Exercise 2, § 6

Let V be a finite dimensional real Hilbert space over $\mathbf{R}$. Give the space $\mathbf{S}(V^*)$ of polynomials on V the scalar product defined in Topological Vector Spaces, Chap. V, §3, no. 3, formulas (13) and (14) (cf. Exerc. 1). Denote by $\gamma$ the canonical gaussian measure on V (Integration, Chap. IX, §6, nos. 4 to 6); thus, if $(x_i)_{1\leq i\leq n}$ is an orthonormal basis of $V^*$, we have $d\gamma (x) =$ $(2\pi )^{-n/2}e^{-(x|x)/2}dx_1. . . dx_n$.

a) Let $q$ be the element of $\mathbf{S}^2(V)$ that defines the scalar product on $V^*$, and let $\Delta :\mathbf{S}(V^*)\rightarrow \mathbf{S}(V^*)$ be the operator given by taking the inner product with $q($Algebra, Chap. III, §11, no. 9). Show that, for every orthonormal basis

$(x_i)_{1\leq i\leq n}$ of $V^*$ and every polynomial $P\in \mathbf{S}(V^*)$, we have $\Delta (P) =\frac{1}{2}\sum_{i=1}^n\frac{\partial^2P}{\partial x_i^2}$.

b) For $P\in \mathbf{S}(V^*)$, put $P^*= P*\gamma$, so that $P^*(x) =\int_VP(x-y)d\gamma (y)$ for

$x\in V$. Prove the equality $P^*=\sum_{n=1}^{\infty}\frac{\Delta^n(P)}{n!}=e^{\Delta}(P)$.

(Reduce to proving the analogous formula for the function $x \rightarrow e^{i(x|u)}$, for $u\in V.$)

c) Prove the formula $\int P^*(ix)Q^*(ix)d\gamma (x) = (P|Q)$ for P and Q in $\mathbf{S}(V^*)$ (identified with a subspace of $\mathbf{S}_{\mathbf{C}}((V\otimes \mathbf{C})^*)$).

d) If P is a homogeneous polynomial on V such that $\Delta (P) = 0$, then $\int_VP(x)^2d\gamma (x) = (P|P)$.

e) Let W be a finite group of automorphisms of V generated by reflections; denote by H the set of reflections in W. For $h\in H$, let $e_h\in V$ and $f_h\in V^*$ be such that $h(x) =x+f_h(x)e_h$. Show that the polynomial $P =\prod_{h\in H}f_h$ satisfies $\Delta (P) = 0$ (use Chap. V, §5, no. 4, Prop. 5).