Parabolic Whittaker functions and topological field theories I

Abstract

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...

Geometry of canonical bases and mirror symmetry

“…The latter solves the quantum Toda lattice integrable system [Ko]. Therefore it provides, generalising Givental's work [Gi2] for G = GL m in non-equivariant setting, the integral presentation of the special solution of equivariant quantum cohomology D -module for the flag variety B L studied in [GiK], [Gi3], [GKLO], [GLO1]- [GLO3], [L], [R1], [R2].…”

Baxter Operator Formalism for Macdonald Polynomials

“…This interpretation leads to establishing a deep relation between topological field theories and Archimedean algebraic geometry [GLO5], [GLO6], [GLO7]. The construction of the Baxter operator formalism for the Macdonald-Ruijsenaars integrable system allows to define a (q, t)-generalization of the local Archimedean L-factors associated with principle series representations of GL ℓ+1 (qgeneralization of local L-factors was introduced previously in [GLO2]).…”

“…One can expect that these generalized L-factors shall be related with principal series representations of loop groups associated with GL ℓ+1 . Taking into account the results of [GLO5], [GLO6], [GLO7] one should look for a higher dimensional topological field theory interpretation of the Macdonald polynomials and the associated Baxter operator formalism. We are going to discuss this interpretation in the future publications.…”

On parabolic Whittaker functions II

“…, x N ) to the quantum parabolic Toda lattice was defined in [GLO2] as a certain matrix element in principal series representation, and it was referred to as parabolic Whittaker function, or Gr m,N -Whittaker function. It was conjectured in [GLO2] that the specialized parabolic Whittaker function, Ψ (m,N ) λ (x, 0, . .…”

“…, 0). Our basic tool is a generalization of the classical Whittaker model for principal series representations of U (gl N ), which extends the construction of generalized Whittaker model from [GLO2]. This result should be compared with the Mellin-Barnes integral representation for Ψ (m,N ) λ (x, 0, .…”