New Exercises for Representations and Invariants of the Classical Groups

11. Assume that (ρ, V ) is an irreducible regular representation of the linear algebraic group G. Fix v∗ ∈ V ∗ with v∗ 6= 0. For v ∈ V let φv ∈ Aff(G) be the representative function φv(g) = 〈v∗, ρ(g)v〉. Let E = {φv : v ∈ V } and let T : V → E be the map Tv = φv. Prove that T is a bijective linear map and that Tρ(g) = R(g)T for all g ∈ G, where R(g)f(x) = f(xg) for f ∈ Aff(G). Thus every irreducible regular representation of G is equivalent to a subrepresentation of (R,Aff(G)).