Whittaker vector of deformed Virasoro algebra and Macdonald symmetric functions

TL;DR

This paper proves a conjecture providing an explicit formula for the Whittaker vector of the deformed Virasoro algebra using Macdonald symmetric functions, with connections to quantum algebra and geometric interpretation.

Contribution

It offers a rigorous proof of the explicit Whittaker vector formula involving Macdonald functions and links it to the Fock space and geometric Hilbert scheme interpretations.

Findings

Explicit formula for Whittaker vector expressed via Macdonald functions

Use of Ding-Iohara-Miki quantum algebra currents in the proof

Geometric interpretation related to Hilbert schemes

Abstract

We give a proof of Awata and Yamada's conjecture for the explicit formula of Whittaker vector of the deformed Virasoro algebra realized in the Fock space. The formula is expressed as a summation over Macdonald symmetric functions with factored coefficients. In the proof we fully use currents appearing in the Fock representation of Ding-Iohara-Miki quantum algebra. We also mention an interpretation of Whittaker vector in terms of the geometry of the Hilbert schemes of points on the affine plane.