In this chapter, we will work not with \(\mathrm{GL}(n, \mathbb{C})\) but with its compact subgroup U(n). As in the previous chapters, we will consider elements of \(\mathcal{R}_{k}\) as generalized characters on S k . If \(\mathbf{f} \in \mathcal{R}_{k}\), then \(f ={ \mathrm{ch}}^{(n)}(\mathbf{f}) \in \varLambda _{k}^{(n)}\) is a symmetric polynomial in n variables, homogeneous of weight k. Then \(\psi _{f}: \mathrm{U}(n)\longrightarrow \mathbb{C}\), defined by ( 33.6), is the function on U(n) obtained by applying f to the eigenvalues of g ∈U(n). We will denote \(\psi _{f} ={ \mathrm{Ch}}^{(n)}(\mathbf{f})\). Thus, Ch(n) maps the additive group of generalized characters on S k to the additive group of generalized characters on U(n). It extends by linearity to a map from the Hilbert space of class functions on S k to the Hilbert space of class functions on U(n).