Let U be a maximal unipotent
subgroup of a connected semisimple group G and U′ the derived group of U. If X is
an affine G-variety, then the algebra of U′-invariants, k[X]U′, is finitely generated
and the quotient morphism π : X → X∕∕U′ =Speck[X]U′ is well-defined.
In this article, we study properties of such quotient morphisms, e.g. the
property that all the fibres of π are equidimensional. We also establish an
analogue of the Hilbert-Mumford criterion for the null-cones with respect to
U′-invariants.