A Fourier transform for the quantum Toda lattice
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Author(s)
Lonergan, Gus
Date Issued
June 5, 2018
Publisher
Springer International Publishing
Version
Author's final manuscript
Abstract
Abstract
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
$$D(T)^W$$
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
$${{\mathrm{\mathfrak {t}}}}^*$$
t
∗
descend to
$${{\mathrm{\mathfrak {t}}}}^*//W$$
t
∗
/
/
W
if they descend to
$${{\mathrm{\mathfrak {t}}}}^*/\langle s_i\rangle $$
t
∗
/
⟨
s
i
⟩
for every simple reflection
$$s_i$$
s
i
of W.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00029-018-0419-x
