Langlands Duality for Representations of Quantum Groups
نویسندگان
چکیده
We establish a correspondence (or duality) between the characters and the crystal bases of finite-dimensional representations of quantum groups associated to Langlands dual semi-simple Lie algebras. This duality may also be stated purely in terms of semi-simple Lie algebras. To explain this duality, we introduce an “interpolating quantum group” depending on two parameters which interpolates between a quantum group and its Langlands dual. We construct examples of its representations, depending on two parameters, which interpolate between representations of two Langlands dual quantum groups. 2000 Mathematics Subject Classification: 17B37 (17B10, 81R50).
منابع مشابه
Langlands Duality for Finite-dimensional Representations of Quantum Affine Algebras
We describe a correspondence (or duality) between the q-characters of finite-dimensional representations of a quantum affine algebra and its Langlands dual in the spirit of [6, 4]. We prove this duality for the Kirillov–Reshetikhin modules. In the course of the proof we introduce and construct “interpolating (q, t)-characters” depending on two parameters which interpolate between the q-characte...
متن کاملGauge Theory and Langlands Duality
In the late 1960s Robert Langlands launched what has become known as the Langlands Program with the ambitious goal of relating deep questions in Number Theory to Harmonic Analysis [L]. In particular, Langlands conjectured that Galois representations and motives can be described in terms of the more tangible data of automorphic representations. A striking application of this general principle is...
متن کاملA Note on Quantum Geometric Langlands Duality, Gauge Theory, and Quantization of the Moduli Space of Flat Connections
Montonen-Olive duality implies that the categories of A-branes on the moduli spaces of Higgs bundles on a Riemann surface C for groups G and G are equivalent. We reformulate this as a statement about categories of B-branes on the quantized moduli spaces of flat connections for groups GC and GC. We show that it implies the statement of the Quantum Geometric Langlands duality with a purely imagin...
متن کاملLocal Langlands Duality and a Duality of Conformal Field Theories
We show that the numerical local Langlands duality for GLn and the T-duality of twodimensional quantum gravity arise from one and the same symmetry principle. The unifying theme is that the local Fourier transform in both its `-adic and complex incarnation gives rise to symmetries of arithmetic and geometric local Langlands parameters.
متن کاملA Quantization Procedure of Fields Based on Geometric Langlands Correspondence
We expose a new procedure of quantization of fields, based on the Geometric Langlands Correspondence. Starting from fields in the target space, we first reduce them to the case of fields on one-complex-variable target space, at the same time increasing the possible symmetry group G. Use the sigma model and momentum maps, we reduce the problem to a problem of quantization of trivial vector bundl...
متن کامل