Mackey restriction formula

For $H,K\le G$ and a $K$-representation $V$,where ${}^{x}V(xk{x}^{-1})=V(k)$.

To prove the formula, write induction as $\mathbb{C}[G]{\otimes }_{\mathbb{C}[K]}V$ and decompose the $(H,K)$-bimoduleFor $L_x=H\cap xK{x}^{-1}$, the mapis a well-defined $H$-equivariant isomorphism. Taking the direct sum over the double cosets proves the result.

For an irreducible $H$-representation $V$, the induced representation ${\mathrm{Ind}}_H^{G}V$ is irreducible exactly when, for every nonidentity double coset representative $x\in H\backslash G/H$, the restrictions of $V$ and its $x$-conjugate to $H\cap xH{x}^{-1}$ have inner product zero.

This is the finite special-linear-group instance of the general Bruhat decomposition.

For the upper-triangular subgroup $B$ of $S{L}_2({\mathbb{F}}_p)$ andthe two double cosets are $B$ and $BwB$, and $B\cap wB{w}^{-1}=T$, the diagonal subgroup.

For one-dimensional representations $\theta ,\phi$ of the upper-triangular subgroup $B\le S{L}_2({\mathbb{F}}_p)$,where ${\theta }^w(t)=\theta (w^{-1}tw)$.

Every linear character of the upper-triangular subgroup $B\le S{L}_2({\mathbb{F}}_q)$ with $q\ge 4$ is obtained from a multiplicative character $\theta$ of ${\mathbb{F}}_q^{\times }$. Its induction to $S{L}_2({\mathbb{F}}_q)$ is irreducible exactly when ${\theta }^2\ne 1$.