Exercise 10, § 7

Let $\Gamma$ be a subgroup of P containing Q. Such a subgroup is stable under W.

a) Show that, if $\lambda \in P_{++}\cap \Gamma$, every weight of $E(\lambda )$ belongs to $\Gamma$.

b) Let $R_{\Gamma}(\mathfrak{g})$ be the subgroup of $R(\mathfrak{g})$ with basis the $[\lambda ]$, with $\lambda \in P_{++}\cap \Gamma$. If E is a $\mathfrak{g}$-module, $[E]\in R_{\Gamma}(\mathfrak{g})$ if and only if the weights of E belong to $\Gamma$. Deduce that $R_{\Gamma}(\mathfrak{g})$ is a subring of $R(\mathfrak{g})$.

c) Show that the homomorphism ch$: R_{\Gamma}(\mathfrak{g})\rightarrow \mathbf{Z}[\Gamma ]$ is an isomorphism from $R_{\Gamma}(\mathfrak{g})$ to the subring of $\mathbf{Z}[\Gamma ]$ consisting of the elements invariant under W. (Use Th. 2 (ii).)

d) Describe $R(\mathfrak{g})$ and $R_{\Gamma}(\mathfrak{g})$ explicitly when $\mathfrak{g}=\mathfrak{s}\mathfrak{l}(2, k)$ and $\Gamma = Q$.