Japan. J. Math. 10, 43--96 (2015)

Mackey's theory of $\tau$-conjugate representations for finite groups

T. Ceccherini-Silberstein, F. Scarabotti, F. Tolli

Abstract: 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 $g\mapsto g^{-1}$). 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.