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SUMMARY:Leonid Rybnikov (Harvard)
DTSTART:20230202T195000Z
DTEND:20230202T205000Z
DTSTAMP:20260423T022922Z
UID:GPRTatNU/73
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/GPRTatNU/73/
 ">Kashiwara crystals from maximal commutative subalgebras</a>\nby Leonid R
 ybnikov (Harvard) as part of Geometry\, Physics\, and Representation Theor
 y Seminar\n\n\nAbstract\nShift of argument subalgebras is a family of maxi
 mal commutative subalgebras in the universal enveloping algebra U(g) param
 etrized by regular elements of the Cartan subalgebra of a reductive Lie al
 gebra 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 b
 ases themselves because exists for arbitrary semisimple Lie algebra g). I 
 will define a natural structure of a Kashiwara crystal on the spectra of t
 he 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-P
 rocesi closure of the complement of the root hyperplane arrangement in the
  Cartan subalgebra. In particular\, this gives a topological description o
 f 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.\n
LOCATION:https://researchseminars.org/talk/GPRTatNU/73/
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