The unitary irreducible representations of SL¯ (3,R)

The unitary irreducible representations of the universal covering group, $\overline{\mathrm{SL}}$ (3,R), of the SL (3,R) group are analyzed by means of the methods developed by Harish−Chandra and Kihlberg. We have found a single closed expression for the matrix elements of the noncompact generators for an arbitrary unitary representation of the $\overline{\mathrm{SL}}$ (3,R) group. The irreducibility of the representations is achieved by using the little group technique and the scalar product for each irreducible Hilbert space is explicitly given. Contraction (in the Inönü and Wigner sense) of the $\overline{\mathrm{SL}}$ (3,R) unitary irreducible representations to the corresponding representations of the T₅ σ SU (2) group is discussed.