Abstract
Let U_{ε}(g) be the standard simply connected version of the Drinfeld–Jumbo quantum group at an odd mth root of unity ε. The center of U_{ε}(g) contains a huge commutative subalgebra isomorphic to the algebra Z_{G }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 U_{ε}(g) on which Z_{G }acts according to a non–trivial character η_{g}
given by evaluation of regular functions at g ∈ G. Then V is a representation of
the ﬁnite–dimensional algebra U_{ηg}= U_{ε}(g)/U_{ε}(g)Ker η_{g}. We show that in this case,
under certain restrictions on m, Uη_{g} contains a subalgebra Uη_{g }(m_{−}) of dimension m^{1/2 dim O}, where ^{O} is the conjugacy class of g, and Uη_{g }(m_{−}) has
a one–dimensional representation C_{χg}. We also prove that if V is not trivial then
the space of Whittaker vectors Hom_{Uηg }(m_{−}) (Cχ_{g },V) is not trivial and the algebra
W_{ηg }=End_{Uηg }(U_{ηg} ⊗_{Uηg} (m_{−}) C_{χg}) naturally acts on it which gives rise to a Schur–type duality between representations of the algebra U_{ηg }and of the algebra W_{ηg}
called a qW algebra.
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 U_{ε}(g) on which Z_{G }acts according to a non–trivial character η_{g}
given by evaluation of regular functions at g ∈ G. Then V is a representation of
the ﬁnite–dimensional algebra U_{ηg}= U_{ε}(g)/U_{ε}(g)Ker η_{g}. We show that in this case,
under certain restrictions on m, Uη_{g} contains a subalgebra Uη_{g }(m_{−}) of dimension m^{1/2 dim O}, where ^{O} is the conjugacy class of g, and Uη_{g }(m_{−}) has
a one–dimensional representation C_{χg}. We also prove that if V is not trivial then
the space of Whittaker vectors Hom_{Uηg }(m_{−}) (Cχ_{g },V) is not trivial and the algebra
W_{ηg }=End_{Uηg }(U_{ηg} ⊗_{Uηg} (m_{−}) C_{χg}) naturally acts on it which gives rise to a Schur–type duality between representations of the algebra U_{ηg }and of the algebra W_{ηg}
called a qW algebra.
Original language  English 

Pages (fromto)  6394 
Number of pages  32 
Journal  Journal of Algebra 
Volume  511 
Early online date  20 Jun 2018 
DOIs  
Publication status  Published  1 Oct 2018 
Keywords
 quantum group
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Alexey Sevostyanov
Person: Academic