BIQUOTIENT ACTIONS ON UNIPOTENT LIE GROUPS

Abstract

We consider pairs (V, H) of subgroups of a connected unipotent complex Lie group G for which the induced V × H-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 ℂn. Furthermore, a global slice isomorphic to ℂn 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.