The Character of the Principal Series of Representations of the Real Unimodular Group

The character of the principal series of representations of SL(n,R) is evaluated by using Gel'fand and Naimark's definition of character. This representation is realized in the space of functions defined on the right coset space of SL(n,R) with respect to the subgroup of real triangular matrices. This form of the representations considerably simplifies the problem of determination of the integral kernel of the group ring which is fundamental in the Gel'fand-Naimark theory of character. An important feature of the principal series of representations is that the `elliptic' elements of SL(n,R) do not contribute to its character.