matrices). This is the setting in which the Peter–Weyl theorem and the Borel–Weil theorem are formulated. The former demonstrates the existence of a Fourier decomposition of functions on a compact group into characters of finite-dimensional representations. The latter theorem, providing more explicit representations, makes use of the unitarian trick to yield representations of complex non-compact groups, e.g. The following exemplifies action of the Lorentz group and the rotation subgroup on some function spaces. Euclidean rotations The subgroup of three-dimensional Euclidean rotations has an infinite-dimensional representation on the Hilbert space where are the spherical harmonics. An arbitrary square integrable function one the unit sphere can be expressed as where the are generalized Fourier coefficients. The Lorentz group action restricts to that of and is expressed as where the are obtained from the representatives of odd dimension of the generators of rotation. The Möbius group The identity component of the Lorentz group is isomorphic to the Möbius group . This group can be thought of as conformal mappings of either the complex plane or, via stereographic projection, the Riemann sphere. In this way, the Lorentz group itself can be thought of as acting conformally on the complex plane or on the Riemann sphere. In the plane, a Möbius transformation characterized by the complex numbers acts on the plane according to and can be represented by complex matrices since multiplication by a nonzero complex scalar does not change . These are elements of and are unique up to a sign (since give the same ), hence The Riemann P-functions The Riemann P-functions, solutions of Riemann's differential equation, are an example of a set of functions that transform among themselves under the action of the Lorentz group. The Riemann P-functions are expressed as where the are complex constants. The P-function on the right hand side can be expressed using standard hypergeometric functions. The connection is The set of constants in the upper row on the left hand side are the regular singular points of the Gauss' hypergeometric equation. Its exponents, i. e. solutions of the indicial equation, for expansion around the singular point are and ,corresponding to the two linearly independent solutions, and for expansion around the singular point they are and . Similarly, the exponents for are and for the two solutions. One has thus where the condition (sometimes called Riemann's identity) on the exponents of the solutions of Riemann's differential equation has