A Fourier transform for the quantum Toda lattice

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We answer a question of V. Drinfeld by constructing an ‘algebraic Fourier transform’ for the quantum Toda lattice of a complex reductive algebraic group G, which extends the classical ‘algebraic Fourier transform’ for its subalgebra $$D(T)^W$$ of Weyl group invariant differential operators on a maximal torus. The proof is contained in Sect. 2 and relies on a result of Bezrukavnikov–Finkelberg realizing the quantum Toda lattice as the equivariant homology of the dual affine Grassmannian; the Fourier transform boils down to nothing more than the duality between homology and cohomology. In Sect. 3, we compare our result with a related result of V. Ginzburg, and explain the apparent discrepancy by showing that W-equivariant quasicoherent sheaves on $${{\mathrm{\mathfrak {t}}}}^*$$ descend to $${{\mathrm{\mathfrak {t}}}}^*//W$$ if they descend to $${{\mathrm{\mathfrak {t}}}}^*/\langle s_i\rangle $$ for every simple reflection $$s_i$$ of W.