CHAPTER VII - REPRESENTATIONS OF THE GROUP OF REAL UNIMODULAR MATRICES IN TWO DIMENSIONS

Abstract

This chapter discusses the representation theory of a group of real matrices with a determinant in two dimensions. The group of real matrices has several interesting properties that differ considerably from those of the group of complex matrices. The most important are a more complicated structure of the representations at integer points and the existence of representations acting on analytic function spaces. An analytic representation with index s = – 1, –2, … possesses the invariant functional. There exist two bilinear functionals invariant on Fs, whose kernels are the associated homogeneous generalized functions (x1 –x2)-s-1ln(x1 – x2 – i0) and (x1 – x2)-s-1ln(x1 – x2 +i0).