More precisely, Selberg's conjecture is essentially the generalized Ramanujan conjecture for the group GL 2 over the rationals at the infinite place, and says that the component at infinity of the corresponding representation is a principal series representation of GL 2 ( "'R "') ( rather than a complementary series representation ).
22.
For non-discrete irreducible representations the formal restriction of Harish-Chandra's character formula need not give the decomposition under the maximal compact subgroup : for example, for the principal series representations of SL 2 the character is identically zero on the non-singular elements of the maximal compact subgroup, but the representation is not zero on this subgroup.
