Bessel functions on finite groups (Q1346099)

Let \(U\), \(K\), \(G\) be finite groups, \(U\subseteq K\subseteq G\), \(\vartheta\) a linear character of \(U\), \(e_\vartheta=\vartheta(1)|U|^{-1}\sum \vartheta(u^{-1})u\) the corresponding primitive central idempotent, and let \(\psi\) be an irreducible character of \(G\). The corresponding Bessel function \(J\) on \(G\) is defined by \(J(g)=\psi (ge_\vartheta)\) and is equal to \(J(g)=|U|^{-1} \sum_u \vartheta (u^{-1}) \psi (gu)\). The authors prove the following Theorem: Let \(\varphi\) be an irreducible character of \(K\) which is induced by \(\vartheta\) and assume the restriction of \(\psi\) to \(K\) is \(\varphi\), i.e., \(\vartheta^K=\varphi=\psi_K\). Furthermore let \(g_1, \dots, g_r\) be a complete set of right coset representatives for \(U\) in \(K\) and for each \(g \in G\) define the matrix \(M(g)=(J(g_i gg^{-1}_j))\). Then \(M\) is a representation of \(G\) with character \(\psi\).