Variation of Tannaka groups of perverse sheaves in family

Abstract: Let $k$ be a field of characteristic $0$, let $S$ be a smooth, geometrically connected variety over $k$, with generic point $\eta$, and $f:\mathbb{X}\rightarrow S$ a morphism separated and of finite type. Fix a prime $\ell$. Let $\mathbb{P}$ be an $f$-universally locally acyclic relative perverse $\overline{\mathbb{Q}}\ell$-sheaf on $\mathbb{X}/S$. We prove that if for some (equivalently, every) geometric point $\bar \eta$ over $\eta$ the restriction $\mathbb{P}|{\mathbb{X}{\bar \eta}}$ is simple as a perverse $\overline{\mathbb{Q}}\ell$-sheaf on $\mathbb{X}{\bar \eta}$, then there is a non-empty open subscheme $U\subset S$ such that, for every geometric point $\bar s$ on $U$, the restriction $\mathbb{P}|{\mathbb{X}{\bar s}}$ is simple as a perverse $\overline{\mathbb{Q}}\ell$-sheaf on $\mathbb{X}{\bar s}$. When $f:\mathbb{X}\rightarrow S$ is an abelian scheme, we give applications of this result to the variation with $s\in S$ of the Tannaka group of $\mathbb{P}|{\mathbb{X}_{\bar s}}$.