Let g be a semisimple Lie algebra over an algebraically closed field k of characteristic 0. Let V be a simple finite-dimensional g-module and let y\in V be a highest weight vector. It is a classical result of B. Kostant that the algebra of functions on the closure of the orbit of y under the simply connected group which corresponds to g is quadratic (i.e. the closuree of the orbit is a quadratic cone). In the present paper we extend this result of Kostant to the case of the quantized universal enveloping algebra U_q(g). The result uses certain information about spectrum of braiding operators for U_q(g) due to Reshetikhin and Drinfeld.