The aim of the present paper is to expose two contributions of Mackey, together with a more recent result of Kawanaka and Matsuyama, generalized by Bump and Ginzburg, on the representation theory of a finite group equipped with an involutory anti-automorphism (e.g. the anti-automorphism (Formula presented.). Mackey’s first contribution is a detailed version of the so-called Gelfand criterion for weakly symmetric Gelfand pairs. Mackey’s second contribution is a characterization of simply reducible groups (a notion introduced by Wigner). The other result is a twisted version of the Frobenius–Schur theorem, where “twisted” refers to the above-mentioned involutory anti-automorphism.
Ceccherini Silberstein, T., Scarabotti, F., Tolli, F. (2015). Mackey’s theory of τ-conjugate representations for finite groups. JAPANESE JOURNAL OF MATHEMATICS. NEW SERIES, 10(1), 43-96 [10.1007/s11537-014-1390-8].