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 ).
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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.
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