Exercise 2, § 7

Let $G =\mathbf{S}\mathbf{O}(2l+ 1,\mathbf{R})$, with $l\geq 1$; we use the notations of Exerc. 2 of §4. Denote by $\sigma$ the representation of G on $\mathbf{C}^{2l+1}$ obtained from the identity representation by extension of scalars. a) The $\mathbf{Z}$-module X(T) has basis $\varpi_1, . . . , \varpi_{l-1},2\varpi_l$, as well as $\varepsilon_1, . . . , \varepsilon_l$.

b) Denote by $\eta_i$ the element $e^{\varepsilon_i}+e^{-\varepsilon_i}$ of $\mathbf{Z}[X(T)]$. Show that every element of $\mathbf{Z}[X(T)]^W$ can be written as $P(\eta_1, . . . , \eta_l)$ where P is a symmetric polynomial in $l$ variables with integer coefficients.

c) Show that the representations $\wedge \wedge^r\sigma (r\leq 2l+ 1)$ are irreducible. For $r\leq l$, prove the equality

Ch($\wedge \wedge^r\sigma$ ) $=s_r(\eta_1, . . . , \eta_l) + (l-r+ 2)s_{r-2}(\eta_1, . . . , \eta_l) +\cdots +$

+ $s_{r-2k}(\eta_1, . . . , \eta_l) +\cdots$,

where the $s_k$ are the elementary symmetric polynomials in $l$ variables.

d) Show that the homomorphism $u:\mathbf{Z}[X_1, . . . ,X_l]\rightarrow R(G)$ such that $u(X_i) =$ $[\wedge \wedge^i\sigma ]$ is an isomorphism.