A centerless representation of the Virasoro algebra associated with the unitary circular ensemble

Abstract. We consider the 2-dimensional Toda lattice tau functions τn(t, s; η, θ) deforming the probabilities τn(η, θ) that a randomly chosen matrix from the unitary group U(n), for the Haar measure, has no eigenvalues within an arc (η, θ) of the unit circle. We show that these tau functions satisfy a centerless Virasoro algebra of constraints, with a boundary part in the sense of Adler, Shiota and van Moerbeke. As an application, we obtain a new derivation of a differential equation due to Tracy and Widom, satisfied by these probabilities, linking it to the Painlevé VI equation.