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Positive Casimir and Central Characters of Split Real Quantum Groups

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Abstract

We describe the generalized Casimir operators and their actions on the positive representations \({\mathcal{P}_\lambda}\) of the modular double of split real quantum groups \({\mathcal{U}_{q\tilde{q}}(\mathfrak{g}_\mathbb{R})}\). We introduce the notion of virtual highest and lowest weights, and show that the central characters admit positive values for all parameters \({\lambda}\). We show that their image defines a semi-algebraic region bounded by real points of the discriminant variety independent of q, and we discuss explicit examples in the lower rank cases.

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Authors and Affiliations

  1. Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo, Kashiwa, Chiba, 277-8583, Japan

    Ivan C. H. Ip

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  1. Ivan C. H. Ip
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Correspondence to Ivan C. H. Ip.

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Communicated by N. Reshetikhin

Dedicated to the memory of my grandfather N. Y. Wong (1926–2015)

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Ip, I.C.H. Positive Casimir and Central Characters of Split Real Quantum Groups. Commun. Math. Phys. 344, 857–888 (2016). https://doi.org/10.1007/s00220-016-2639-2

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