Representation matrix elements and Clebsch–Gordan coefficients of the semisimple Lie groups

Abstract

We give a general theory of matrix elements (ME’s) of the unitary irreducible representations (UIR’s) of linear semisimple Lie groups and of reductive Lie groups. This theory connects together the following things, (1) MEUIR’s of all the representation series of a noncompact Lie group, (2) MEUIR’s of compact and noncompact forms of the same complex Lie group. The theory presented is based on the results of the theory of the principal nonunitary series representations and on a theorem which states that ME’s of the principal nonunitary series representations are entire analytic functions of continuous representation parameters. The principle of analytic continuation of Clebsch–Gordan coefficients (CGC’s) of finite dimensional representations to CGC’s of the tensor product of a finite and an infinite dimensional representation and to CGC’s of the tensor product of two infinite dimensional representations is proved. ME’s for any UIR of the group U(n) and of the group U(n,1) are obtained. The explicit expression for all CGC’s summed over the multiplicity of the irreducible representation in the tensor product decomposition is derived.