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.

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