Joint eigenfunctions of the quantum Toda lattice and the quantum cohomology of the flag varieties

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We considered the quantization of the Toda lattice. Let E be 2n dimensional Euclid space. Let $F$ be the space of Fourier integrals with smooth rapidly decreasing functions. We also consider a certain dual space of F, F'. We see that F' has D-module structure. We see that E has foliation by iso level set of Toda lattice. The Fourier integrals are divided into direction of moduli space and that of iso level sets. We call the restriction of the Fourier integrals to the iso level set of Toda lattice as Radon transform. We can define Radon transform of F'. We show that if we restrict the category of the ring of differential, the D-module structure of F' is preserved. As application we show that the integrability of the quantum Toda lattice.