We propose a Givental-type stationary phase integral representation for the restricted Grm,N-Whittaker function, which is expected to describe the (S 1×U N)-equivariant Gromov-Witten invariants of the Grassmann variety Grm,N. Our key tool is a generalization of the Whittaker model for principal series U(gl N)-modules, and its realization in the space of functions of totally positive unipotent matrices. In particular, our construction involves a representation theoretic derivation of the Batyrev-Ciocan-Fontanine-Kim-van Straten toric degeneration of Grm,N.
@article{bwmeta1.element.doi-10_2478_s11533-011-0129-5,
author = {Sergey Oblezin},
title = {On parabolic Whittaker functions II},
journal = {Open Mathematics},
volume = {10},
year = {2012},
pages = {543-558},
zbl = {1246.53117},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0129-5}
}
Sergey Oblezin. On parabolic Whittaker functions II. Open Mathematics, Tome 10 (2012) pp. 543-558. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-011-0129-5/
[1] Astashkevich A., Sadov V., Quantum cohomology of partial flag manifolds , Comm. Math. Phys., 1995, 170(3), 503–528 http://dx.doi.org/10.1007/BF02099147 | Zbl 0865.14027
[2] Batyrev V.V., Ciocan-Fontanine I., Kim B., van Straten D., Conifold transitions and mirror symmetry for Calabi-Yau complete intersections in Grassmannians, Nuclear Phys. B, 1998, 514(3), 640–666 http://dx.doi.org/10.1016/S0550-3213(98)00020-0 | Zbl 0896.14025
[3] Batyrev V.V., Ciocan-Fontanine I., Kim B., van Straten D., Mirror symmetry and toric degenerations of partial flag manifolds, Acta Math., 2000, 184(1), 1–39 http://dx.doi.org/10.1007/BF02392780 | Zbl 1022.14014
[4] Gerasimov A., Kharchev S., Lebedev D., Oblezin S., On a Gauss-Givental representation of quantum Toda chain wave function, Int. Math. Res. Not., 2006, #96489 | Zbl 1142.17019
[5] Gerasimov A., Lebedev D., Oblezin S., Parabolic Whittaker functions and topological field theories I, Commun. Number Theory Phys., 2011, 5(1), 135–201 | Zbl 1252.81115
[6] Gerasimov A., Lebedev D., Oblezin S., New integral representations of Whittaker fucntions for classical Lie groups, preprint available at http://arxiv.org/abs/0705.2886 | Zbl 1267.17007
[7] Givental A.B., Homological geometry and mirror symmetry, In: Proceedings of the International Congress of Mathematicians, Zürich, August 3–11, 1994, Birkhäuser, Basel, 1995, 472–480
[8] Givental A., Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture, In: Topics in Singularity Theory, Amer. Math. Soc. Transl. Ser. 2, 180, American Mathematical Society, Providence, 1997, 103–115 | Zbl 0895.32006
[9] Givental A., Kim B., Quantum cohomology of flag manifolds and Toda lattices, Comm. Math. Phys., 1995, 168(3), 609–641 http://dx.doi.org/10.1007/BF02101846 | Zbl 0828.55004
[10] Kim B., Quantum cohomology of partial flag manifolds and a residue formula for their intersection pairings, Internat. Math. Res. Notices, 1995, 1, 1–15 http://dx.doi.org/10.1155/S1073792895000018 | Zbl 0849.14019
[11] Lustzig G., Total positivity in reductive groups, In: Lie Theory and Geometry, Progr. Math., 123, Birkhäuser, Boston, 1994, 531–568
[12] Oblezin S., On parabolic Whittaker functions, preprint available at http://arxiv.org/abs/1011.4250