Adjoint isotypic modules have the zero-weight rank

Write $C=S\mathrm{/}(R_{+}S)$ for the coinvariant algebra of the adjoint action. By [F1], $S_d=(R_{+}S{)}_d\oplus {\mathcal{H}}_d$ in every degree, with both summands $\mathfrak{g}$-stable; hence the quotient map restricts to an isomorphism $\mathcal{H}\to C$ of graded $\mathfrak{g}$-modules, and each ${\mathcal{H}}_d$ is finite dimensional. In particular the multiplicity series $H_E(q):={\sum }_{d\ge 0}{\dim }_{\mathbb{C}}{\mathrm{Hom}}_{\mathfrak{g}}(E,{\mathcal{H}}_d)q^d$ has nonnegative integer coefficients.

By [F2] the ring $R$ is a polynomial algebra $\mathbb{C}[p_1,\dots ,p_r]$ on homogeneous generators $p_i$ of degrees $d_i$, and the Hilbert series $\mathrm{Hilb}(R,q)={\prod }_i(1-q^{d_i}{)}^{-1}$ arises from the graded algebra structure of $R$ on $S$. Also ${\prod }_i{d}_i=\mathrm{\mid }W\mathrm{\mid }$.

Since $R$ acts on $S$ by $R$-linear maps commuting with $\mathfrak{g}$ and acts trivially on $E$, evaluating on $E$ gives an isomorphism of graded $R$-modules ${\mathrm{Hom}}_{\mathfrak{g}}(E,S)\cong R{\otimes }_{\mathbb{C}}{\mathrm{Hom}}_{\mathfrak{g}}(E,\mathcal{H})$. Indeed $R\otimes \mathcal{H}$ is free over $R$ on $\mathcal{H}$ and the adjoint action is $R$-linear, so ${\mathrm{Hom}}_{\mathbb{C}}(E,R\otimes \mathcal{H})\cong R\otimes {\mathrm{Hom}}_{\mathbb{C}}(E,\mathcal{H})$ as $R$-modules and taking $\mathfrak{g}$-invariants commutes with the graded direct sum. Consequently ${\mathrm{Hom}}_{\mathfrak{g}}(E,S)$ is a free graded $R$-module; it has finite rank exactly when ${\mathrm{Hom}}_{\mathfrak{g}}(E,\mathcal{H})$ is finite dimensional, and then its rank is ${\dim }_{\mathbb{C}}{\mathrm{Hom}}_{\mathfrak{g}}(E,\mathcal{H})=H_E(1)$.

The graded character of $S$ is $\mathrm{ch}S={\prod }_{\alpha \in \mathrm{\Phi }}(1-q{e}^{\alpha }{)}^{-1}\cdot (1-q{)}^{-r}$: the weights of $\mathfrak{g}$ under the adjoint action are the roots $\alpha \in \mathrm{\Phi }$ together with $0$ with multiplicity $r$, and the symmetric algebra on a finite-dimensional module with weights $\mu$ has character ${\prod }_{\mu }(1-q{e}^{\mu }{)}^{-1}$ in the completed character ring, the geometric series being interpreted coefficientwise in $\mathbb{C}[P][[q]]$, whose coefficients have finite support.

The simple characters are orthonormal for the constant-term form $⟨\varphi ,\psi ⟩=\frac{1}{\mathrm{\mid }W\mathrm{\mid }}\mathrm{CT}(\varphi {\psi }^{\ast }{\prod }_{\alpha \in \mathrm{\Phi }}(1-e^{\alpha })),{\psi }^{\ast }={\sum }_{\mu }(\dim {\psi }_{\mu })e^{-\mu },$ where $\mathrm{CT}$ takes the coefficient of $e^0$: by [F4], ${\prod }_{\alpha \in \mathrm{\Phi }}(1-e^{\alpha })=A(\rho )A(\rho {)}^{\ast }$, so the pairing of ${\chi }_E$ and ${\chi }_F$ equals $\frac{1}{\mathrm{\mid }W\mathrm{\mid }}\mathrm{CT}(A(E+\rho )A(F+\rho {)}^{\ast })$, and the constant term selects the pairs with $w(E+\rho )=w^{\mathrm{\prime }}(F+\rho )$; each dominant orbit has exactly $\mathrm{\mid }W\mathrm{\mid }$ terms and the signs multiply to $1$, so the value is ${\delta }_{EF}$.

Realize constant terms as matrix coefficients on ${\mathrm{\ell }}^2(P)$, $P$ the weight lattice, with orthonormal basis ${\delta }_{\mu }$: identify $e^{\mu }$ with ${\delta }_{-\mu }$ and put $U_{\alpha }{\delta }_{\mu }={\delta }_{\mu -\alpha }$, so multiplication by $e^{\alpha }$ corresponds to $U_{\alpha }$. For $0\le q<1$ the operators $F_{\alpha }(q)=(I-U_{\alpha })(I-q{U}_{\alpha }{)}^{-1}$ are bounded and commute, and $F_{-\alpha }(q)=F_{\alpha }(q{)}^{\ast }$; hence $T(q)={\prod }_{\alpha \in \mathrm{\Phi }}{F}_{\alpha }(q)={\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}{F}_{\alpha }(q)F_{\alpha }(q{)}^{\ast }$ is self-adjoint, and $\mathrm{\parallel }{F}_{\alpha }(q)\mathrm{\parallel }\le 2$ for $q\ge \frac{1}{2}$: the inverse is the norm-convergent Neumann series ${\sum }_{n\ge 0}{q}^n{U}_{\alpha }^n$, and for every vector $v$, $\mathrm{\parallel }(I-q{U}_{\alpha })v{\mathrm{\parallel }}^2-q\mathrm{\parallel }(I-U_{\alpha })v{\mathrm{\parallel }}^2=(1-q{)}^2\mathrm{\parallel }v{\mathrm{\parallel }}^2\ge 0$. Applying this inequality to $(I-q{U}_{\alpha }{)}^{-1}v$ gives $\mathrm{\parallel }{F}_{\alpha }(q)\mathrm{\parallel }\le q^{-1\mathrm{/}2}\le 2$. Moreover $(F_{\alpha }(q)-I){\delta }_0=-(1-q){\sum }_{n\ge 0}{q}^n{\delta }_{-(n+1)\alpha }$ has squared norm $\frac{1-q}{1+q}\to 0$, so expanding ${\prod }_{\alpha }{F}_{\alpha }(q)-I$ as a telescoping sum with one factor $F_{\alpha }(q)-I$ per term and all other factors bounded by $2$ gives $\mathrm{\parallel }T(q){\delta }_0-{\delta }_0\mathrm{\parallel }\to 0$ as $q\to 1^{-}$.

By [F3] each finite-dimensional ${\mathcal{H}}_d$ decomposes into isotypic components, so $\mathrm{ch}\mathcal{H}={\sum }_E{H}_E(q){\chi }_E$ in the completed character ring, the sum over isomorphism classes of finite-dimensional simple $\mathfrak{g}$-modules; both sides are formal power series whose coefficients are finite characters.

Since $R$ has trivial $\mathfrak{g}$-action, the decomposition $S=R\otimes \mathcal{H}$ of [F1] tensors characters: $\mathrm{ch}S=\mathrm{Hilb}(R,q)\cdot \mathrm{ch}\mathcal{H}$. Combining with steps 1.2, 1.4 and 1.7, $\mathrm{ch}\mathcal{H}={\prod }_i[d_i{]}_q\cdot {\prod }_{\alpha \in \mathrm{\Phi }}(1-q{e}^{\alpha }{)}^{-1},[d_i{]}_q=1+q+\cdots +q^{d_i-1}\mathrm{.}$

Interpreting a finitely supported character as the finite combination of shifts $\psi \mapsto {\sum }_{\mu }{\psi }_{\mu }{U}_{\mu }$ identifies the constant term of step 1.6 with a matrix coefficient: $\mathrm{CT}\left(\psi {\prod }_{\alpha \in \mathrm{\Phi }}\frac{1-e^{\alpha }}{1-q{e}^{\alpha }}\right)=⟨\psi T(q){\delta }_0,{\delta }_0⟩\mathrm{.}$

Applying the orthonormality of step 1.5 to the expansion 2.1 gives, for each finite-dimensional simple $E$, with ${\chi }_E^{\ast }$ as in step 1.5 and the matrix-coefficient form of step 2.2, $H_E(q)={\prod }_i[d_i{]}_q\cdot \frac{1}{\mathrm{\mid }W\mathrm{\mid }}⟨{\chi }_E^{\ast }T(q){\delta }_0,{\delta }_0⟩\mathrm{.}$ This identity is coefficientwise in $q$; for $\mathrm{\mid }q\mathrm{\mid }<1$ the finitely many Neumann series converge in operator norm, so the constant-term power series also converges to the displayed matrix coefficient.

Letting $q\to 1^{-}$ in step 3.1, the matrix coefficient tends to $⟨{\chi }_E^{\ast }{\delta }_0,{\delta }_0⟩=\dim E_0$ by the second part of step 1.6, and ${\prod }_i[d_i{]}_q\to {\prod }_i{d}_i=\mathrm{\mid }W\mathrm{\mid }$ by step 1.2; hence ${\lim }_{q\to 1^{-}}{H}_E(q)=\dim E_0$.

The coefficients $\dim {\mathrm{Hom}}_{\mathfrak{g}}(E,{\mathcal{H}}_d)$ of $H_E(q)$ are nonnegative integers and $H_E(q)$ is increasing for $0\le q<1$; the finite limit $\dim E_0$ of step 4.1 therefore forces ${\mathrm{Hom}}_{\mathfrak{g}}(E,{\mathcal{H}}_d)=0$ for all but finitely many $d$ and ${\sum }_d\dim {\mathrm{Hom}}_{\mathfrak{g}}(E,{\mathcal{H}}_d)=\dim E_0$. Hence ${\mathrm{Hom}}_{\mathfrak{g}}(E,\mathcal{H})$ is finite dimensional of dimension $\dim E_0$, and by step 1.3 the module ${\mathrm{Hom}}_{\mathfrak{g}}(E,S)$ is a finitely generated graded free $R$-module of rank $\dim E_0$.

The identification of step 1.1 gives ${\mathrm{Hom}}_{\mathfrak{g}}(E,C)\cong {\mathrm{Hom}}_{\mathfrak{g}}(E,\mathcal{H})$ as graded vector spaces, so $\dim {\mathrm{Hom}}_{\mathfrak{g}}(E,C)=\dim E_0$; tensoring the isomorphism of step 1.3 with $R\mathrm{/}{R}_{+}=\mathbb{C}$ and using $S{\otimes }_R(R\mathrm{/}{R}_{+})=C$ shows that ${\mathrm{Hom}}_{\mathfrak{g}}(E,S){\otimes }_R(R\mathrm{/}{R}_{+})\to {\mathrm{Hom}}_{\mathfrak{g}}(E,C)$ is an isomorphism. This is the asserted compatibility with the natural quotient map $S\to C=S\mathrm{/}(R_{+}S)$.

By [F5] the symmetrization $\mathrm{sym}$ is filtered, $\mathfrak{g}$-equivariant with associated graded the identity, and the symmetrized lifts of a homogeneous basis of $\mathcal{H}$ form a $Z$-basis of $U(\mathfrak{g})$; thus $U(\mathfrak{g})=Z\cdot \mathrm{sym}(\mathcal{H})\cong Z{\otimes }_{\mathbb{C}}\mathrm{sym}(\mathcal{H})$ with $Z$ acting trivially under the adjoint action. Applying the argument of step 1.3 with $Z$ in place of $R$ and $\mathrm{sym}(\mathcal{H})$ in place of $\mathcal{H}$ gives an isomorphism of $Z$-modules ${\mathrm{Hom}}_{\mathfrak{g}}(E,U(\mathfrak{g}))\cong Z{\otimes }_{\mathbb{C}}{\mathrm{Hom}}_{\mathfrak{g}}(E,\mathcal{H})$; by step 5.1 this is a free $Z$-module of rank $\dim E_0$.

For every central character with kernel $\mathfrak{m}\subseteq Z$ the reduction satisfies ${\mathrm{Hom}}_{\mathfrak{g}}(E,U(\mathfrak{g}){\otimes }_{Z}Z\mathrm{/}\mathfrak{m})\cong (Z\mathrm{/}\mathfrak{m}){\otimes }_{\mathbb{C}}{\mathrm{Hom}}_{\mathfrak{g}}(E,\mathcal{H})$, of dimension $\dim E_0$ over $Z\mathrm{/}\mathfrak{m}$; in the polynomial family the reductions ${\mathrm{Hom}}_{\mathfrak{g}}(E,S){\otimes }_{R}R\mathrm{/}\mathfrak{m}$ have the same rank $\dim E_0$, and the special case $\mathfrak{m}=R_{+}$, with $S{\otimes }_R(R\mathrm{/}{R}_{+})=C$, is step 6.1. Hence every central reduction has adjoint $E$-multiplicity $\dim E_0$.

For later use we record that the argument also applies to a finite direct sum: if $E={⨁}_j{E}_j$ is a finite direct sum of finite-dimensional modules, then $E_0={⨁}_j(E_j{)}_0$ and ${\mathrm{Hom}}_{\mathfrak{g}}(E,-)\cong {⨁}_j{\mathrm{Hom}}_{\mathfrak{g}}(E_j,-)$, so all ranks add and the conclusions hold with $\dim E_0={\sum }_j\dim (E_j{)}_0$. The Axiom of Choice is inherited from [F1], [F2] and [F5], namely from the Kostant decomposition, the Chevalley restriction and the free-over-center theorem; the multiplicity computation itself uses only finite-dimensional linear algebra, finite sums over the finite Weyl group and single root-lattice shifts.