 Solution

The Mackey restriction formula says that for subgroups $H,K\le G$ and a $K$-representation $W$,where ${}^{x}W(xk{x}^{-1})=W(k)$.

For $H=K=B$, the Bruhat decomposition of SL2 over a finite field givesThus $B\backslash G/B$ has representatives $1,w$. The identity double coset contributes $\theta$, whilecontributes an induced representation from the diagonal subgroup. ThereforewhereFor $t(a)=\mathrm{diag}(a,a^{-1})$, one has $w^{-1}t(a)w=t(a^{-1})$.