Quantum deformation of Whittaker modules and the Toda lattice.
Let \(\mathfrak{g}\) be a complex simple finite-dimensional Lie algebra. It this very interesting paper the author constructs a quantum analogue of the Whittaker model of the center of \(U(\mathfrak{g})\), which was introduced in the classical case by \textit{B. Kostant} in [Invent. Math. 48, 101--184 (1978; Zbl 0405.22013)]. As the positive part of the the quantum group \(U_h(\mathfrak{g})\) does not have nonsingular characters, the author uses a new realization of the quantum groups, developed by him in \textit{A. Sevostyanov} [Commun. Math. Phys. 204, No. 1, 1--16 (1999; Zbl 0955.17011)]. Using the Whittaker model of the center of \(U_h(\mathfrak{g})\), the author introduces quantum deformations of Whittaker modules and studies the deformed quantum Toda lattice.
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