The algebraic integrability of the quantum toda lattice and the radon transform

We study the maximal commutative ring of partial differential operators which includes the quantum completely integrable system defined by the quantum Toda lattice. Kostant shows that the image of the generalized Harish-Chandra homomorphism of the center of the enveloping algebra is commutative (Kostant in Invent. Math. 48:101-184, 1978). We demonstrate the commutativity of the ring of partial differential operators whose principal symbols are N-invariant. Our commutative ring includes the commutative system of Kostant (Invent. Math. 48:101-184, 1978). The main tools in this paper are Fourier integral operators and Radon transforms.