Langlands Duality and Invariant Differential Operators

TL;DR

The paper investigates the connection between Langlands duality and invariant differential operators within Harish-Chandra's representation theory, proposing a canonical construction for a semisimple group $G$ based on parabolic induction and generalized Verma modules. It organizes reducible elementary representations into multiplets and constructs intertwining differential operators from singular vectors using the BGG reducibility condition. The analysis specializes to $G=SL(2n,\mathbb{R})$, with detailed multiplet structures computed for $n=2$ ($sl(4)$) and $n=3$ ($sl(6)$), illustrating Knapp–Stein duality, restricted Weyl-group actions, and the transition from integral to differential intertwiners. In the real form $su(2,2)$, the case $m_1=m_3=1$ exhibits an electromagnetic duality, highlighting the physical relevance of the Langlands-dual invariant-operator picture. Overall, the work provides a representation-theoretic bridge between Langlands duality and invariant differential operators, demonstrated through explicit, duality-rich cases.

Abstract

Langlands duality is one of the most influential topics in mathematical research. It has many different appearances and influential subtopics. Yet there is a topic that until now seems unrelated to the Langlands program. That is the topic of invariant differential operators. That is strange since both items are deeply rooted in Harish-Chandra's representation theory of semisimple Lie groups. In this paper we start building the bridge between the two programs.