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# V.Balaji (Chennai Mathematical Institute), Geometric Methods in Representation Theory Seminar

## November 29, 2017 @ 3:30 pm - 4:30 pm

**Title:*** 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 Vi of a group Γ are such that (dimVi − 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 Vi are semi-simple representations of a group scheme G. In my talk I present the recent work of mine with Deligne and Parameswaran where we consider the case when 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 G of reductive groups G. As an application, we obtain an analogue of classical Luna’s ́etale slice theorem in positive characteristics.