Abstract Vikraman Balaji

Vikraman Balaji will speak on

On semi-simplicity of tensor products in positive characteristics

Abstract: We work over an algebraically closed field $k$ of characteristic ${p > 0}$. In 1994, Serre showed that if semi-simple representations ${V_i}$ of a group ${Γ}$ are such that ${∑(dimV_i−1)< p}$, then their tensor product is semi-simple. In the late nineties, Serre generalized this theorem comprehensively to the case where ${Γ}$ is a subgroup of ${G(k)}$, for ${G}$ a reductive group, and answered the question of “complete reducibility” of ${Γ}$ in ${G}$,(Seminaire Bourbaki, 2003). In 2014, Deligne generalized the results of Serre (of 1994) to the case when the ${V_i}$ are semi-simple representations of a group scheme ${\mathfrak{G}}$. In my talk I present the recent work of mine with Deligne and Parameswaran where we consider the case when ${\mathfrak{G}}$ is a subgroup scheme of a reductive group ${G}$ and generalize the results of Serre and Deligne. A key result is a structure theorem on “doubly saturated” subgroup schemes ${\mathfrak{G}} $ of reductive groups ${G}$. As an application, we obtain an analogue of clasical Luna's étale slice theorem in postive characteristics.