Representation Theory over Tropical Semifield and Langlands Correspondence

TL;DR

This paper introduces elementary analogs of Whittaker functions and L-factors using symplectic volumes and tropical geometry, revealing dual representations and connections to Langlands duality and mirror symmetry.

Contribution

It constructs elementary Whittaker functions as equivariant volumes and matrix elements, linking Archimedean and non-Archimedean cases via tropical semifield limits.

Findings

Elementary Whittaker functions as equivariant volumes

Dual description as matrix elements of monoid representations

Elementary L-factors match previously known L-factors

Abstract

Recently we propose a class of infinite-dimensional integral representations of classical gl(n+1)-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(n+1)-equivariant symplectic volumes of appropriate Kahler U(n+1)-manifolds. We demonstrate that the functions thus defined have a dual description as matrix elements of representations of monoids GL(n+1,R), R being the tropical semifield. We also show that the elementary Whittaker functions can be obtained from the non-Archimedean Whittaker functions over Q_p 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 Q_1. 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.