Two pairs (v,C) give the same limit of discrete series representation if and only if they are conjugate under the Weyl group of K. Just as for discrete series representations v gives the infinitesimal character. There are at most |WG|/|WK| limit of discrete series representations with any given infinitesimal character.
Two sets (v,C) can have the same upper bound for representation in terms of discrete series if and only if they are conjugate under the Weyl group of K. This concept relates to the infinitesimal character provided by v for discrete series representations. The maximum number of such representations is limited by the size of the Weyl group of K.