Langlands Duality and Invariant Differential Operators: the Case SL(2n+1)

Recently we started building a bridge between two cases of Langlands duality. The latter 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. We started with the case of the group ~$SL(2n)$. In the present paper we deal with the group $SL(2n+1)$. The two mentioned groups are similar, but their representation theories is rather different.