Let $L=sl_{2}(\mathbb{C})$ be the special linear Lie algebra with basis $e,f,h$;
Let $V_{n}$ be its irreducible $(n+1)$-dimensional representation;
Finally let $Hom(V_{m},V_{n})\cong V^{*}_{m}\otimes V_{n}$ and assume $m\geq n$.
Describe the $h$-weights of $Hom(V_{m},V_{n})$ and count the dimension of the weight spaces, for weights $l\in\mathbb{Z}$ such that $m − n \leq l \leq m + n$.
I think that we can suppose to take a basis $(v_{0},...,v_{m})$ of $V_{m}$ seeing $V_{n}\subset V_{m}$ with basis $(v_{o},...,v_{n})$;
Now we know $h.v_{j}=(\lambda-2j)v_{j}$ and using the induced action on $V_{m}^{*}$ and after on $V_{m}^{*}\otimes V_{n}$ I get that the weights are of the form $2(k-j)$ with $0\leq k\leq m$ and $0\leq j\leq n$.
Maybe all its wrong and I appreciate some suggestion to how to find these weights;
Thanks very much!