1.Let $\pi$ be an integrable representation of a Kac Moody algebra $g(A)$ on a vector space $V$. For $i = 1,2 ,...,n$ set $r_i^{\pi} = (exp fi)(exp (-ei))(exp fi)$. Then how to prove that $r_i^{\pi}(V_{\lambda}) = V_{r_i(\lambda)}$. Where $r_i$ is a fundamental reflection.
2.How to prove that $w(\alpha_i) = \alpha_j \implies w( \alpha_i^{\vee}) = \alpha_j^{\vee}$ for a $w \in W$?
First doubt is Lemma 3.8 from Kac's book on infinite dimensional Lie algebras and second doubt is statement (3.10.3) in the same book.
Thanks for your valuable time.
