Exercise 6, § 4

Let $G$ be a finite-dimensional Lie group, $g$ its Lie algebra, $g^*$ the dual vector space of $g$, $\sigma$ the adjoint representation of $G$ on $g$ and $\rho$ the representation of $G$ on $g^*$ defined by $\rho(g) = {}^t \sigma(g)^{-1}$ for all $g \in G$.

(a) $L(\rho)(x) = -{}^t(\mathrm{ad}\ x)$ for all $x \in g$.

(b) Let $f \in g^*$. Let $G_f$ denote the stabilizer of $f$ in $G$; it is a Lie subgroup of $G$ and $g_f = L(G_f)$ is the set of $x \in G$ such that ${}^t(\mathrm{ad}\ x)f = 0$.

(c) For $x,\ y$ in $g$, we write $B_f(x, y) = f([x, y])$. Show that $B_f$ is an alternating bilinear form on $g$, that the mapping $s_f$ of $g$ into $g^*$ left associated with $B_f$ is the mapping $x \mapsto {}^t(\mathrm{ad}\ x)f$ and that the orthogonal of $g$ with respect to $B_f$ is $g_f$. Let $\beta_f$ denote the non-degenerate alternating bilinear form on $g/g_f$ (which is therefore of even dimension) derived from $B_f$ when passing to the quotient. We denote by $\beta_f'$ the inverse form of $B_f$ on $(g/g_f)^*$.

(d) Let $\Omega$ denote an orbit of $G$ on $g^*$; we give it the manifold structure derived from that on $G/G_f$ by transport of structure, where $f$ is an arbitrary point of $\Omega$. Then $G$ operates analytically on $\Omega$ on the left. Let $D$ be the associated law of infinitesimal operation. For all $a \in g$ and all $f \in \Omega$,

The tangent subspace $T_f(\Omega)$ is the image of $s_f$, that is the orthogonal of $g_f$ in $g^*$, and is canonically identified with $(g/g_f)^*$, so that $s_f$ defines a canonical isomorphism $r_f$ of $T_f(\Omega)$ onto its dual. The field $f \mapsto \beta_f'$ is an analytic differential form $\omega$ of degree 2 on $\Omega$, which is invariant under $G$. For $f \in \Omega,\ a \in g,\ b \in g,\ \omega(D_a(f), D_b(f)) = f([a, b])$.

(e) Show that $d\omega = 0$.

(f) If $\alpha$ is a differential form of degree 1 on $\Omega$, the isomorphisms $r_f$ allow us to identify $\alpha$ with a vector field. Let $A$ be the set of analytic functions on $\Omega$ with values in $K$. For $\phi, \psi$ in $A$, we write $[\phi, \psi] = \omega(d\phi, d\psi) \in A$. Show that $A$ has thus a Lie algebra structure.

(g) For all $a \in g$, let $\psi_a$ be the function $f \mapsto f(a)$ on $\Omega$. Show that $a \mapsto \psi_a$ is a homomorphism of $g$ into the Lie algebra $A$ and that the form $d\psi_a$ is identified, by means of the isomorphisms $r_f$, with the vector field $D_a$.

(h) Let $U$ be an open subset of $g^*$ and $\phi$ an analytic function on $U$. For all $f \in U$, the differential $d_f \phi$ of $\phi$ at $f$ is identified with an element of $g$. Suppose that $X\phi = 0$ for every vector field $X$ defined by the action of $G$ on $g^*$. Show that then, for all $f \in U$, $d_f \phi$ belongs to the centre of $g_f$.

(i) For all $f \in g^*$, we write $r_f = \dim g_f$. Let $r = \inf_{f \in g^*} r_f$. Show that the set $V$ of $f \in g^*$ such that $r_f = r$ is open and dense in $g^*$. Show that, for all $f \in V$, $g_f$ is commutative. (Construct $r$ functions $\phi$ satisfying the conditions of (h) in a neighbourhood of $f$ and such that their differentials at $f$ are linearly independent.)