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 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.Research supported by SFB/TR12 "Symmetrien und Universalität in mesoskopischen Systemen" of the Deutsche Forschungsgemeinschaft.