The for 0 < t < 2, the complementary series is defined on L 2 ( C ) for the inner product ( f , g ) t = ∬ ∬ f ( z ) g ( w ) ¯ ¯ | z − − w | 2 − − t d z d w , with the action given by π π t ( a b c d ) − − 1 f ( z ) = | c z + d | − − 2 − − t f ( a z + b c z + d ) .