Representations of quantum groups at roots of unity, Whittaker vectors and q-W algebras

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Let Uε(g) be the standard simply connected version of the Drinfeld–Jumbo quantum group at an odd m-th root of unity ε. The center of Uε(g) contains a huge commutative subalgebra isomorphic to the algebra ZG of regular functions
on (a finite covering of a big cell in) a complex connected, simply connected algebraic group G with Lie algebra g. Let V be a finite–dimensional representation of Uε(g) on which ZG acts according to a non–trivial character ηg
given by evaluation of regular functions at g ∈ G. Then V is a representation of
the finite–dimensional algebra Uηg= Uε(g)/Uε(g)Ker ηg. We show that in this case,
under certain restrictions on m, Uηg contains a subalgebra Uηg (m) of dimension m1/2 dim O, where O is the conjugacy class of g, and Uηg (m) has
a one–dimensional representation Cχg. We also prove that if V is not trivial then
the space of Whittaker vectors HomUηg (m) (Cχg ,V) is not trivial and the algebra
Wηg =EndUηg (UηgUηg (m) Cχg) naturally acts on it which gives rise to a Schur–type duality between representations of the algebra Uηg and of the algebra Wηg
called a q-W algebra.
Original languageEnglish
Pages (from-to)63-94
Number of pages32
JournalJournal of Algebra
Early online date20 Jun 2018
Publication statusPublished - 1 Oct 2018


  • quantum group


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