Biquotient actions on unipotent Lie groups

Details Biquotient actions on unipotent Lie groups Biquotient actions on unipotent Lie groups

2007-01-03

arxiv.org/abs/math/0701109v1

Int. J. Math. 19 (2008) no. 8, 997--1008

Mathematics

Complex Variables

Scientific paper

We consider pairs (V,H) of subgroups of a connected unipotent complex Lie group G for which the induced VxH-action on G by multiplication from the left and from the right is free. We prove that this action is proper if the Lie algebra g of G is 3-step nilpotent. If g is 2-step nilpotent then there is a global slice of the action that is isomorphic to C^n. Furthermore, a global slice isomorphic to C^n exists if dim V = 1 = dim H or dim V = 1 and g is 3-step nilpotent. We give an explicit example of a 3-step nilpotent Lie group and a pair of 2-dimensional subgroups such that the induced action is proper but the corresponding geometric quotient is not affine.

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