been used to define . The first set of constants on the left hand side in , denotes the regular singular points of Riemann's differential equation. The second set, , are the corresponding exponents at for one of the two linearly independent solutions, and, accordingly, are exponents at for the second solution. Define an action of the Lorentz group on the set of all Riemann P-functions by first setting where are the entries in for a Lorentz transformation. Define where is a Riemann P-function. The resulting function is again a Riemann P-function. The effect of the Möbius transformation of the argument is that of shifting the poles to new locations, hence changing the critical points, but there is no change in the exponents of the differential equation the new function satisfies. The new function is expressed as where Infinite-dimensional unitary representations History The Lorentz group and its double cover also have infinite dimensional unitary representations, studied independently by , and at the instigation of Paul Dirac. This trail of development begun with where he devised matrices and necessary for description of higher spin (compare Dirac matrices), elaborated upon by , see also , and proposed precursors of the Bargmann-Wigner equations. In he proposed a concrete infinite-dimensional representation space whose elements were called expansors as a generalization of tensors. These ideas were incorporated by Harish–Chandra and expanded with expinors as an infinite-dimensional generalization of spinors in his 1947 paper. The Plancherel formula for these groups was first obtained by Gelfand and Naimark through involved calculations. The treatment was subsequently considerably simplified by and , based on an analogue for of the integration formula of Hermann Weyl for compact Lie groups. Elementary accounts of this approach can be found in and . The theory of spherical functions for the Lorentz group, required for harmonic analysis on the hyperboloid model of 3-dimensional hyperbolic space sitting in Minkowski space is considerably easier than the general theory. It only involves representations from the spherical principal series and can be treated directly, because in radial coordinates the Laplacian on the hyperboloid is equivalent to the Laplacian on This theory is discussed in , , and the posthumous text of . Principal series for SL(2, C) The principal series, or unitary principal series, are the unitary representations induced from the one-dimensional representations of the lower triangular subgroup of Since the one-dimensional representations of correspond to the representations