Exercise 5, § 7

Let $k$ be a commutative field of characteristic zero, V a vector space over $k$, and $\Psi$ a separating quadratic form on V. Let $\Gamma \in \mathbf{S}^2(V^*)$ (resp. $\Gamma^*\in \mathbf{S}^2(V)$) be the element ssociated to $\Psi$ (resp. the inverse form of $\Psi$ ). Denote by Q the endomorphism of $\mathbf{S}(V)$ given by taking the product with $\Gamma^*$, by $\Delta$ the endomorphism of $\mathbf{S}(V)$ given by taking the inner product with $\Gamma$, and by $h$ the endomorphism of $\mathbf{S}(V)$ that reduces on $\mathbf{S}^r(V)$ to multiplication by $-\frac{n}{2}-r$. a) If $(x_i)_{1\leq i\leq n}$ is an orthonormal basis of V, then $Q(P) =\frac{1}{2}(\sum x^2_i).P$ and $\Delta (P) =\frac{1}{2}\sum\frac{\partial^2P}{\partial x^{2i}}$ for $P\in \mathbf{S}(V)$.

b) Prove the formulas $[\Delta ,Q] =-h,[h, \Delta ] = 2\Delta ,[h,Q] =-2Q$.

c) Let $H_r$ be the subspace of $\mathbf{S}^r(V)$ annihilated by $\Delta$ (“harmonic homogeneous polynomials of degree $r$”). Deduce from b) that there is a direct sum decomposition, stable under $\mathbf{O}(\Psi )$, given by

$\mathbf{S}^r(V) = H_r\oplus$ QH$_{r-2}\oplus Q^2H_{r-4}\oplus \cdots$.

d) Show that the representation $H_r$ is irreducible (cf. Chap. VIII, §13, no. 3, (IV)).

e) Take $k=\mathbf{C}$, and $V =\mathbf{C}^n$ equipped with the usual quadratic form $(n\geq 3)$. We thus obtain irreducible representations $\tau_r$ of $\mathbf{S}\mathbf{O}(n,\mathbf{R})$; with the notations of Exerc. 2 of §4, show that the highest weight of $\tau_r$ is $r\varpi_1$.

f) Let $\Gamma$ be the Casimir element of G obtained from the Killing form. Prove the formula $\Gamma_{\mathbf{S}(V)}$ = $\frac{1}{2n-4}(-4Q\Delta +(H +\frac{n}{2}I)((H +(2-\frac{n}{2})I))$. Calculate $\widetilde{\Gamma}(\tau_r)$ and deduce the value of the form $Q_{\Gamma}$ (Prop. 4).