Mathematics Colloquia and Seminars

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Shift of argument subalgebras is a family of maximal commutative subalgebras in the universal enveloping algebra U(g) parametrized by regular elements of the Cartan subalgebra of g. According to Vinberg, the Gelfand-Tsetlin subalgebra in U(gl_n) is a limit case of such family, so one can regard the eigenbases for such commutative subalgebras in finite-dimensional g-modules as a deformation of the Gelfand-Tsetlin basis (which is more general than Gelfand-Tsetlin bases themselves because exists for arbitrary semisimple Lie algebra g). I will define a natural structure of a Kashiwara crystal on the spectra of the shift of argument subalgebras of U(g) in finite-dimensional g-modules. This gives a topological description of the inner cactus group action on a g-crystal, as a monodromy of an appropriate covering of the De Concini-Procesi closure of the complement of the root hyperplane arrangement in the Cartan subalgebra. In particular, this gives a topological description of the Berenstein-Kirillov group (generated by Bender-Knuth involutions on the Gelfand-Tsetlin polytope) and of its relation to the cactus group due to Chmutov, Glick and Pylyavskyy.