Serre's notion of complete reducibility for algebraic groups and geometric invariant theory

The aim of this lecture series is to give an introduction to J.-P. Serre's concept of G-complete reducibility for reductive algebraic groups and to outline recent new advances in the theory stemming from methods from geometric invariant theory. Let G be a connected reductive linear algebraic group. In the mid 1990s J.-P. Serre introduced the notion of a G-completely reducible subgroup of G, in order to generalise to arbitrary reductive groups the representation-theoretic idea of a completely reducible subgroup of GL(V). This philosophy has been developed by Serre, Tits, Liebeck, Seitz, and others over the last 10 years, where the aim is to extend standard results from representation theory to algebraic groups by replacing representations from H to GL(V) with morphisms from H to G, where the target group is an arbitrary reductive algebraic group. Following Serre [3], [4], we say that a closed subgroup H of G is G-completely reducible provided that whenever H is contained in a parabolic subgroup P of G, it is contained in a Levi subgroup of P. In the case G = GL(V), where V a finite-dimensional k-vector space, a subgroup H is G-completely reducible exactly when V is a semisimple H-module, so Serre's concept faithfully generalises the notion of complete reducibility from representation theory. Following R.W. Richardson, we say that H is strongly reductive in G provided H is not contained in any proper parabolic subgroup of C_G(S), where S is a maximal torus of C_G(H), [2]. In the recent paper [1], a new criteria for G-complete reducibility was given; it is shown that a subgroup of G is G-completely reducible if and only if it is strongly reductive in G. This equivalence gives a bridge between the G-completely reducible setting, where techniques are mainly algebraic, and the theory of character varieties, where geometric ideas play an important part. Using work of Richardson on group actions on varieties, one can easily deduce a number of new results on G-completely reducible subgroups. The series of lectures will cover the following topics: - An introduction to algebraic groups, especially reductive algebraic groups, parabolic subgroups. - An introduction to Serre's notion of G-complete reducibility. - Basic results on strongly reductive and G-completely reducible subgroups of a reductive group G. - Some geometric invariant theory, introduction to Kempf-Rousseau theory. - Normal subgroups of G-completely reducible subgroups (Clifford Theory) and commuting G-completely reducible subgroups. - Converses to Clifford's Theorem for G-completely reducible subgroups. The material will be accessible to advanced postgraduate students with a good background in Lie Theory. Bibliography [1] M. Bate, B. Martin, G. Röhrle, A Geometric Approach to Complete Reducibility, Inv. math. 161, no. 1, (2005), 177--218. [2] R. W. Richardson, Conjugacy classes of n-tuples in Lie algebras and algebraic groups, Duke Math. J. 57, (1988), no. 1, 1--35. [3] J.-P. Serre, The notion of complete reducibility in group theory, Moursund Lectures, University of Oregon, 1998. [4] J.-P. Serre, Complète Réductibilité, Séminaire Bourbaki, 56ème année, 2003-2004, n 932.