Why is Lorentz Group in 3D SL(2, R)?

Hi,



While reading "Superspace: One Thousand and One Lessons in Supersymmetry" by Gates et al. I came across the following paragraph:

Quote:

Our three-dimensional notation is as follows: In three-dimensional spacetime (with signature -++) the Lorentz group is SL(2, R) (instead of SL(2, C)) and the corresponding fundamental representation acts on a real Majorana two-component spinor [itex]\psi^\alpha = (\psi^+, \psi^-)[/itex].

Maybe I haven't understood what exactly they're trying to say here, but



1. Why is the Lorentz Group SL(2, R) instead of SL(2, C)?

2. Why is the two-component spinor real? (Well I guess this follows from question 1).



Any response will be much appreciated!



PS - The book is available freely from http://ift.tt/1AC4ZZT.



Thanks!