Serre's notion of complete reducibility and geometric invariant theory

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In this proposed research we intend to further investigate J-P. Serre's notion of G-complete reducibility by means of geometric invariant theory, concentrating on rationality and building theoretic questions. The specific principal objectives are as follows. Firstly, we want to generalize Richardson's algebraic characterization of the closed Gorbits in Gn, the n-fold cartesian product of G with itself, under simultaneous conjugation to the action of an arbitrary reductive subgroup of G on Gn. This in turn will lead to a generalization of Serre's notion of G-complete reducibility. Secondly, we require a suitable concept of optimality by combining Kempf's instability notion with Hesselink's idea of uniform instability. Apart from being of independent interest, this will then be used to address building theoretic questions. In particular, here we pursue a uniform geometric approach towards the so called Center Conjecture due to J. Tits. Further, we will study rationality questions of G-complete reducibility by geometric means, specifically here we will address a general problem posed by Serre concerning the behavior of G-complete reducibility under separable field extensions.