On parabolic Whittaker functions II

Sergey Oblezin

Open Mathematics (2012)

  • Volume: 10, Issue: 2, page 543-558
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Sergey Oblezin. "On parabolic Whittaker functions II." Open Mathematics 10.2 (2012): 543-558. <http://eudml.org/doc/269332>.

@article{SergeyOblezin2012,
abstract = {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.},
author = {Sergey Oblezin},
journal = {Open Mathematics},
keywords = {Quantum cohomology; Whittaker model; Stationary phase; Toric degeneration; quantum cohomology; stationary phase; toric degeneration},
language = {eng},
number = {2},
pages = {543-558},
title = {On parabolic Whittaker functions II},
url = {http://eudml.org/doc/269332},
volume = {10},
year = {2012},
}

TY - JOUR
AU - Sergey Oblezin
TI - On parabolic Whittaker functions II
JO - Open Mathematics
PY - 2012
VL - 10
IS - 2
SP - 543
EP - 558
AB - 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.
LA - eng
KW - Quantum cohomology; Whittaker model; Stationary phase; Toric degeneration; quantum cohomology; stationary phase; toric degeneration
UR - http://eudml.org/doc/269332
ER -

References

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  1. [1] Astashkevich A., Sadov V., Quantum cohomology of partial flag manifolds F n 1 , . . . , n k , Comm. Math. Phys., 1995, 170(3), 503–528 http://dx.doi.org/10.1007/BF02099147 Zbl0865.14027
  2. [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 Zbl0896.14025
  3. [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 Zbl1022.14014
  4. [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 Zbl1142.17019
  5. [5] Gerasimov A., Lebedev D., Oblezin S., Parabolic Whittaker functions and topological field theories I, Commun. Number Theory Phys., 2011, 5(1), 135–201 Zbl1252.81115
  6. [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 Zbl1267.17007
  7. [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. [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 Zbl0895.32006
  9. [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 Zbl0828.55004
  10. [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 Zbl0849.14019
  11. [11] Lustzig G., Total positivity in reductive groups, In: Lie Theory and Geometry, Progr. Math., 123, Birkhäuser, Boston, 1994, 531–568 
  12. [12] Oblezin S., On parabolic Whittaker functions, preprint available at http://arxiv.org/abs/1011.4250 

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