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We investigate the Brezin-Gross-Witten model, a tau-function of the KdVhierarchy, and its natural one-parameter deformation, the generalizedBrezin-Gross-Witten tau-function. In particular, we derive the Virasoroconstraints, which completely specify the partition function. We solve them interms of the cut-and-join operator. The Virasoro constraints lead to the loopequations, which we solve in terms of the correlation functions. Explicitexpressions for the coefficients of the tau-function and the free energy arederived, and a compact formula for the genus zero contribution is conjectured.A family of polynomial solutions of the KdV hierarchy, given by the Schurfunctions, is obtained for the half-integer values of the parameter. Thequantum spectral curve and its classical limit are discussed.

Cut-and-join description of generalized Brezin-Gross-Witten model