Semisimplicity, purity and Mumford-Tate conjecture for hyper-Kähler varieties

We prove the Mumford-Tate conjecture in every degree for hyper-Kähler varieties over fields finitely generated over $\mathbb{Q}$. The proof establishes semisimplicity of $\mathrm{\ell }$-adic cohomology by eliminating the unipotent radical of the algebraic monodromy group of total cohomology. We also prove the weight-monodromy conjecture for hyper-Kähler varieties over $p$-adic fields. For hyper-Kähler varieties over number fields with $b_2\ge 4$, we establish, after suitable finite extensions, strong compatibility of the associated Weil-Deligne representations valued in Mumford-Tate groups and its integral refinement away from finitely many primes.