Twisted Fock module of toroidal algebra via DAHA and vertex operators

Abstract

We construct the twisted Fock module of quantum toroidal ${\mathfrak{gl}}_1$ algebra with a slope $n^{\prime }/n$ using vertex operators of quantum affine ${\mathfrak{gl}}_n$. The proof is based on the $q$-wedge construction of an integrable level-one $U_q({\widehat{\mathfrak{gl}}}_n)$-module and the representation theory of double affine Hecke algebra. The results are consistent with Gorsky-Neguţ conjecture (Kononov-Smirnov theorem) on stable envelopes for Hilbert schemes of points in the plane and can be viewed as a manifestation of $({\mathfrak{gl}}_1,{\mathfrak{gl}}_n)$-duality.

References in corpus (1)

• Kernel function and quantum algebras