Exercise 6, § 9

Let $x \rightarrow x^*$ be the involution of $k[P]$ that takes $e^p$ to $e^{-p}$ for all $p\in P$. a) Put $D =d^*d=\prod_{\alpha\in R}(1-e^{\alpha})$. Show that $d^*= (-1)^Nd$, where N = Card(R$_+$), and hence that $D = (-1)^Nd^2$.

b) Define two linear forms $\varepsilon$ and I on $k[P]$ by the formulas:

$\varepsilon (1) = 1,\varepsilon (e^p) = 0$ if $p\in P$ -- $\{0\}$

and

$I(f) =\frac{1}{m}\varepsilon (D.f)$, where $m=$ Card(W).

Show, by using the formula $d= J(e^{\rho})$, that I(1) = 1.

c) Let $\lambda \in P_{++}$ and $c_{\lambda}=$ ch $E(\lambda ) = J(e^{\lambda+\rho})/d$. Show that $I(c_{\lambda}) = 0$ if $\lambda \not= 0$. (Same method as for $b$)$.)$

d) Show that I takes integer values on the subalgebra $\mathbf{Z}[P]^W=$ ch $R(\mathfrak{g})$ of $k[P]$. If E is a $\mathfrak{g}$-module, the dimension of the space of invariants of $\mathfrak{g}$ in E is equal to I(ch E). (Reduce to the case in which E is simple and use $b$) and $c).)$ e) We have dim $E =\sum_{\lambda\in P_{++}}I(c^*_{\lambda}$ch $E)d(\lambda )$, where $d(\lambda ) =$ dim $E(\lambda )$. In particular:

$I(c^*_{\lambda}c_\mu) =\delta_{\lambda \mu}$ if $\lambda , \mu\in P_{++}$.

f) If $\lambda , \mu, \nu \in P_{++}$, the integer $m(\lambda , \mu, \nu )$ of Prop. 2 is equal to $I(c_{\lambda}c_\mu c^*_{\nu})$. Deduce the identity

(Apply $e$) to the $\mathfrak{g}$-module $E = E(\lambda )\otimes E(\mu).)$