Exercise 3, § 7

Let $G =\mathbf{S}\mathbf{O}(2l,\mathbf{R})$, with $l\geq 2$; we use the notations of Exerc. 2 of §4. Denote by $\sigma$ the representation of G on $\mathbf{C}^{2l}$ obtained from the identity representation by extension of scalars.

a) The $\mathbf{Z}$-module X(T) has basis $\varepsilon_1, . . . , \varepsilon_l$.

b) Denote by $\eta_i$ the element $e^{\varepsilon_i}+e^{-\varepsilon_i}$ of $\mathbf{Z}[X(T)]$ and by $\delta$ the element

$\prod_{i=1}^l(e^{\varepsilon_i}-e^{-\varepsilon_i})$. Show that every element of $\mathbf{Z}[X(T)]^W$ can be written as

$P(\eta_1, . . . , \eta_l) + Q(\eta_1, . . . , \eta_l)\delta$, where P and Q are symmetric polynomials in $l$ variables with coefficients in $\frac{1}{2}\mathbf{Z}$.

c) Show that the representations $\wedge \wedge^r\sigma$ are irreducible for $r\not=l$; the representation $\wedge \wedge^l\sigma$ is the direct sum of two subrepresentations $\tau^+$ and $\tau^-$, of highest weights $2\varpi_l$ and $2\varpi_{l-1}$, respectively (see Chap. VIII, §13, Exerc. 10).

d) Show that, for $r\leq l$, the element Ch($\wedge \wedge^r\sigma$ ) is given by the formula in Exerc. $2c)$ and that

Ch($\tau^+$) $=\frac{1}{2}(\delta +$ Ch($\wedge \wedge^l\sigma$ )) Ch($\tau^-$) $=\frac{1}{2}(-\delta +$ Ch($\wedge \wedge^l\sigma$ )).

e) Show that the homomorphism $u:\mathbf{Z}[X_1, . . . ,X_l; Y]\rightarrow R(G)$ such that $u(X_i) = [\wedge \wedge^i\sigma ]$ and $u(Y) = [\tau^+]$ is surjective, and that its kernel is generated by $Y^2-$ YX$_l+ A$, with $A\in \mathbf{Z}[X_1, . . . ,X_l]$.