On parabolic Whittaker functions II

• Autorzy: Sergey Oblezin
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Quantum cohomology; Whittaker model; Stationary phase; Toric degeneration

Kategorie tematyczne

• 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods
• 57T15: Homology and cohomology of homogeneous spaces of Lie groups
• 14M15: Grassmannians, Schubert varieties, flag manifolds
• 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 2
• Strony: 543-558

Daty

wydano

2012-04-01

online

2012-01-18

Twórcy

autor

Sergey Oblezin

• Institute for Theoretical and Experimental Physics

Bibliografia

• [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
• [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
• [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
• [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
• [5] Gerasimov A., Lebedev D., Oblezin S., Parabolic Whittaker functions and topological field theories I, Commun. Number Theory Phys., 2011, 5(1), 135–201
• [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
• [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
• [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
• [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
• [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

Identyfikatory

DOI

10.2478/s11533-011-0129-5