Cup Products in Automorphic Cohomology

Cup Products in Automorphic Cohomology - Matthew Kerr

Matthew Kerr
Washington University in St. Louis
March 30, 2012

In three very interesting and suggestive papers, H. Carayol introduced new aspects of complex geometry and Hodge theory into the study of non-classical automorphic representations -- in particular, those involving the totally degenerate limits of discrete series. This talk is based on two joint projects which aim to put Carayol's work into a more general context, while hewing to his over-riding theme of producing arithmetic structures on the cohomology of non-algebraic generalizations of Shimura varieties.

The first of these, with P. Griffiths and M. Green, studies Penrose transforms and cup products in the coherent cohomology of arithmetic-group quotients of certain classifying spaces for Hodge structures with given symmetries. These spaces, called Mumford-Tate domains, are homogeneous spaces for a reductive Lie group G. Our talk will focus mainly on the co-compact setting, and describe an extension of Carayol's main result on "virtual surjectivity" of cup products (for G=SU(2,1)) to the case G=Sp_4.

Time permitting, we shall briefly discuss cuspidal automorphic cohomology for non-co-compact quotients, and describe recent work with G. Pearlstein on their boundary components (with G=G_2 as a key example).