A new class of infinite dimensional representations of the Yangians Y (g) and Y (b) corresponding to a complex semisimple algebra g and its Borel subalgebra b ⊂ g is constructed. It is based on the generalization of the Drinfeld realization of Y (g), g = gl(N ) in terms of quantum minors to the case of an arbitrary semisimple Lie algebra g. The Poisson geometry associated with the constructed representations is described. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of G-monopoles defined as the components of the space of based maps of P 1 into the generalized flag manifold X = G/B. Thus the constructed representations of the Yangian may be considered as a quantization of the moduli space of the monopoles.

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

We identify q-deformed gl ℓ+1 -Whittaker functions with a specialization of Macdonald polynomials. This provides a representation of q-deformed gl ℓ+1 -Whittaker functions in terms of Demazure characters of affine Lie algebra gl ℓ+1 . We also define a system of dual Hamiltonians for q-deformed gl ℓ+1 -Toda chains and give a new integral representation for q-deformed gl ℓ+1 -Whittaker functions. Finally an expression of q-deformed gl ℓ+1 -Whittaker function as a matrix element of a quantum torus algebra is derived.

Abstract:In this paper we introduce Baxter integral Q-operators for finite-dimensional Lie algebras gl +1 and so 2 +1 . Whittaker functions corresponding to these algebras are eigenfunctions of the Q-operators with the eigenvalues expressed in terms of Gammafunctions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G = G L( + 1) proved earlier by Stade. We also identify eigenvalues of the Baxter Q-operator acting on Whittaker functions with local Archimedean L-factors.