Representation Theory over Tropical Semifield and Langlands Duality

• Published in: Communications in mathematical physics Vol. 320; no. 2; pp. 301 - 346
• Main Authors: Gerasimov, Anton A., Lebedev, Dimitri R.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.06.2013
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Whittaker Function; Equivariant Cohomology; Toda Chain; Integral Representation; Coadjoint Orbit
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-013-1705-2

Recently we propose a class of infinite-dimensional integral representations of classical -Whittaker functions and local Archimedean local L -factors using two-dimensional topological field theory framework. The local Archimedean Langlands duality was identified in this setting with the mirror symmetry of the underlying topological field theories. In this note we introduce elementary analogs of the Whittaker functions and the Archimedean L -factors given by U ℓ+1 -equivariant symplectic volumes of appropriate Kähler U ℓ+1 -manifolds. We demonstrate that the functions thus defined have a dual description as matrix elements of representations of monoids being the tropical semifield. We also show that the elementary Whittaker functions can be obtained from the non-Archimedean Whittaker functions over by taking the formal limit p → 1. Hence the elementary special functions constructed in this way might be considered as functions over the mysterious field . The existence of two representations for the elementary Whittaker functions, one as an equivariant volume and the other as a matrix element, should be considered as a manifestation of a hypothetical elementary analog of the local Langlands duality for number fields. We would like to note that the elementary local L -factors coincide with L -factors introduced previously by Kurokawa.