 Determinant separation for positive-characteristic SL2

Choose a torus-invariant functional nonzero on a fixed vector, and map the representation to right-torus-invariant matrix coefficients on $S{L}_2$. These functions lie in a finite filtration by polynomials of bidegree $(n,n)$ in the two matrix columns. For $n=q-1$, all binomial coefficients are nonzero in characteristic $p$, so the invariant tensor given by the determinant to power $n$ induces an invertible map from the symmetric power's dual to itself. The filtration piece becomes its endomorphism space, with the constant function corresponding to the identity. The endomorphism determinant is an invariant degree-$q$ polynomial equal to one there; composing it with the original representation map proves geometric reductivity.