Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 19I / a / Solution

The Mackey's theorem, or Mackey restriction formula, says that if $H,K\le G$, $W$ is a complex $K$-representation, and $D$ is a set of representatives for the double cosets $H\backslash G/K$, thenHere ${}^{x}W$ is the representation of the conjugate subgroup $xK{x}^{-1}$ defined by

For the proof, use the group-algebra modelThe partition of $G$ into double cosets gives the direct-sum decomposition of $(H,K)$-bimodulesPut $L_x=H\cap xK{x}^{-1}$. For each $x$, defineIf $\ell =xk{x}^{-1}\in L_x$, thenso $T_x$ is well defined. It is visibly $H$-equivariant and surjective, and choosing representatives for $H/L_x$ shows that both sides have dimension $[H:L_x]\dim W$; hence it is an isomorphism. Summing the $T_x$ proves the formula.

The Frobenius reciprocity theorem states that for an $H$-representation $U$ and a $G$-representation $V$,For the corresponding character of a representation, this is