On parabolic Whittaker functions II

Oblezin, Sergey

doi:10.2478/s11533-011-0129-5

Open Mathematics

2012

DOI: 10.2478/s11533-011-0129-5

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On parabolic Whittaker functions II

Sergey Oblezin

Abstract: We derive a Givental-type stationary phase integral representation for the specified Gr m,N -Whittaker function introduced in [GLO2], which presumably describes the S 1 × U N -equivariant Gromov-Witten invariants of Grassmann variety Gr m,N . Our main tool is a generalization of Whittaker model for principal series U(gl N )-modules. In particular, our construction includes a representation theory interpretation of the Batyrev-Ciocan-Fontanine-Kim-van Straten toric degeneration of Grassmannian, providing a dire… Show more

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Cited by 2 publications

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On a matrix element representation of the GKZ hypergeometric functions

2023

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We develop a representation theory approach to the study of generalized hypergeometric functions of Gelfand, Kapranov and Zelevisnky (GKZ). We show that the GKZ hypergeometric functions may be identified with matrix elements of non-reductive Lie algebras $${\mathfrak L}_N$$ L N of oscillator type. The Whittaker functions associated with principal series representations of $$\mathfrak {gl}_{\ell +1}({\mathbb {R}})$$ gl ℓ + 1 ( R ) being special cases of GKZ hypergeometric functions thus admit along with a standard matrix element representations associated with reductive Lie algebra $$\mathfrak {gl}_{\ell +1}({\mathbb {R}})$$ gl ℓ + 1 ( R ) , another matrix element representation in terms of $${\mathfrak L}_{\ell (\ell +1)}$$ L ℓ ( ℓ + 1 ) .

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On a matrix element representation of the GKZ hypergeometric functions

2023

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The B-model connection and mirror symmetry for Grassmannians

Marsh

Rietsch

2020

Advances in Mathematics

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We consider the Grassmannian X = Gr n−k (C n ) and describe a 'mirror dual' Landau-Ginzburg model (X • , Wq :X • → C), whereX • is the complement of a particular anti-canonical divisor in a Langlands dual GrassmannianX, and we express W succinctly in terms of Plücker coordinates. First of all, we show this Landau-Ginzburg model to be isomorphic to the one proposed by the second author in [73]. Secondly we show it to be a partial compactification of the Landau-Ginzburg model defined earlier by Eguchi, Hori, and Xiong. Finally we construct inside the Gauss-Manin system associated to Wq a free submodule which recovers the trivial vector bundle with small Dubrovin connection defined out of Gromov-Witten invariants of X. We also prove a T -equivariant version of this isomorphism of connections. We conjecture that the free submodule above is isomorphic to the entire Gauss-Manin system (a tameness property for Wq). Our results imply a special case of [73, Conjecture 8.1]. They also imply [6, Conjecture 5.2.3] and resolve [67, Problem 13].

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On parabolic Whittaker functions II

Cited by 2 publications

References 10 publications

On a matrix element representation of the GKZ hypergeometric functions

On a matrix element representation of the GKZ hypergeometric functions

The B-model connection and mirror symmetry for Grassmannians

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