Representations of the Lie algebra \(sl(2)\) in \(\ell\)-adic cohomologies (Q1081651)

Let X be a smooth projective algebraic variety defined over a global field K. Then the Galois group G of K acts on the \(\ell\)-adic cohomology \(H^ n_{\ell}(X)\). The action induces an action of a compact \(\ell\)- adic Lie group on \(H^ n_{\ell}(X)\). The author studies the special case of a simple Lie algebra g of rank 1 acting on \(H^ n_{\ell}(X)\) and proves under certain restrictions, that the simple submodules have dimension \(\leq n+1\).