The spectral decomposition of graph adjacency matrices is an essential ingredient in the design of graph signal processing (GSP) techniques. When the adjacency matrix has multi-dimensional eigenspaces, it is desirable to base GSP constructions on a particular eigenbasis (the `preferred basis'). In this paper, we provide an explicit and detailed representation-theoretic account for the spectral decomposition of the adjacency matrix of a Cayley graph, which results in a preferred basis. Our method applies to all (not necessarily quasi-Abelian) Cayley graphs, and provides descriptions of eigenvalues and eigenvectors based on the coefficient functions of the representations of the underlying group. Next, we use such bases to build frames that are suitable for developing signal processing on Cayley graphs. These are the Frobenius--Schur frames and Cayley frames, for which we provide a characterization and a practical recipe for their construction.

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