De Concini-Kac filtration and Gelfand-Tsetlin characters for quantum gl_N

Details De Concini-Kac filtration and Gelfand-Tsetlin characters for quantum gl_N De Concini-Kac filtration and Gelfand-Tsetlin characters for quantum gl_N

2012-03-21

arxiv.org/abs/1203.4793v1

Mathematics

Representation Theory

18 pages

Scientific paper

It was shown by the first author and Ovsienko that the universal enveloping algebra of $\mathfrak{gl}_N$ is a Galois order, that is, it has a hidden invariant skew group structure. We extend this result to the quantized case and prove that $U_q(\mathfrak{gl}_N)$ is a Galois order over its Gelfand-Tsetlin subalgebra. This leads to a parameterization of finite families of isomorphism classes of irreducible Gelfand-Tsetlin modules for $U_q(\mathfrak{gl}_N)$ by the characters of Gelfand-Tsetlin subalgebra. In particular, any character of the Gelfand-Tsetlin subalgebra extends to an irreducible Gelfand-Tsetlin module over $U_q(\mathfrak{gl}_N)$ and, moreover, extends uniquely when such character is generic. We also obtain a proof of the fact that the Gelfand-Tsetlin subalgebra of $U_q(\mathfrak{gl}_N)$ is maximal commutative, as previously conjectured by Mazorchuk and Turowska.

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