If G is unimodular, an irreducible unitary representation ρ of G is in the discrete series if and only if one (and hence all) matrix coefficient with v, w non-zero vectors is square-integrable on G, with respect to Haar measure.
Harish-Chandra's classification of the discrete series representations of a semisimple connected Lie group is given as follows.
In the case where G is not compact, the representations have infinite dimension, and the notion of character is therefore more subtle to define since it is a Schwartz distribution (represented by a locally integrable function), with singularities.
Harish-Chandra's regularity theorem implies that the character of a discrete series representation is a locally integrable function on the group.
There is such a representation for every pair (v,C) where v is a vector of L + ρ orthogonal to some root of G but not orthogonal to any root of K corresponding to a wall of C, and C is a Weyl chamber of G containing v. (In the case of discrete series representations there is only one Weyl chamber containing v so it is not necessary to include it explicitly.)