Abstract
Let G be a reductive group acting on an affine variety X, let x ∈ X be a point whose G-orbit is not closed, and let S be a G-stable closed subvariety of X which meets the closure of the G-orbit of x but does not contain x. In this paper we study G. R. Kempf’s optimal class Ω G (x; S) of cocharacters of G attached to the point x; in particular, we consider how this optimality transfers to subgroups of G.
Suppose K is a G-completely reducible subgroup of G which fixes x, and let H = C G (K)0. Our main result says that the H-orbit of x is also not closed, and the optimal class Ω H (x; S) for H simply consists of the cocharacters in Ω G (x; S) which evaluate in H. We apply this result in the case that G acts on its Lie algebra via the adjoint representation to obtain some new information about cocharacters associated with nilpotent elements in good characteristic.
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Bate, M. Optimal Subgroups and Applications to Nilpotent Elements. Transformation Groups 14, 29–40 (2009). https://doi.org/10.1007/s00031-008-9038-5
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DOI: https://doi.org/10.1007/s00031-008-9038-5