• DOI:10.1090/S0002-9947-10-04901-9
• Corpus ID: 1697765

Complete reducibility and separability

@article{Bate2007CompleteRA,
  title={Complete reducibility and separability},
  author={Michael Bate and Benjamin Martin and Gerhard R{\"o}hrle and Rudolf Tange},
  journal={Transactions of the American Mathematical Society},
  year={2007},
  volume={362},
  pages={4283-4311}
}

• M. Bate, B. Martin, +1 author R. Tange
• Published 24 September 2007
• Mathematics
• Transactions of the American Mathematical Society

Let G be a reductive linear algebraic group over an algebraically closed field of characteristic p > 0. A subgroup of G is said to be separable in G if its global and infinitesimal centralizers have the same dimension. We study the interaction between the notion of separability and Serre's concept of G-complete reducibility for subgroups of G. The separability hypothesis appears in many general theorems concerning G-complete reducibility. We demonstrate that many of these results fail without this hypothesis. On the other hand, we prove that if G is a connected reductive group and p is very good for G, then any subgroup of G is separable; we deduce that under these hypotheses on G, a subgroup H of G is G-completely reducible provided the Lie algebra of G is semisimple as an H-module.Recently, Guralnick has proved that if H is a reductive subgroup of G and C is a conjugacy class of G, then the intersection of C and G is a finite union of H-conjugacy classes. For generic p -- when certain extra hypotheses hold, including separability -- this follows from a well-known tangent space argument due to Richardson, but in general, it rests on Lusztig's deep result that a connected reductive group has only finitely many unipotent conjugacy classes. We show that the analogue of Guralnick's result is false if one considers conjugacy classes of n-tuples of elements from H for n > 1.