Representation Theory over Tropical Semifield and Langlands Correspondence

Details Representation Theory over Tropical Semifield and Langlands Correspondence Representation Theory over Tropical Semifield and Langlands Correspondence

2010-11-10

arxiv.org/abs/1011.2462v2

Mathematics

Algebraic Geometry

43 pages, Sections 6 and 7 are extended

Scientific paper

Previously a class of infinite-dimensional integral representations of the classical gl_{\ell+1}-Whittaker functions and local Archimedean local L-factors was constructed using two-dimensional topological field theory framework. The local Archimedean Langlands duality was identified in this setting with mirror symmetry of the underlying topological field theories. In this note we introduce elementary analogs of the Whittaker functions and the Archimedean L-factors identifying them with U_{\ell+1}-equivariant symplectic volumes of appropriate K\"ahler U_{\ell+1}-manifolds. We demonstrate that thus defined functions have dual descriptions as matrix elements of representations of monoids GL_{\ell+1}(\CR), \CR being the tropical semifield. The existence of two representations for the elementary Whittaker functions, one as an equivariant volume and another as a matrix element should be considered as an elementary analog of the Langlands duality. The elementary Whittaker functions can be also obtained from the non-Archimedean Whittaker functions over Q_p by taking a formal limit p\to 1. Thus the constructed elementary special functions might be considered as functions over a mysterious field Q_1. We also note that elementary version of local L-factors coincides with L-factors introduced by Kurokawa.

Affiliated with

Also associated with

No associations

Rating

Representation Theory over Tropical Semifield and Langlands Correspondence does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Representation Theory over Tropical Semifield and Langlands Correspondence, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Representation Theory over Tropical Semifield and Langlands Correspondence will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-614282