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 = M A N 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 S L ( 2 n , R ) , which is currently prominently used via Langlands duality. In that case, we have M = S L ( n , R ) × S L ( n , R ) . We classify the induced representations implementing P = M A N . We find out and classify the reducible cases. Using our procedure, we classify the invariant differential operators in this case.