The residual Eisenstein cohomology of $Sp_{4}$ over a totally real number field

Abstract:

Let $G=Sp_4/k$ be the $k$-split symplectic group of $k$-rank 2, where $k$ is a totally real number field. In this paper we compute the Eisenstein cohomology of $G$ with respect to any finite–dimensional, irreducible, $k$-rational representation $E$ of $G_\infty =R_{k/\mathbb {Q}}G(\mathbb {R})$, where $R_{k/\mathbb {Q}}$ denotes the restriction of scalars from $k$ to $\mathbb {Q}$. This approach is based on the work of Schwermer regarding the Eisenstein cohomology for $Sp_4/\mathbb {Q}$, Kim’s description of the residual spectrum of $Sp_4$, and the Franke filtration of the space of automorphic forms. In fact, taking the representation theoretic point of view, we write, for the group $G$, the Franke filtration with respect to the cuspidal support, and give a precise description of the filtration quotients in terms of induced representations. This is then used as a prerequisite for the explicit computation of the Eisenstein cohomology. The special focus is on the residual Eisenstein cohomology. Under a certain compatibility condition for the coefficient system $E$ and the cuspidal support, we prove the existence of non–trivial residual Eisenstein cohomology classes, which are not square–integrable, that is, represented by a non–square–integrable residue of an Eisenstein series.