These notes give a beautiful exposition of the theory of representations of the group \({\rm GL}(2,K)\), where \(K\) is a finite field. In 71 well-organized pages, the author manages to cover a remarkable amount of material clearly, concisely, and with many details. The table of contents goes like this:Preliminaries (induced representations of finite groups and the conjugacy classes of \({\rm GL}(2,K)\), etc.); The representations of \({\rm GL}(2,K)\) (inducing representations from the upper triangular subgroup, construction of the cuspidal representations of \({\rm GL}(2)\) via characters of the quadratic extension of \(K\), the small Weil group and the small reciprocity law); \(\Gamma\)-functions and Bessel functions (Whittaker models, computation of \(\Gamma\)-factors, and computation of the character table for \({\rm GL}(2,K)\)).

The reviewer heartily recommends these notes for anyone interested in either entering this research area or teaching a self-contained introduction to the theory of group representations. Although many of the proofs given exploit the fact that \(K\) is finite, in presenting the material the author definitely has in mind the current research being done in the theory of (infinite-dimensional) representations of \({\rm GL}(2)\) (and more general groups) over a local (as opposed to a finite) field \(K\).