Langlands Duality and Invariant Differential Operators

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 has seemed unrelated to the Langlands program. That is the topic of invariant differential operators. It 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. We first give a short review of our method of constructing invariant differential operators. A cornerstone in our program is the induction of representations from parabolic subgroups P=MAN of semisimple Lie groups. The connection to the Langlands program is through the subgroup M, which other authors use in the context of the Langlands program. Next we consider the group SL(2n,R), which is currently prominently used via Langlands duality. In that case, we have M=SL(n,R)×SL(n,R). We classify the induced representations implementing P=MAN. We find out and classify the reducible cases. Using our procedure, we classify the invariant differential operators in this case.