Using the identification between the Riemann sphere minus a point and C , the principal series can be defined directly on L 2 ( C ) by the formula π π ν ν , k ( a b c d ) − − 1 f ( z ) = | c z + d | − − 2 − − i ν ν ( c z + d | c z + d | ) − − k f ( a z + b c z + d ) .