Products in the cohomology of Shimura varieties: some calculations. (Q1763537)

Let \(S\) be a Shimura variety. \textit{T. N. Venkataramana} [``Cohomology of compact locally symmetric spaces'', Compos. Math. 125, 221--253 (2001; Zbl 0983.11027)] proved a criterion for two cohomology classes \(\omega\), \(\eta \in H^*(S)\) to have a virtually non-zero cup product (virtually means that there exists some (specified in the paper) coverings of \(S\) such that the inverse images of \(\omega\), \(\eta\) with respect to these coverings have a non-zero cup product). The aim of the paper under review is to make the condition of this criterion explicit in the case when the reductive group associated to \(S\) is \(\text{SU}(p,q)\). The answer is given in terms of partitions of numbers. It is shown that strongly primitive cohomology classes in \(H^*(S)\) are parametrized by some pairs of partitions \(\lambda, \mu\). These parts are called compatible, and the corresponding subspace of \(H^*(S)\) is denoted by \(H^{\lambda, \mu}(S)\). The main theorem states that two non-zero elements in \(H^{\lambda, \mu}(S)\), \(H^{\alpha, \beta}(S)\) have a virtually non-zero cup product, if a simple geometric condition (expressed in terms of Young diagrams of \(\lambda, \mu, \alpha, \beta\)) holds. Particularly, this condition implies that if the sum of degrees of \(\omega\), \(\eta\) is \(\leq p+q-1\) then they have always a virtually non-zero cup product.