega,\rho)\) and a \({\mathbb{C}}\)-valued continuous function \(\phi\) on \(\tilde G_{0,{\mathbb{Q}}}\setminus \tilde G_{0,f}/\tilde U_ f\) (\(\tilde U_ f\) is a certain subgroup of \(\tilde G_{0,f})\), a Dirichlet series \(Z(s;\Phi,\phi)\) is defined. Theorem 1 of the present paper asserts that this series satisfies a functional equation. When \(S_ 0\) is an orthogonal sum of two submatrices \(S_ 0'\) and \(S_ 0''\), the Eisenstein series \(E(\phi:s,y)\) is defined associated with \(\phi\). Theorem 2 asserts that \(E(\phi;s- m/2,g)\) satisfies a Rankin-Selberg type identity. All details and proofs will appear elsewhere.
scientific article; zbMATH DE number 4008490