Let $G$ be the algebraic group $\mathrm{GL}_n$ (over $\mathbb{C}$ say). We let $T$ be the diagonal torus, $B$ the Borel subgroup of upper triangular matrices and $U$ its unipotent radical. We write $B^{\mathrm{op}}$ and $U^{\mathrm{op}}$ for the opposite groups. We identify $X^\ast(T)$ with $\mathbb{Z}^n$ in the usual way and we fix the set of positive root given by $B$. If $\lambda \in X^\ast(T)$ is a weight of $T$ we extend it to $B$ by $\lambda(U)=1$ (and similarly for $B^{\mathrm{op}}$).

Let $\lambda$ be a dominant weight and consider the representation of $G$ given by $$ V_\lambda := \left\{ f \colon G \to \mathbb{A}^1 \mbox{ such that } f(b^-g)=\lambda(b^-)f(g) \mbox{ for all } (b^-,g) \in B^{\mathrm{op}} \times G \right\}, $$ where $G$ acts (on the left) on $V_\lambda$ via $(g.f)(x)=f(xg)$. Then $V_\lambda$ is (if I understand correctly the theory) the irreducible representation of $G$ with highest (w.r.t. $B$) weight $\lambda$.

Consider now this other representation $$ W_\lambda := \left\{ f \colon G \to \mathbb{A}^1 \mbox{ such that } f(gb)=\lambda(b)f(g) \mbox{ for all } (g,b) \in G \times B \right\}, $$ where $G$ acts (on the left) on $W_\lambda$ via $(g.f)(x)=f(g^{-1}x)$.

What is the highest weight of $W_\lambda$? I guess it is $\lambda'$, where, if $\lambda$ corresponds to $(\lambda_1,\ldots,\lambda_n)$ then $\lambda'$ corresponds to $(-\lambda_n,\ldots,-\lambda_1)$, but I am not sure. Can you provide an explicit isomorphism between $W_\lambda$ and $V_{\lambda'}$? Thank you!