  We consider a system with symmetries whose configuration space is a compact
Lie group, acted upon by inner automorphisms. The classical reduced phase space
of this system decomposes into connected components of orbit type subsets. To
investigate hypothetical quantum effects of this decomposition one has to
construct the associated costratification of the Hilbert space of the quantum
system in the sense of Huebschmann. In the present paper, instead of the
decomposition by orbit types, we consider the related decomposition by
reflection types (conjugacy classes of reflection subgroups). These two
decompositions turn out to coincide e.g. for the classical groups SU(n) and
Sp(n). We derive defining relations for reflection type subsets in terms of
irreducible characters and discuss how to obtain from that the corresponding
costratification of the Hilbert space of the system. To illustrate the method,
we give explicit results for some low rank classical groups.
