Let G be a semisimple simply connected algebraic Lie group over complex numbers. Following Gerasimov, Lebedev and Oblezin, we use the q-Toda integrable system obtained by the quantum group version of the Kostant-Whittaker reduction to define the notion of q-Whittaker functions. This is a family of invariant polynomials on the maximal torus T in G depending on a dominant weight of G, whose coefficients are rational functions in the variable q. For a conjecturally the same (but a priori different) definition of the q-Toda system these functions were studied by I.Cherednik. For G=SL(N) these functions were extensively studied by Gerasivom, Lebedev and Oblezin. We show that when G is simply laced, the Whittaker function is equal to the character of the global Weyl module. When G is not simply laced a twisted version of the above result holds. Our proofs are algebro-geometric.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.