Eisenstein cohomology - explicit computation and relation to Franke's trace formula

Question

Let $G$ be a reductive group over $\mathbb{Q}_p$. Let $X_G$ be a locally symmetric space associated to the group $G$, and let $\partial X_G$ be the Borel-Serre boundary of $X_G$. The space $\partial X_G$ has a stratification by locally symmetric spaces $X_P$ associated to parabolic subgroups of $X_G$. Assuming I understand correctly, the cohomology of any local system (of sufficiently regular weight, with rational coefficients) on $\partial X_G$ should come from the cohomology of local systems on the $X_P$'s. Kostant's theorem further expresses the cohomology of local systems on $X_P$ in terms of the cohomology of local systems on $X_M$, where $M$ is the Levi subgroup of $P$. A cohomology class on $\partial X_G$ will appear in either the ordinary cohomology or the compactly supported cohomology of $X_G$. If $G$ has discrete series, then $X_G$ should only have cohomology in degree $\ge q$ and compactly supported cohomology in degree $\le q$, where $\dim X_G = 2q$.

Given all of this information, if I have a cusp form on $X_M$, it seems that I should be able to write down very explicitly the Eisenstein classes associated to it. However, when I look at the trace formulas of Franke or Goresky-MacPherson, they do not look quite like I would expect. More specifically, the issue of whether a cohomology class of $X_P$ appears in the ordinary or compactly supported cohomology of $X_G$ does not seem to come up. Is this consideration implicitly there somehow, or is there something else going on? Does anything that I said in the first paragraph seem incorrect?

More concretely, if I have a cusp form on some Levi subgroup of $G=Sp(4)$, what Eisenstein classes on $X_G$ does it contribute to? Can the answer be seen from any formulas in Franke's paper Harmonic Analysis on Weighted $L_2$ Spaces?

Answer 1

After rereading section 4 of On the Eisenstein Cohomology of Arithmetric Groups by Schwermer and Li, I think I understand this now.

If $\lambda$ is sufficiently regular, then there is an isomorphism $$H^i(X_P,V_{\lambda}) \cong \bigoplus_{w \in W^M} H^{i+\dim \mathfrak{n}-\ell(w) }(X_M,V_{w(\lambda+\rho)-\rho})$$ where $V_{\lambda}$ denotes a local system of weight $\lambda$, $W^M$ is the set of elements $w$ of the Weyl group of $G$ such that $w^{-1}$ sends positive roots of $M$ to positive roots of $G$, and $\rho$ is half the sum of the positive roots. A class will show up in the cohomology (resp. compactly supported cohomology) of $X_G$ if the restriction of $w \lambda$ to the center of $M$ is positive (resp. negative). I believe that Franke uses the notation $\mathfrak{a}_{\mathcal{R}}^+$ for the set of weights whose restriction to the center of $M$ is positive. The relation between these criteria and the degree of the cohomology class can be found in Theorem 4.5 in Schwermer and Li's paper.

For $G=Sp(4)$:

If $P$ is either of the maximal parabolics, then cusp forms on $X_M$ occur in degree $1$ of the cohomology. They appear in degree $1,2,3,4$ of the cohomology of $X_P$. The degree $3,4$ classes occur in the same degree in the cohomology of $X_G$. The degree $1,2$ classes occur in one degree higher in the compactly supported cohomology of $X_G$.

If $P$ is the minimal parabolic, then cusp forms on $X_M$ occur in degree $0$ of the cohomology. They appear in degree $0,1,1,2,2,3,3,4$ of the cohomology of $X_P$. The degree $4$ class occurs in the same degree in the cohomology of $X_G$, and the degree $0$ class occurs in the degree $2$ compactly supported cohomology of $X_G$. The other classes become zero in the cohomology of $\partial X_G$. One can check this as follows. If $P'$ is a maximal parabolic and $M'$ is the corresponding Levi, then Eisenstein series coming from cusp forms on $X_M$ appear in degree $1$ in $X_{M'}$ and in degrees $1,2,3,4$ in $X_{P'}$. Then we can apply Mayer-Vietoris. The degree $1,2,3$ classes on $X_P$ cancel with classes in one of the $X_P'$, and so they do not show up in the cohomology of $\partial X_G$. The degree $0$ class does not show up in either $X_{P'}$ and so it shows up in degree $1$ of the cohomology of $\partial X_G$. The degree $4$ class shows up in both of the $X_{P'}$, and so it shows up in degree $4$ of the cohomology of $\partial X_G$.