Representation theory of the reflection equation algebra II: Theory of shapes

We continue our study of the representations of the Reflection Equation Algebra (=REA) on Hilbert spaces, focusing again on the REA constructed from the R-matrix associated to the standard q-deformation of GL(N,C) for 0<q<1. We consider the Poisson structure appearing as the classical limit of the R-matrix, and parametrize the symplectic leaves explicitly in terms of a type of matrix we call a shape matrix. We then introduce a quantized version of the shape matrix for the REA, and show that each irreducible representation of the REA has a unique shape.