of the diagonal matrices, with non-zero complex entries and , they thus have the form for an integer, real and with . The representations are irreducible; the only repetitions, i.e. isomorphisms of representations, occur when is replaced by . By definition the representations are realized on sections of line bundles on which is isomorphic to the Riemann sphere. When , these representations constitute the so-called spherical principal series. The restriction of a principal series to the maximal compact subgroup of can also be realized as an induced representation of using the identification , where is the maximal torus in consisting of diagonal matrices with . It is the representation induced from the 1-dimensional representation , and is independent of . By Frobenius reciprocity, on they decompose as a direct sum of the irreducible representations of with dimensions with a non-negative integer. Using the identification between the Riemann sphere minus a point and the principal series can be defined directly on by the formula Irreducibility can be checked in a variety of ways: The representation is already irreducible on . This can be seen directly, but is also a special case of general results on irreducibility of induced representations due to François Bruhat and George Mackey, relying on the Bruhat decomposition where is the Weyl group element . The action of the Lie algebra of can be computed on the algebraic direct sum of the irreducible subspaces of can be computed explicitly and the it can be verified directly that the lowest-dimensional subspace generates this direct sum as a -module. Complementary series for The for , the complementary series is defined on for the inner product with the action given by The representations in the complementary series are irreducible and pairwise non-isomorphic. As a representation of , each is isomorphic to the Hilbert space direct sum of all the odd dimensional irreducible representations of . Irreducibility can be proved by analyzing the action of on the algebraic sum of these subspaces or directly without using the Lie algebra. Plancherel theorem for SL(2, C) The only irreducible unitary representations of are the principal series, the complementary series and the trivial representation. Since acts as on the principal series and trivially on the remainder, these will give all the irreducible unitary representations of the Lorentz group, provided is taken to be even. To decompose the left regular representation of on only the principal