A talk on Sep 16, 2020 at the WIS Representation theory and Algebraic Geometry seminar

Abstract: By old results with Millson, the generating series forthe cohomology classes of special cycles on orthogonal Shimura varieties over atotally real field areHilbert-Siegel modular forms. These forms arise via theta series.

Usingthis result and the Siegel-Weil formula, we show that the products in the subringof cohomology generated by the special cycles are controlled by the Fouriercoefficients of triple pullbacks of certain Siegel-Eisenstein series.

As aconsequence, there are comparison isomorphisms between special subringsfor differentShimura varieties. In the case in which the signature of the quadraticspace V is (m,2)at an even number d_+ of archimedean places, the comparison gives a`combinatorial model' forthe special cycle ring in terms of the associated totally positive definitespace.

Link to slides:
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