### Abstract

We introduce a new set of q-difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebra g[ u] KR-modules [FL1] for g= A _{r}. These operators are generalizations of the Kirillov-Noumi [KN] Macdonald raising operators, in the dual q-Whittaker limit t → ∞. They form a representation of the quantum Q-system of type A, see [DFK3]. This system is a subalgebra of a quantum cluster algebra, and is also a discrete integrable system whose conserved quantities, analogous to the Casimirs of U_{q}(sl_{r} _{+} _{1}) , act as difference operators on the above family of symmetric polynomials. The characters in the special case of products of fundamental modules are class I q-Whittaker functions, or characters of level-1 Demazure modules or Weyl modules. The action of the conserved quantities on these characters gives the difference quantum Toda equations [E]. For other modules, the action of the conserved quantities is thus a generalization of these difference equations in the case of an arbitrary (higher level) tensor products of KR-modules.

Original language | English (US) |
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Pages (from-to) | 391-424 |

Number of pages | 34 |

Journal | Transformation Groups |

Volume | 23 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2018 |

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology