On the Boundary and Intersection Motives of Genus 2 Hilbert-Siegel Varieties

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We study genus 2 Hilbert-Siegel varieties, i.e. Shimura varieties S_K corresponding to the group \mathrm{G}\mathrm{Sp}_{4,F} over a totally real field F , along with the relative Chow motives ^\lambda\mathcal{V} of abelian type over S_K obtained from irreducible representations V_\lambda of \mathrm{G}\mathrm{Sp}_{4,F} . We analyse the weight filtration on the degeneration of such motives at the boundary of the Baily-Borel compactification and we find a criterion on the highest weight \lambda , potentially generalisable to other families of Shimura varieties, which characterizes the absence of the middle weights 0 and 1 in the corresponding degeneration. Thanks to Wildeshaus' theory, the absence of these weights allows us to construct Hecke-equivariant Chow motives over \mathbb{Q} , whose realizations equal interior (or intersection) cohomology of S_K with V_{\lambda} -coefficients. We give applications to the construction of homological motives associated to automorphic representations.