Nilpotent-orbit tensor covariants extend and polarize

The zero-dimensional case. If $V=0$, its orbit closure is a point and its boundary is empty by [F1]. For $d=0$, $V^{\otimes d}=\mathbb{C}$ and every fixed operator is a scalar, extended by the constant polynomial map; the empty permutation and empty tensor product give the asserted normal form. For $d>0$, $V^{\otimes d}=0$ and its endomorphism space has only the zero operator, extended by the zero map; the normal form is the empty sum. Thus both assertions hold without a largest Jordan block. Henceforth $V\mathrm{\ne }0$, so its largest block length $l$ exists.

Hartogs lifting. Let $A=\mathbb{C}[X]$ and let $f\in \mathrm{Frac}(A)$ be regular on $O$, hence regular off a closed subset of $X\setminus O$; since $\mathrm{\partial }X$ has codimension $\ge 2$ in $X$ by the normality lemma [F1], $f$ is regular in codimension $1$ and therefore extends to $A$ by [F2]. Applying this coordinatewise to maps into a finite-dimensional vector space shows that every regular map $O\to W$ into a finite-dimensional vector space extends uniquely to a regular map $X\to W$.

The equivariant map on $O$. Fix $B\in \mathrm{End}(V^{\otimes d}{)}^{G_e}$ and define, for $g\in \mathrm{G}\mathrm{L}(V)$, $F(ge{g}^{-1}):=(g^{\otimes d})B(g^{-1}{)}^{\otimes d}\in \mathrm{End}(V^{\otimes d})$. This is well defined: if $g_{1}e{g}_1^{-1}=g_{2}e{g}_2^{-1}$, then $g_2^{-1}{g}_1\in G_e$, so $(g_2^{-1}{g}_1{)}^{\otimes d}B(g_2^{-1}{g}_1{)}^{-\otimes d}=B$ by $G_e$-invariance, which rearranges to $(g_1^{\otimes d})B(g_1^{-1}{)}^{\otimes d}=(g_2^{\otimes d})B(g_2^{-1}{)}^{\otimes d}$. The map $F$ is $\mathrm{G}\mathrm{L}(V)$-equivariant by construction: $F(hX{h}^{-1})=(h^{\otimes d})F(X)(h^{-1}{)}^{\otimes d}$ for $X\in O$.

Regularity of $F$ on $O$. The ranks of $X^0,X^1,\dots ,X^{l-1}$ are constant on $O$. Cover $O$ by open charts where fixed minors used to select bases of the kernels and images, their quotient complements and Jordan top vectors are nonzero. Gaussian elimination on such a chart gives a Jordan basis for $X$ with entries regular in $X$. Let $g(X)$ carry a fixed Jordan basis for $e$ to this chosen Jordan basis for $X$, with the same ordered string lengths. Then $g(X)eg(X{)}^{-1}=X$. Restricting to the open chart where the Jordan-basis determinant is nonzero makes both $g(X)$ and $g(X{)}^{-1}$ regular, since their entries have only the fixed nonvanishing minors and determinant as denominators. Thus $X\mapsto (g(X{)}^{\otimes d})B(g(X{)}^{-1}{)}^{\otimes d}$ is the map of step 1.3 and is regular on each chart. These charts cover $O$; on overlaps $g_2(X{)}^{-1}{g}_1(X)\in G_e$, so the expressions agree by $G_e$-invariance of $B$. Hence $F$ is regular on $O$.

Extension to $X$. By steps 1.2 and 2.1 the map $F\text{\hspace{0.17em}⁣}:O\to \mathrm{End}(V^{\otimes d})$ is a regular equivariant map on the dense orbit; by step 1.2 it extends uniquely to a regular map $F\text{\hspace{0.17em}⁣}:X\to \mathrm{End}(V^{\otimes d})$. Since $\mathbb{C}[X]=\mathbb{C}[\mathrm{End}(V)]\mathrm{/}I(X)$ and $X$ is an affine closed subset of the affine space $\mathrm{End}(V)$, the coordinate functions of $F$ are restrictions of polynomial functions on $\mathrm{End}(V)$; fixing representatives gives a polynomial map $\mathrm{\Phi }\text{\hspace{0.17em}⁣}:\mathrm{End}(V)\to \mathrm{End}(V^{\otimes d})$ extending $F$.

Equivariance of the polynomial lift. The extension $F$ on $X$ is equivariant: for every $g$, the two regular maps expressing equivariance agree on the dense orbit by step 1.3 and therefore on $X$. Let $E_K$ be the finite-dimensional rational module of polynomial maps $operatornameEnd(V)ooperatornameEnd(V^{otimesd})$ of degree at most $K$, for $KgedegPhi$, with action $(gcdotPsi)(Y)=g^{otimesd}Psi(g^{-1}Yg)(g^{-1}{)}^{otimesd}$. Let $I_{K}subset{E}_K$ consist of maps vanishing on $X$. Then $I_K$ is stable and the class of the lift from step 3.1 in $E_K\mathrm{/}{I}_K$ is invariant. Scalar matrices act trivially on $E_K$: they conjugate neither the input nor the output. For $Nge2$, Weyl's theorem [F3] applied to the semisimple Lie algebra $mathfraksl(V)$ gives an $mathfraksl(V)$-stable complement to $I_K$. This complement is stable under $mathrmSL(V)$, since elementary matrices $exp(t{E}_{ij})$ generate that group and their rational representation acts by exponentiating the differential; together with scalar matrices they generate $mathrmGL(V)$ over $mathbbC$. For $N=1$ the whole group acts trivially, so any vector-space complement suffices. The quotient map restricted to this complement is thus a $mathrmGL(V)$-isomorphism and lifts the invariant class to an invariant $Ph{i}^{\mathrm{\prime }}in{E}_K$. It agrees with $F$ on $X$ and satisfies $Ph{i}^{\mathrm{\prime }}(e)=B$, proving (1).

Polarization. Write $\mathrm{\Phi }={\sum }_k{\mathrm{\Phi }}_k$ for its homogeneous components of degree $k$; each ${\mathrm{\Phi }}_k$ is equivariant. Fix $k$ and assume for the structure argument that $\mathrm{\Phi }$ is homogeneous of degree $k$. Polarize the $k$ matrix variables $X_1,\dots ,X_k$ of $\mathrm{\Phi }$ and pair the output in $\mathrm{End}(V^{\otimes d})$ with $d$ independent matrices $Y_1,\dots ,Y_d$ using the nondegenerate trace pairing on each tensor factor; this produces a multilinear $\mathrm{G}\mathrm{L}(V)$-invariant $P(X_1,\dots ,X_k;Y_1,\dots ,Y_d)\in \mathbb{C}$ in $(k+d)$ matrix arguments.

Tensor invariants are permutation contractions. By the Schur-Weyl description [F3], every multilinear invariant of the matrix arguments is a linear combination of contractions: each argument contributes one upper and one lower index, an invariant pairs each upper index with a lower index of another argument, and the invariant is a product over the cycles of the resulting permutation of traces of the corresponding cyclically ordered product of matrices.

Covariant normal form. Depolarize in the $k$ identical variables $X_i=X$ and read each contraction cycle. A cycle that uses only $X$-arguments contributes a scalar invariant polynomial in $X$; a cycle that uses some $Y$-arguments alternates blocks of $X$'s and single $Y$'s, so its trace is precisely the trace pairing of $Y_1\otimes \cdots \otimes Y_d$ with a place permutation of $X^{h_1}\otimes \cdots \otimes X^{h_d}$. By nondegeneracy of the trace pairing, the operator identity $\mathrm{\Phi }(X)={\sum }_h{c}_h(X){\sigma }_h(X^{h_1}\otimes \cdots \otimes X^{h_d})$ follows, with $h_i\ge 0$, ${\sigma }_h$ a place permutation and $c_h$ a scalar invariant polynomial (a polynomial in the traces $\mathrm{tr}(X^m)$); summing over the homogeneous components gives assertion (2).

Evaluation at $e$. Since $e^l=0$ for the largest Jordan block length $l$ of $e$, evaluating the normal form at $X=e$ kills every term with some $h_i\ge l$; the surviving terms are place permutations applied to a tensor product of nonnegative powers $e^{h_1}\otimes \cdots \otimes e^{h_d}$ multiplied by the scalar invariants $c_h(e)$, which are complex constants; since $\mathrm{tr}(e^m)=0$ for every positive $m$, the scalar invariant polynomials satisfy $c_h(e)=c_h(0)$. This gives the concrete description of the $G_e$-fixed endomorphisms used by the double centralizer theorem.