On the subring of special cycles

Abstract

Abstract By old results with Millson, the generating series for the cohomology classes of special cycles on orthogonal Shimura varieties over a totally real field are Hilbert–Siegel modular forms. These forms arise via theta series. Using this result and the Siegel–Weil formula, we show that the products in the subring of cohomology generated by the special cycles are controlled by the Fourier coefficients of triple pullbacks of certain Siegel–Eisenstein series. As a consequence, there are comparison isomorphisms between special subrings for different Shimura varieties. In the case in which the signature of the quadratic space V is ( m , 2 ) (m,2) at an even number d + d_{+} of archimedean places, the comparison gives a “combinatorial model” for the special cycle ring in terms of the associated totally positive definite space.