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*?



Eisenstein cohomology - explicit computation and relation to Franke's trace formula
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 $\partial X\_G$.


