Whittaker Vector of Deformed Virasoro Algebra and Macdonald Symmetric Functions

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.