Beyond the Large N Limit: Non-linear W(infinity) as symmetry of the SL(2,R)/U(1) coset model

We show that the symmetry algebra of the $SL(2,R)_{k}/U(1)$ coset model is a non-linear deformation of $W_{\infty}$, characterized by $k$. This is a universal $W$-algebra which linearizes in the large $k$ limit and truncates to $W_{N}$ for $k=-N$. Using the theory of non-compact parafermions we construct a free field realization of the non-linear $W_{\infty}$ in terms of two bosons with background charge. The $W$-characters of all unitary $SL(2,R)/U(1)$ representations are computed. Applications to the physics of 2-d black hole backgrounds are also discussed and connections with the KP approach to $c=1$ string theory are outlined.