Superintegrability in $\beta$-deformed Gaussian Hermitian matrix model from W-operators

This paper is devoted to the phenomenon of superintegrability. This phenomenon is manifested in the existence of a formula for character averages, expressed through the same characters at special points and of its various generalization. In this paper we develop a method of proving such formulas from first principle from Virasoro constraints and W-representation. We apply it to prove the formula for the Jack functions averages -- appropriate analogue of characters for the $\beta$-deformed Hermitian Gaussian matrix model. We also sketch the construction of W-operators from Calogero-Ruijsenaars Hamiltonians.