series are required. This immediately yields the decomposition on the subrepresentations the left regular representation of the Lorentz group, and the regular representation on 3-dimensional hyperbolic space. (The former only involves principal series representations with k even and the latter only those with .) The left and right regular representation and are defined on by Now if is an element of , the operator defined by is Hilbert–Schmidt. Define a Hilbert space by where and denotes the Hilbert space of Hilbert–Schmidt operators on Then the map defined on by extends to a unitary of onto . The map satisfies the intertwining property If are in then by unitarity Thus if denotes the convolution of and and then The last two displayed formulas are usually referred to as the Plancherel formula and the Fourier inversion formula respectively. The Plancherel formula extends to all By a theorem of Jacques Dixmier and Paul Malliavin, every smooth compactly supported function on is a finite sum of convolutions of similar functions, the inversion formula holds for such . It can be extended to much wider classes of functions satisfying mild differentiability conditions. Classification of representations of The strategy followed in the classification of the irreducible infinite-dimensional representations is, in analogy to the finite-dimensional case, to assume they exist, and to investigate their properties. Thus first assume that an irreducible strongly continuous infinite-dimensional representation on a Hilbert space of is at hand. Since is a subgroup, is a representation of it as well. Each irreducible subrepresentation of is finite-dimensional, and the representation is reducible into a direct sum of irreducible finite-dimensional unitary representations of if is unitary. The steps are the following: Choose a suitable basis of common eigenvectors of and . Compute matrix elements of and . Enforce Lie algebra commutation relations. Require unitarity together with orthonormality of the basis. Step 1 One suitable choice of basis and labeling is given by If this were a finite-dimensional representation, then would correspond the lowest occurring eigenvalue of in the representation, equal to , and would correspond to the highest occurring eigenvalue, equal to . In the infinite-dimensional case, retains this meaning, but does not. For simplicity, it is assumed that a given occurs at most once in a given representation (this is the case for finite-dimensional representations), and it can be shown that the assumption is possible to avoid (with a slightly more complicated calculation) with