Abstract
Let
V
and
W
be finite dimensional real vector spaces and let
and
be finite subgroups. Assume for simplicity that the actions contain no reflections. Let
Y
and
Z
denote the real algebraic varieties corresponding to
and
, respectively. If
V
and
W
are quasi-isomorphic, i.e., if there is a linear isomorphism
L
:
V
→
W
such that
L
sends
G
-orbits to
H
-orbits and
L
−1
sends
H
-orbits to
G
-orbits, then
L
induces an isomorphism of
Y
and
Z
. Conversely, suppose that
f
:
Y
→
Z
is a germ of a diffeomorphism sending the origin of
Y
to the origin of
Z
. Then we show that
V
and
W
are quasi-isomorphic, This result is closely related to a theorem of Strub [8], for which we give a new proof. We also give a new proof of a result of Kriegl et al. [3] on lifting of biholomorphisms of quotient spaces.