Invariant symmetric polynomials on a matrix Lie algebra

Let $G$ be either a real matrix Lie group $G\subseteq {\mathrm{GL}}_n(\mathbb{R})$ with real Lie algebra $\mathfrak{g}$, or a complex matrix Lie group $G\subseteq {\mathrm{GL}}_n(\mathbb{C})$ with complex Lie algebra $\mathfrak{g}$. In the real case let $K\in \{\mathbb{R},\mathbb{C}\}$ and use polynomials in real linear coordinates with coefficients in $K$. In the complex case take $K=\mathbb{C}$ and use polynomials in complex linear coordinates (with no conjugate-coordinate variables). A homogeneous degree-$k$ polynomial $P:\mathfrak{g}\to K$ is $G$-invariant when $P({\mathrm{Ad}}_{g}A)=P(A)(g\in G,A\in \mathfrak{g}),$ where the adjoint action is the one in Conjugation and the adjoint representation of a Lie group. For $k\ge 1$, its polarization is the unique symmetric multilinear map $P_k:{\mathfrak{g}}^k\to K$ with $P(A)=P_k(A,\dots ,A)\mathrm{.}$ Here multilinearity is over $\mathbb{R}$ when $\mathfrak{g}$ is a real Lie algebra and over $\mathbb{C}$ when it is a complex Lie algebra; thus a $\mathbb{C}$-valued polynomial on a real Lie algebra still has a real-multilinear polarization. A complex matrix Lie group may also be regarded as a real Lie group, in which case the underlying real Lie algebra and real-multilinear convention apply; this is also the convention for real-valued coordinate polynomials on a complex matrix Lie algebra, such as $z\mapsto \mathrm{Re}z$ on ${\mathfrak{g}\mathfrak{l}}_1(\mathbb{C})$.

The polarization can be computed by $P_k(A_1,\dots ,A_k)=\frac{1}{k!}[t_1\cdots t_k]P(t_1{A}_1+\cdots +t_k{A}_k),$ where the bracket extracts the coefficient of $t_1\cdots t_k$. A homogeneous degree-$k$ coordinate polynomial makes this coefficient symmetric and multilinear in the $A_j$; setting every $A_j=A$ gives the coefficient $k!P(A)$, and the same extraction proves uniqueness. Since $K$ has characteristic zero, division by $k!$ is valid. For $k=0$, the invariant polynomials are constants, viewed as symmetric $0$-linear forms $P_0\in K$. Finite sums of these homogeneous invariant polynomials form the invariant polynomial algebra; sums and products remain invariant because the adjoint action respects addition and multiplication of scalar values.

Polarization preserves invariance: simultaneous application of ${\mathrm{Ad}}_g$ to the arguments leaves every term in the coefficient formula unchanged. Conversely, if a symmetric $P_k$ is invariant under simultaneous adjoint action, its diagonal $A\mapsto P_k(A,\dots ,A)$ is a $G$-invariant polynomial. Differentiating that multilinear invariance along $g(t)=\exp (tX)$ gives the infinitesimal identity ${\sum }_{j=1}^k{P}_k(A_1,\dots ,[X,A_j],\dots ,A_k)=0(X\in \mathfrak{g})\mathrm{.}$ Indeed, for matrices, ${\frac{d}{dt}\mid }_0{\mathrm{Ad}}_{\exp (tX)}{A}_j=[X,A_j]$, so the chain rule gives exactly the displayed sum. For $k=0$ the sum is empty and equals zero.

For ${\mathfrak{g}\mathfrak{l}}_r(\mathbb{C})$, every coefficient of $\det (I-tA)$ is invariant under ${\mathrm{GL}}_r(\mathbb{C})$, since $I-t(gA{g}^{-1})=g(I-tA)g^{-1}$ and determinant is unchanged by conjugation. Its restriction to $\mathfrak{u}(r)$ is therefore invariant for the adjoint action of $U(r)$. For the Pfaffian, use only $\mathfrak{s}\mathfrak{o}(2m)$ with $G=SO(2m)$ and a fixed oriented orthonormal frame. For $A=(a_{ij})\in \mathfrak{s}\mathfrak{o}(2m)$, define ${\alpha }_A=\frac{1}{2}{\sum }_{i,j}{a}_{ij}{e}^i\wedge e^j$ and set $\frac{{\alpha }_A^m}{m!}=\mathrm{Pf}(A)e^1\wedge \cdots \wedge e^{2m}\mathrm{.}$ This is a homogeneous polynomial of degree $m$. If $S\in O(2m)$, the induced action on the top exterior power multiplies the oriented volume by $\det S$; hence $\mathrm{Pf}(SA{S}^{\mathsf{T}})=(\det S)\mathrm{Pf}(A)$. It is invariant under $SO(2m)$, and a reversal of orientation changes its sign. The convention gives $\mathrm{Pf}\text{\hspace{0.17em}⁣}\left(\begin{array}{cc}0& a\\ -a& 0\end{array}\right)=a$; for $m=0$ it gives the empty-matrix Pfaffian $1$.