New integral representations of Whittaker functions for classical Lie groups

Gerasimov, A.; Лебедев, Д. В.; Oblezin, Sergey

doi:10.1070/rm2012v067n01abeh004776

Cited by 33 publications

(43 citation statements)

References 25 publications

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“…Finally, let us stress that the main driving force of the whole project including this note and the previous ones [20][21][22][23] is to uncover the proper geometric description of Archimedean places in arithmetic geometry. The results of this note imply that the infinite-dimensional symplectic geometry could be a proper setting to discuss quantum field theory models for Archimedean arithmetic geometry seriously.…”

Section: Resultsmentioning

confidence: 99%

Archimedean $L$-factors and topological field theories I

Gerasimov

Лебедев

Oblezin

2011

Communications in Number Theory and Physics

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We propose a functional integral representation for Archimedean L-factors given by products of Γ-functions. The corresponding functional integral arises in the description of type-A equivariant topological linear sigma model on a disk. The functional integral representation provides in particular an interpretation of the Γ-function as an equivariant symplectic volume of an infinitedimensional space of holomorphic maps of the disk to C. This should be considered as a mirror dual to the classical Euler integral representation of the Γ-function. We give an analogous functional integral representation of q-deformed Γ-functions using a three-dimensional equivariant topological linear sigma model on a handlebody. In general, upon proper ultra violent completion, the topological sigma model considered on a particular class of threedimensional spaces with a compact Kähler target space provides a quantum field theory description of a K-theory version of GromovWitten invariants. IntroductionArchimedean local L-factors were introduced to simplify functional equations of global L-functions. From the point of view of arithmetic geometry these factors complete the Euler product representation of global L-factors by taking into account Archimedean places of the compactified spectrum of global fields. A known construction of non-Archimedean local L-factors is rather transparent and uses characteristic polynomials of the image of the Frobenius homomorphism in finite-dimensional representations of the local Weil-Deligne group closely related to the local Galois group. On the other hand, Archimedean L-factors are expressed through products of Γ-functions and thus are analytic objects avoiding simple algebraic interpretation. Moreover, Archimedean Weil-Deligne groups are rather mysterious objects in comparison with their non-Archimedean counterparts. For instance, in the case of the field of complex numbers the corresponding Galois group is trivial while the Weil-Deligne group is isomorphic to the multiplicative group C * of 57

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Section: Resultsmentioning

confidence: 99%

Archimedean $L$-factors and topological field theories I

Gerasimov

Лебедев

Oblezin

2011

Communications in Number Theory and Physics

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“…We expect that Gr(m, + m) analogs of the correlation functions of the topological quantum field theories considered in the previous sections are given by the integral expressions (5.1). Note also that the Givental-type integral representation for Whittaker functions associated with classical groups was constructed in [15]. This provides a Landau-Ginzburg model description of the mirror dual to a type-A topological sigma models on the flag manifolds associated with the classical groups.…”

Section: Discussionmentioning

confidence: 99%

Parabolic Whittaker functions and topological field theories I

Gerasimov

Лебедев

Oblezin

2011

Communications in Number Theory and Physics

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First, we define a generalization of the standard quantum Toda chain inspired by a construction of quantum cohomology of partial flags spaces GL +1 /P , P a parabolic subgroup. Common eigenfunctions of the parabolic quantum Toda chains are generalized Whittaker functions given by matrix elements of infinite-dimensional representations of gl +1 . For maximal parabolic subgroups (i.e., for P such that GL +1 /P = P ) we construct two different representations of the corresponding parabolic Whittaker functions as correlation functions in topological quantum field theories on a two-dimensional disk. In one case the parabolic Whittaker function is given by a correlation function in a type-A equivariant topological sigma model with the target space P . In the other case, the same Whittaker function appears as a correlation function in a type-B equivariant topological Landau-Ginzburg model related with the type-A model by mirror symmetry. This note is a continuation of our project of establishing a relation between twodimensional topological field theories (and more generally topological string theories) and Archimedean (∞-adic) geometry. From this perspective the existence of two, mirror dual, topological field theory representations of the parabolic Whittaker functions provide a quantum field theory realization of the local Archimedean Langlands duality for Whittaker functions. The established relation between the Archimedean Langlands duality and mirror symmetry in two-dimensional topological quantum field theories should be considered as a main result of this note. IntroductionIn [17,18] we propose two-dimensional topological field theories as a proper framework for a description of the Archimedean completion of arithmetic schemes (∞-adic geometry according to [28]). In particular, we give a representation of local Archimedean L-factors (we include local epsilon-factor in the definition of the L-factors) in terms of two-dimensional topological field theories. It is well-known that local L-factors allow two types of 135 136 Anton Gerasimov, Dimitri Lebedev and Sergey Oblezin constructions -"arithmetic" construction based on representation theory of the Weil-Deligne group of the local field and "automorphic" construction relying on representation theory of reductive groups over local field (see, e.g., [1,6,7]). The equivalence of these constructions for various types of L-factors is a subject of the local Langlands duality. In an interpretation suggested in [17,18] the "arithmetic" construction of local Archimedean L-factors is naturally identified with a type-A topological field theory description [17] in terms of equivariant volumes of spaces of holomorphic maps of a disk into complex vector spaces. The "automorphic" construction of the same local L-factors is realized using a type-B topological field theory via periods of holomorphic forms [18]. The Archimedean Langlands duality between these two constructions of the local Archimedean L-factors appears as a mirror duality between underlying type-A and type-B topolo...

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“…In quantum cohomology and mirror symmetry, Givental [Giv97] viewed Whittaker functions as solutions to a certain integrable system, which turned out to be the quantum Toda lattice to which he constructed integral solutions via mirror symmetry. This approach was extended further for general classical groups by Gerasimov-Lebedev-Oblezin [GLO07,GLO08]. From that quantum cohomology setting further integral representations over geometric crystals have also emerged [Lam13], [Rie12].…”

Section: Whittaker Functionsmentioning

confidence: 99%

“…gl n -Whittaker functions. Following Givental [Giv97], see also [GLO07,GLO08], we introduce gl n -Whittaker functions, as integrals on triangular patterns. Let n ≥ 1, and consider a triangular array of depth n…”

Section: Whittaker Functionsmentioning

confidence: 99%

“…Similarly to the gl n case, we will define so 2n+1 -Whittaker functions as integrals of half-triangular arrays. Such definition was first given in [GLO07,GLO08] but also emerged naturally in [Nte17] in the study of a system of interacting particles via intertwining and Markovian dynamics. Let n ≥ 1, and let us consider a half-triangular array of depth 2n…”

Section: So 2n+1 -Whittaker Functionsmentioning

confidence: 99%

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Point-to-line polymers and orthogonal Whittaker functions

Bisi

Zygouras

2019

Trans. Amer. Math. Soc.

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We study a one dimensional directed polymer model in an inverse-gamma random environment, known as the log-gamma polymer, in three different geometries: point-to-line, point-to-half-line and when the polymer is restricted to a half-space with end point lying free on the corresponding half-line. Via the use of A.N.Kirillov's geometric Robinson-Schensted-Knuth correspondence, we compute the Laplace transform of the partition functions in the above geometries in terms of orthogonal Whittaker functions, thus obtaining new connections between the ubiquitous class of Whittaker functions and exactly solvable probabilistic models. In the case of the first two geometries we also provide multiple contour integral formulae for the corresponding Laplace transforms. Passing to the zero-temperature limit, we obtain new formulae for the corresponding last passage percolation problems with exponential weights.

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New integral representations of Whittaker functions for classical Lie groups

Cited by 33 publications

References 25 publications

Archimedean $L$-factors and topological field theories I

Archimedean $L$-factors and topological field theories I

Parabolic Whittaker functions and topological field theories I

Point-to-line polymers and orthogonal Whittaker functions

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