Hodge modules on Shimura varieties, and their degeneration in the Baily-Borel compactification
Let \(j: M \to M^*\) be the Baily-Borel compactification of a Shimura variety \(M\) associated to a reductive group \(G\) over \(\mathbb Q\). Then the boundary \(M^*\setminus M\) has a natural stratification, and each stratum \(i: M_1 \hookrightarrow M^*\) is itself a Shimura variety. On the other hand, any algebraic representation \(\mathbb V\) of \(G\) determines a Hodge module \(\mu (\mathbb V)\) on \(M\). The group \(G_1\) belonging to \(M_1\) is the maximal reductive quotient \(P_1/W_1\) of a certain parabolic subgroup \(P_1\) of \(G\), and the topological inertia group of \(M_1\) in \(M\) is an extension of an arithmetic group \(\overline{H}_C\) by a lattice in \(W_1 (\mathbb Q)\). In this paper the authors express \(i^* j_* \circ \mu\) as a composition of Hochschild cohomology of \(W_1\), abstract cohomology of \(\overline{H}_C\), and the canonical construction on \(M_1\). This result is the Hodge-theoretic analogue of the theorem on \(\ell\)-adic sheaves of \textit{R.~Pink} [Math. Ann. 292, 197-240 (1992; Zbl 0748.14008)].
scientific article; zbMATH DE number 1907818