Thank you both for the reply! I think I get a sense of it now. Without spin-orbital coupling, for any k, \psi_{nk}(r) and \psi_{nk}(r)^* = \psi_{-nk}(r) are degenerate (in H). But Bloch state are simultaneous eigenstates for both H and translation T_R , and only at -k = k + G are...
How to prove that wave function at \Gamma point can always be a real function? I know it is not true for general k point, but for \Gamma and other high symmetry point like X, is there a simple proof?
Thanks!