### Abstract

Let U

on (a ﬁnite covering of a big cell in) a complex connected, simply connected algebraic group

given by evaluation of regular functions at g ∈ G. Then V is a representation of

the ﬁnite–dimensional algebra U

under certain restrictions on m, Uη

a one–dimensional representation C

the space of Whittaker vectors Hom

W

called a q-W algebra.

_{ε}(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 U_{ε}(g) contains a huge commutative subalgebra isomorphic to the algebra Z_{G }of regular functionson (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*U*η_{g }(m_{−}) hasa one–dimensional representation C

_{χg}. We also prove that if V is not trivial thenthe space of Whittaker vectors Hom

_{Uηg }(m_{−}) (Cχ_{g },V) is not trivial and the algebraW

_{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 q-W algebra.

Original language | English |
---|---|

Pages (from-to) | 63-94 |

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