17 research outputs found
Parabolic Whittaker Functions and Topological Field Theories I
First, we define a generalization of the standard quantum Toda chain inspired
by a construction of quantum cohomology of partial flags spaces GL(\ell+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(\ell+1). For maximal parabolic
subgroups (i.e. for P such that GL(\ell+1)/P=\mathbb{P}^{\ell}) 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 \mathbb{P}^{\ell}. 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
two-dimensional topological field theories (and more generally topological
string theories) and Archimedean (\infty-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.Comment: Section 1 is extended and Appendices are added, 23 page
On parabolic Whittaker functions
We derive a Mellin-Barnes integral representation for solution to generalized
(parabolic) quantum Toda lattice introduced in \cite{GLO}, which presumably
describes the -equivariant Gromov-Witten invariants of
Grassmann variety.Comment: 14 page
Normalizers of maximal tori and real forms of Lie groups
For a complex reductive Lie group Tits defined an extension of
the corresponding Weyl group . The extended group is supplied with an
embedding into the normalizer of the maximal torus such
that together with generate . We give an interpretation of
the Tits classical construction in terms of the maximal split real form
, leading to a simple topological
description of . We also propose a different extension of the
Weyl group associated with the compact real form . This results into a presentation of the normalizer of maximal
torus of the group extension by the
Galois group . We also describe explicitly
the adjoint action of and on the Lie algebra of .Comment: 17 page
Baxter operator formalism for Macdonald polynomials
We develop basic constructions of the Baxter operator formalism for the
Macdonald polynomials associated with root systems of type A. Precisely we
construct a dual pair of mutually commuting Baxter operators such that the
Macdonald polynomials are their common eigenfunctions. The dual pair of Baxter
operators is closely related to the dual pair of recursive operators for
Macdonald polynomials leading to various families of their integral
representations. We also construct the Baxter operator formalism for the
q-deformed gl(l+1)-Whittaker functions and the Jack polynomials obtained by
degenerations of the Macdonald polynomials associated with the type A_l root
system. This note provides a generalization of our previous results on the
Baxter operator formalism for the Whittaker functions. It was demonstrated
previously that Baxter operator formalism for the Whittaker functions has deep
connections with representation theory. In particular the Baxter operators
should be considered as elements of appropriate spherical Hecke algebras and
their eigenvalues are identified with local Archimedean L-factors associated
with admissible representations of reductive groups over R. We expect that the
Baxter operator formalism for the Macdonald polynomials has an interpretation
in representation theory of higher-dimensional arithmetic fields.Comment: 22 pages, typos are fixe
Normalizers of maximal tori and real forms of Lie groups
Given a complex connected reductive Lie group G with a maximal torus H⊂G, Tits defined an extension WTG of the corresponding Weyl group WG. The extended group is supplied with an embedding into the normalizer NG(H) such that WTG together with H generate NG(H). In this paper we propose an interpretation of the Tits classical construction in terms of the maximal split real form G(R)⊂G, which leads to a simple topological description of WTG. We also consider a variation of the Tits construction associated with compact real form U of G. In this case we define an extension WUG of the Weyl group WG, naturally embedded into the group extension U˜:=U⋊Γ of the compact real form U by the Galois group Γ=Gal(C/R). Generators of WUG are squared to identity as in the Weyl group WG. However, the non-trivial action of Γ by outer automorphisms requires WUG to be a non-trivial extension of WG. This gives a specific presentation of the maximal torus normalizer of the group extension U˜. Finally, we describe explicitly the adjoint action of WTG and WUG on the Lie algebra of G
On q-deformed gl(l+1)-Whittaker function II
A representation of a specialization of a q-deformed class one lattice
gl(\ell+1}-Whittaker function in terms of cohomology groups of line bundles on
the space QM_d(P^{\ell}) of quasi-maps P^1 to P^{\ell} of degree d is proposed.
For \ell=1, this provides an interpretation of non-specialized q-deformed
gl(2)-Whittaker function in terms of QM_d(\IP^1). In particular the (q-version
of) Mellin-Barnes representation of gl(2)-Whittaker function is realized as a
semi-infinite period map. The explicit form of the period map manifests an
important role of q-version of Gamma-function as a substitute of topological
genus in semi-infinite geometry. A relation with Givental-Lee universal
solution (J-function) of q-deformed gl(2)-Toda chain is also discussed.Comment: Extended version submitted in Comm. Math. Phys., 24 page