I'm working in my research with the infinite dimensional (admissible) irreducible representations of $\mathrm{SL}(2,\mathbb{C})$ introduced by Harish-Chandra in his paper ["Infinite Irreducible Representations of the Lorentz Group"](http://www.jstor.org/stable/97833). I'm interested in particular in the *non-unitary* ones.


As the paper was written before Harish-Chandra switched to mathematics, it is not completely rigorous (as he later admitted himself). In particular, although one can show that these representations are admissible $(\mathfrak{g},K)$-modules, it is not clear that they integrate to representations of the group. Does anyone know of a reference on the treatment of these representations in the language of $(\mathfrak{g},K)$-modules (which Harish-Chandra hadn't introduced yet) which addresses my concerns?



Infinite-dimensional admissible representations of SL(2,C)
Though I'm not at all a specialist in this area, I'd certainly urge you to look into some of the relatively modern mathematical textbooks. For example, David Vogan's 1981 book *Representations of Real Reductive Lie Groups* (Progress in Mathematics, Birkhauser, Boston) might still be a useful source even though he focuses mostly on the real groups and uses your group over $\mathbb{R}$ as an example in his first chapter. At any rate, he does adopt the language of $(\mathfrak{g},K)$-modules throughout. (The book looks much larger than it really is, due to being a photocopy of typescript in the pre-TeX manner.)


There are also useful books by Knapp, in particular his (also large) book *Representation Theory of Semisimple Groups: An Overview Based on Examples*, Princeton, 1986. He frequently uses the rank 1 groups over $\mathbb{R}$ and $\mathbb{C}$ as examples along the way. 


There is also a somewhat idiosyncratic book by Lang based on a course he gave and dealing with $\mathrm{SL}\_2(\mathbb{R})$ (1975, republished by Springer in 1985). The experts don't seem to find Lang's approach attractive, however. But one drawback to most of the textbook literature is the tendency to be very general, which might make it hard to extract your special case. 


