Representation theory of the real Gelfand order and real Harish-Chandra modules for ${\mathsf{S}\mathsf{L}}_2(\text{mathbbm}R)$

In this article we study the principal block of the category of real Harish-Chandra modules for the group ${\mathsf{S}\mathsf{L}}_2(\text{RR})$ and relate it to the category of finite dimensional modules over the so-called real Gelfand order. We describe several distinguished classes of the corresponding indecomposable representations.