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
