Compact Lie groups: Euler constructions and generalized Dyson conjecture

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Abstract
In this paper we present a very general method to construct generalized Euler parameterizations for compact simple Lie groups w.r.t. maximally symmetrically embedded simple Lie groups. Our construction is based on a detailed analysis of the geometry of these groups, which moreover gives rise to an interesting connection with certain generalized Dyson integrals. In particular, we obtain a geometry based proof of the generalized Macdonald conjecture correspondent to the root systems associated to all irreducible symmetric spaces.

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