Borel–Weil–Bott theorem and tensor product

Question

Let $G=\operatorname{SL}(2)$ and $V_n$ be the n+1 dimensional irreducible representation of $G$. This gives a representation for $G(O)=\operatorname{Map}(\operatorname{Spec} k[[t]], G)$ and hence an ($n + 1$)-dimensional vector bundle $E_n$ on the affine Grassmannian $\operatorname{Gr}_G=G(K)/G(O)$ where $G(K)= \operatorname{Map}(\operatorname{Spec} k((t)), G)$. By Borel–Weil–Bott theorem for loop group we know that $H^i(\operatorname{Gr}_G,E_n)$ is one dimensional for $n=2i$ and zero otherwise. We have a product $E_m\otimes E_n\longrightarrow E_{m+n}$ which induce maps on their cohomology. Are these induced maps zero for $m,n>0$?