# Is the Galois x Hecke action on cohomology of Shimura varieties semi-simple?

Given a reductive group $G/\mathbf Q$ (+ additional data), and a compact open subgroup $K\subset G(\mathbf A^\infty)$, there is a standard construction that produces a Shimura variety $S$ and if we choose a rational representation $\xi : G\to Aut(V)$, we obtain a certain $l$-adic local system $\mathcal L$ on $S$. Moreover, all of this can be done over some fixed number field $E$.

Question 1: If $S$ is proper, is it true that the $Gal(\overline E/E)\times H(G,K)$ action on $H^*(S\otimes\overline E, L)$ is semisimple? (where $H(G,K)$ is the corresponding Hecke algebra).

Question 2: What happens if $S$ is not proper? Is the action semi-simple at least if we restrict to parabolic cohomology (the image of compactly supported cohomology inside usual cohomology)?

References for places where this is discussed would also be great.

Thanks

• I don't know the answer, but for the non-proper case, it is the intersection cohomology (of min compactification) that people are interested in: this is the one that is related to §L^2§-cohom and hence Lie algebra cohom, and therefore automorphic rep. via trace formula. – shenghao Jun 8 '11 at 12:18

## 1 Answer

To get a Shimura variety, the reductive group $G$ should satisfy some axioms. In fact, you should begin not just with $G$, but with a Shimura datum for $G$.

Leaving that aside, the Hecke action will be semi-simple (if we omit Hecke operators at primes dividing the level); more generally, one could take the limit over all levels, and then get a $G(\mathbb A^{\infty})$-action, which will be semi-simple (e.g. by comparison with automorphic forms; and here I am supposing $S$ compact for the moment).

Each $G(\mathbb A^{\infty})$-rep. will appear with some multiplicity, and this multiplicity space carries the $Gal(\overline{E}/E)$-action. If this action is irreducible, which is the case e.g. for modular curves (admittedly non-compact, but ignore that for the moment!) or Shimura curves, then it is certainly semi-simple. But in more general situations it need not be irreducible, and then its semi-simplicity is a certain case of the Tate conjecture, and I'm pretty sure that nothing will be known about it.

E.g. already in the case of Hilbert modular varieties (or their compact cousins, coming from quat. algebras, if you prefer), in some situations one expects a reducible Galois rep'n, and these are not known to be semi-simple, I believe.

The case of open varieties will be similar, in that not much will be known there either, unless it is forced by irreducibility.