On the Kronecker pairing for mixed cusp forms
Let \(\omega\) be a holomorphic map from the upper half plane \({\mathcal H}\) to itself, and let \(\chi: G\to \Gamma\) be a surjective homomorphism between Fuchsian groups of the first kind. Suppose that \(\omega (g\cdot z)=\chi (g)\cdot \omega (z)\) for all \(g\in G\), and that the map \(G\setminus {\mathcal H}\to \Gamma \setminus {\mathcal H}\) induced by \(\omega\) extends as a holomorphic map \(X_ G\to X_{\Gamma}\) between the compactifications. In this situation one may define mixed cusp forms of type \((\ell,k)\) as holomorphic functions on \({\mathcal H}\) that satisfy \[ f(g\cdot z)=(c_ gz+d_ g)^{\ell}(c_{\gamma}\omega (z)+d_{\gamma})^ kf(z) \] for all \[ g=\left( \begin{matrix} a_ g\quad b_ g\\ c_ g\quad d_ g\end{matrix} \right)\in G,\quad \chi (g)=\left( \begin{matrix} a_{\gamma} & b_{\gamma}\\ c_{\gamma} & d_{\gamma} \end{matrix} \right), \] and that vanish at the cusps of \(G\). Take \(k=0\) to obtain the usual holomorphic cusp forms for \(G\). This definition invites generalization of results known for classical cusp forms to the mixed case. In the usual case the Shimura isomorphism gives, for even \(m\geq 0\), an injection from the space \(S_{2+m}(G)\) of cusp forms of weight \(2+m\) into the cohomology group \(H^ 1(G,S^ m(V))\). Here \(V\) denotes the standard two-dimensional representation of \(G\), and \(S^ m\) the \(m\)-th symmetric power. Hence \(S_{2+m}(G)\) is in duality with the homology group \(H_ 1(G,S^ m(V^*))\), where \(V^*\) is the dual of \(V\). In \textit{S. Katok} and \textit{J. J. Millson}, Trans. Am. Math. Soc. 300, 737--757 (1987; Zbl 0618.10020), this duality is made explicit for arbitrary elements of \(S_{2+m}(G)\) and certain elements of \(H_ 1(G,S^ m(V^*))\) associated to hyperbolic elements of \(G\). The paper under review generalizes this to the case of mixed cusp forms of type \((2,m)\) and the action of \(G\) on \(V\) via \(\chi\).
scientific article; zbMATH DE number 4212142
scientific article