A frustrating cohomology class on the moduli of abelian surfaces

Question

Here's a very frustrating question that I have been stuck on for some time. I believe that my question could fit in a general framework of what happens when you restrict $L^2$-cohomology classes on a Shimura variety to a sub-Shimura variety. However I formulate the question for the special case I am interested in.

Let $A_2$ be the moduli stack of principally polarized abelian surfaces. To the irreducible finite dimensional representation of $\mathrm{Sp}(4)$ of highest weight $a \geq b \geq 0$ we attach a a local system $V_{a,b}$ on $A_2$.

Suppose $(a,b) \neq (0,0)$. One can prove that $H^4_c(A_2,V_{a,b})$ vanishes unless $a=b$ is even, in which case $H^4_c(A_2,V_{2k,2k})$ is pure of Tate type and of the same dimension as the space of cusp forms of weight $4k+4$ for $\mathrm{SL}(2,\mathbf Z)$. The map $H^4_c \to H^4_{(2)}$ to the $L^2$-cohomology is an isomorphism. In terms of automorphic representations, these cohomology classes can be described as follows: for any level 1 cusp form $\pi$ on $\mathrm{GL}(2,\mathbf A)$ of weight $4k+4$ we consider the unique irreducible quotient of $$ \mathrm{Ind}_{P(\mathbf A)}^{\mathrm{GSp}(4,\mathbf A)} \left( \vert \cdot \vert^{1/2} \pi \otimes \vert \cdot \vert^{-1/2} \right)$$ where $P$ denotes the Siegel parabolic subgroup (whose Levi factor is $\mathrm{GL}(2) \times \mathrm{GL}(1)$); this is a discrete automorphic representation for $\mathrm{GSp}(4)$ which contributes a Tate type class to the $L^2$-cohomology in degrees $2$ and $4$.

There is a map $\mathrm{Sym}^2(A_1) \hookrightarrow A_2$ given by taking a pair of elliptic curves to their product. We can also restrict $V_{a,b}$ to $\mathrm{Sym}^2(A_1)$. By determining the branching formula for $\mathrm{SL}(2)^2 \rtimes S_2 \subset \mathrm{Sp}(4)$ we find that the trivial local system occurs as a summand in the restriction of $V_{a,b}$ to $\mathrm{Sym}^2(A_1)$ if and only if $a=b$ is even, in which case it appears with multiplicity $1$. So $H^4_c(\mathrm{Sym}^2(A_1),V_{2k,2k})$ is also pure of Tate type but $1$-dimensional. Again we could think about $L^2$-cohomology and it would not make a difference.

MAIN QUESTION: Is the restriction map $H^4_c(A_2,V_{2k,2k}) \to H^4_c(\mathrm{Sym}^2(A_1),V_{2k,2k})$ nonzero for $k \geq 2$?

Any ideas or pointers at all would be appreciated. I am very ignorant about automorphic representations, Shimura varieties etc. and I am naively hoping that there exists some general method for answering question of this form.

This question arose from the paper http://arxiv.org/abs/1210.5761 . A positive answer would imply that all even cohomology of $\mathcal{\overline{M}}_{2,n}$ is tautological for $n < 20$, and that the Gorenstein conjecture fails on $\mathcal{\overline{M}}_{2,20}$.

Answer 1

The answer to the main question above is indeed positive: the restriction map is nonzero for all $k \geq 2$. This was proved in Section 5 of my paper Tautological rings of spaces of pointed genus two curves of compact type. Compos. Math. 152 (2016), no. 7, 1398–1420.

The idea of the calculation is that these are Eisenstein cohomology classes for the Siegel parabolic subgroup, which corresponds to the 0-dimensional boundary strata in the Baily-Borel compactification. Therefore one can reduce to a calculation on a deleted neighborhood of the 0-dimensional part of the boundary. Such a deleted neighborhood can be described as a fiber bundle over a smaller locally symmetric space. In the end one ends up changing the problem from considering the submanifold $A_1 \times A_1 \subset A_2$ to considering the submanifold inside $A_1$ given by the image of the imaginary axis. This is equivalent to reformulating the problem as an assertion about classical modular symbols.