Relation of Whittaker functions to equivariant Gromov-Witten theory on flag varieties is discussed in
Alexander Givental, Stationary Phase Integrals, Quantum Toda Lattices, Flag Manifolds and the Mirror Conjecture, Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser., 2 vol. 180, Amer.Math.Soc.,Providence, Rhode Island, 1997, pp. 103–115; (arXiv:alg-geom/9612001)
Anton Gerasimov, Dimitri Lebedev, Sergey Oblezin, New Integral Representations of Whittaker Functions for Classical Lie Groups (arXiv:0705.2886)
Sergey Oblezin, On parabolic Whittaker functions (arXiv:1011.4250)
For relation to geometric Langlands duality see Gerasimov-Lebedev-Oblezin 09
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