Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms. (English) Zbl 1133.11037
In [S. S. Kudla and J. J. Millson, Publ. Math., Inst. Hautes Étud. Sci. 71, 121–172 (1990; Zbl 0722.11026)] liftings are constructed from the cohomology with compact supports of locally symmetric spaces associated to the quadratic group \(\mathrm{O}(p, q)\) to the classical holomorphic Siegel modular forms of weight \((p+ q)/2\).
In the present paper this technique, based on theta series attached to the Weil representation of the dual pair \((\mathrm{O}(p, q),\widetilde{\mathrm{Sp}}(n))\), is generalized to the case of cohomology with certain local coefficients. The coefficient bundles considered are flat bundles arising from harmonic tensors. As a result of this lifting process, the authors obtain new vector-valued Siegel modular forms.
In the present paper this technique, based on theta series attached to the Weil representation of the dual pair \((\mathrm{O}(p, q),\widetilde{\mathrm{Sp}}(n))\), is generalized to the case of cohomology with certain local coefficients. The coefficient bundles considered are flat bundles arising from harmonic tensors. As a result of this lifting process, the authors obtain new vector-valued Siegel modular forms.
Reviewer: Anton Deitmar (Tübingen)
MSC:
11F46 | Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms |
11F32 | Modular correspondences, etc. |
11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |
11F75 | Cohomology of arithmetic groups |