Some q-Beta and Mellin-Barnes Integrals on Compact Lie Groups and Lie Algebras

Gustafson, Robert A.

doi:10.2307/2154615

Cited by 33 publications

(59 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Note that due to isomorphism Gr(1, 3) = Gr(2, 3) = P 2 the resulting integral expressions should be equal to (1.24) for = 2 after appropriate identification of the parameters. Below we explicitly verify this equivalence using an integral identity due to Gustafson [23].…”

Section: Appendix D: Explicit Calculations For Gr(2 3)mentioning

confidence: 92%

See 1 more Smart Citation

Parabolic Whittaker functions and topological field theories I

Gerasimov

Лебедев

Oblezin

2011

Communications in Number Theory and Physics

View full text Add to dashboard Cite

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...

show abstract

Section: Appendix D: Explicit Calculations For Gr(2 3)mentioning

confidence: 92%

“…This follows from the limiting form of the identity due to Gustafson [23] (Equation (9.4) with n = 1 and a 4 = ∞)…”

mentioning

confidence: 97%

Parabolic Whittaker functions and topological field theories I

Gerasimov

Лебедев

Oblezin

2011

Communications in Number Theory and Physics

View full text Add to dashboard Cite

show abstract

“…The paper is organized as follows: in sect. 2, after setting the notations, we prove two integral identities which are direct analogs of Gustafson integrals associated with the classical su(N ) and sp(N ) Lie algebras [2]. We analyze analytic properties of these integrals in sect.…”

Section: Introductionmentioning

confidence: 99%

On Complex Gamma-Function Integrals

Derkachov

Manashov²

2020

SIGMA

View full text Add to dashboard Cite

It was observed recently that relations between matrix elements of certain operators in the SL(2, R) spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with SL(2, C) symmetry group and L 2 (C) as a local Hilbert space give rise to a new type of Γ-function integrals. In this work we present a direct calculation of two such integrals. We also analyse properties of these integrals and show that they comprise the star-triangle relations recently discussed in the literature. It is also shown that in the quasi-classical limit these integral identities are reduced to the duality relations for Dotsenko-Fateev integrals. † For a recent progress in constructing the SoV representation for compact chains see refs. [20][21][22]

show abstract

“…Then the last section will illustrate several interesting examples, including those due to Calogero [2, §2.4.5.3], Gustafson [8,9] and Mohlenkamp-Monzón [11]. As demonstrated in these works just cited, the trigonometric identities presented in this paper may find further application in the evaluation of classical multiple hypergeometric series, trigonometric approximation and interpolation.…”

Section: Outline and Introductionmentioning

confidence: 75%

Partial fraction decompositions and trigonometric sum identities

Chu

2007

Proc. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. The partial fraction decomposition method is explored to establish several interesting trigonometric function identities, which may have applications to the evaluation of classical multiple hypergeometric series, trigonometric approximation and interpolation. Outline and introductionRecently, in an attempt to prove, through the Cauchy residue method, Dougall's theorem (Dougall [6, 1907], see [5] also) on the well-poised bilateral hypergeometric 5 H 5 -series, the author found the following trigonometric sum identity:Consider the trigonometric fraction defined byAccording to the Euler formulae2i , this rational function is essentially a fraction in e iz with the degree of the numerator polynomial less than that of the denominator polynomial by one. Therefore we can

show abstract

Some q-Beta and Mellin-Barnes Integrals on Compact Lie Groups and Lie Algebras

Cited by 33 publications

References 0 publications

Parabolic Whittaker functions and topological field theories I

Parabolic Whittaker functions and topological field theories I

On Complex Gamma-Function Integrals

Partial fraction decompositions and trigonometric sum identities

Contact Info

Product

Resources

About