Exercise 11, § 7

The notations are those of no. 7. For any integer $m\geq 1$, denote by $\Psi^m$ the endomorphism of $\mathbf{Z}[\Delta ]$ that takes $e^{\lambda}$ to $e^{m\lambda}$. We have $\Psi^1=$ Id and $\Psi^m\circ \Psi^n=\Psi^{mn}$.

Let E be a finite dimensional $\Delta$-graded vector space. For all $n\geq 0$, denote by $a_nE$ (resp. $s_nE$) the $n$th exterior (resp. symmetric) power of E, equipped with its natural grading.

a) Show that

$n$ ch($s_nE$) $=\sum_{m=1}^n\Psi^m$(ch(E))ch($s_{n-m}E$)

and

$n$ ch($a_nE$) $=\sum_{m=1}^n(-1)^{m-1}\Psi^m$(ch(E))ch($a_{n-m}E$).

Deduce that ch($s_nE$) and ch($a_nE$) can be expressed as polynomials, with rational coefficients, in the $\Psi^m$(ch(E)), $1\leq m\leq n$. For example:

ch($s_2E$) $=\frac{1}{2}$ ch(E)$^2+\frac{1}{2}\Psi^2$(ch(E))

ch($a_2E$) $=\frac{1}{2}$ ch(E)$^2-\frac{1}{2}\Psi^2$(ch(E)).

b) Prove the following identities (in the algebra of formal power series in a variable T, with coefficients in $\mathbf{Q}[\Delta ]$)$:$

$\sum_{n=0}^{\infty}$ ch($s_nE$)$T^n=$ exp$\{\sum_{m=1}^{\infty}\Psi^m$(ch(E))T$^m/m\}$

and

$\sum_{n=0}^{\infty}$ ch($a_nE$)$T^n=$ exp$\{\sum_{m=1}^{\infty}(-1)^{m-1}\Psi^m$(ch(E))T$^m/m\}$.

c) Assume that $\Delta$ is a group, so that $\Psi^m$ can be defined for all $m\in \mathbf{Z}$. Show that, if $E^*$ is the graded dual of E,

ch($E^*$) $=\Psi^{-1}$(ch(E)).

d) Identify $R(\mathfrak{g})$, by means of ch, with a subring of $\mathbf{Z}[P]$. Show that $R(\mathfrak{g})$ is stable under the $\Psi^m,m\in \mathbf{Z}$, and that so is the subring $R_{\Gamma}(\mathfrak{g})$ defined in the preceding exercise.