In 1967 Gelfand and Ponomarev reduced the study of Harish-Chandra modules over the classical Lorentz group to the study of a tame quiver with oriented cycles. Khoroshkin carried out a similar reduction for the Lorentz groups SO(1,n) and their identity components SO0(1,n).
In the first part of my talk, I will give a brief review of Harish-Chandra's correspondence and Lie theory.
The main part will be concerned with derived Auslander-Reiten theory of Khoroshkin quivers and explicit combinatorics of their indecomposable representations. There are four combinatorial classes of indecomposable objects: usual, special, bispecial strings and bands. It turns out that these classes can also be characterized in purely homological or Lie theoretic terms.
At last, we will see examples of the Auslander-Reiten translation, the contragredient dual, projective resolutions and basic invariants of indecomposable representations for the case of SL(2,R) (that is, SO0(1,2)).
This talk is based on joint work with Igor Burban.