Quantum Invariant Measures
We derive an explicit expression for the Haar integral on the quantized algebra of regular functions ℂ q [K] on the compact real form K of an arbitrary simply connected complex simple algebraic group G. This is done in terms of the irreducible ✶-representations of the Hopf ✶-algebra ℂ q [K]. Quantum analogs of the measures on the symplectic leaves of the standard Poisson structure on K which are (almost) invariant under the dressing action of the dual Poisson algebraic group K ✶ are also obtained. They are related to the notion of quantum traces for representations of Hopf algebras. As an application we define and compute explicitly quantum analogs of Harish-Chandra c-functions associated to the elements of the Weyl group of G.
KeywordsExplicit Expression Invariant Measure Algebraic Group Weyl Group Regular Function
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