2·3·1. Let $\varphi$ be the natural representation of $\mathfrak{g}$ over $Q_\chi$ , which induces a ${\unicode{x1D55C}}$ -algebras homomorphism from $Z(\mathfrak{g})$ to $(\text{End}_{{\mathfrak{g}}}Q_{\chi})^{\text{op}}$ such that

\[\begin{array}{lcll}\varphi\;:\;&Z(\mathfrak{g})&\rightarrow&(\text{End}_{\mathfrak{g}}Q_{\chi})^{\text{op}}\\ &x&\mapsto&l_x,\end{array}\]

where $l_x(1_\chi)=x.1_\chi\in Q_\chi$ for any $x\in Z(\mathfrak{g})$ . The map $\varphi$ will play a critical rule in the subsequent arguments.

Recall that the associated graded algebra of $Q_\chi$ under the Kazhdan grading is $\text{gr}(Q_\chi)=\text{gr}(S(\mathfrak{g}))/\text{gr}(I_\chi)$ , and by the PBW theorem we have that $S(\mathfrak{g})=S(\tilde{\mathfrak{p}})\oplus\text{gr}(I_\chi)$ . As $\text{pr}\;:\; S(\mathfrak{g})\rightarrow S(\tilde{\mathfrak{p}})$ is the projection along this direct sum decomposition, this restricts to an isomorphism $\text{gr}(Q_{\chi})\cong S(\tilde{\mathfrak{p}})$ .

The adjoint action of $\mathcal{M}$ as defined in section 2·1 on $Q_\chi$ descends to an adjoint action on $\text{gr}(Q_\chi)$ and this gives a twisted action of $\mathcal{M}$ on $S(\tilde{\mathfrak{p}})$ defined by $\text{tw}(g).f\;:\!=\;\text{pr}(g.f)$ for $g\in\mathcal{M}$ and $f\in S(\tilde{\mathfrak{p}})$ , where $g.f$ denotes the usual adjoint action of g on f in $S(\mathfrak{g})$ . We write $S(\tilde{\mathfrak{p}})^{\mathcal{M}}$ for the invariants with respect to this action.

2·3·2. Frobenius twist

We denote for a scheme X over ${{\unicode{x1D55C}}}$ by $X^{(1)}$ the Frobenius twist of X, which means the same scheme twisted by the Frobenius morphism, i.e., the structure sheaf is exponentiated to the pth power (see [ Reference Jantzen7 , section I·9·1] for the details). This notation will be used through the paper.

Recall that there is a homomorphism $\mathfrak{f}\;:\; S(\mathfrak{g}^{(1)})\rightarrow Z_p(\mathfrak{g})\subset U(\mathfrak{g})$ via sending $x\mapsto x^p-x^{[p]}$ for all $x\in\mathfrak{g}$ . This $\mathfrak{f}$ is a G-equivariant isomorphism, which induces a G-equivariant isomorphism from $\text{Specm}(Z_p(\mathfrak{g}))$ to $(\mathfrak{g}^{(1)})^*$ . Moreover, $\mathfrak{f}|_{S({\mathfrak{h}}^{(1)})}$ becomes a homomorphism from $S({\mathfrak{h}}^{(1)})$ to $U({\mathfrak{h}})=S({\mathfrak{h}})$ . Thus, $\mathfrak{f}|_{S({\mathfrak{h}}^{(1)})}$ gives rise to the induced morphism

$$\mathfrak{F}\;:\;{\mathfrak{h}}^*\rightarrow ({\mathfrak{h}}^{(1)})^*,$$

where the notation $ ({\mathfrak{h}}^{(1)})^*$ (resp. $ (\mathfrak{g}^{(1)})^*$ ) will be often simply denoted by ${\mathfrak{h}}^{(1)*}$ (resp. $\mathfrak{g}^{(1)*}$ ). We have that $\mathfrak{F}(\lambda)(h)=\lambda(h)^p-\lambda(h^{[p]})$ for all $\lambda\in \mathfrak{h}^*$ and $h\in\mathfrak{h}$ .

Note that $\mathfrak{g}^{(1)*}$ is isomorphic to $\mathfrak{g}^{*(1)}\;:\!=\;(\mathfrak{g}^*)^{(1)}$ . In the subsequent, $\text{Specm}(Z_p(\mathfrak{g}))$ is identified with $\mathfrak{g}^{(1)*}$ or $\mathfrak{g}^{*(1)}$ according to the requirement of the context.

2·3·3. Some materials on $Z_0(\mathcal{W})$

We write $I_p\;:\!=\;\varsigma(\mathfrak{m}^{(1)})Z_p(\mathfrak{g})$ for $\varsigma(\mathfrak{m}^{(1)})\;:\!=\;\{x^p-x^{[p]}-\chi(x)^p\mid x\in\mathfrak{m}\}$ . Since the group $\mathcal{M}$ preserves the left ideal $I_\chi$ and $\varsigma$ is G-equivariant, hence $\mathcal{M}$ acts on both $U(\mathfrak{g},e)\cong Q_{\chi}^{\text{ad}\,{\mathfrak{m}}}$ and $Z_p(\mathfrak{g})/I_p$ . Let $U(\mathfrak{g},e)^{\mathcal{M}}$ and $(Z_p(\mathfrak{g})/I_p)^{\mathcal{M}}$ denote the fixed point subspace of $U(\mathfrak{g},e)$ and $Z_p(\mathfrak{g})/I_p$ under the action of $\text{Ad}\,\mathcal{M}$ , respectively.

On the other hand, since $Z_p(\tilde{\mathfrak{p}})\cong\varphi(Z_p(\tilde{\mathfrak{p}})) =\varphi(Z_p(\mathfrak{g}))\cong Z_p(\mathfrak{g})/I_p$ by definition, then we have $Z_p(\tilde{\mathfrak{p}})^{\mathcal{M}} \cong (Z_p(\mathfrak{g})/I_p)^{\mathcal{M}}$ as ${{\unicode{x1D55C}}}$ -algebras. Hence, $Z_0(\mathcal{W})=Z_p(\tilde{\mathfrak{p}})^{\mathcal{M}}$ indeed lies in the center of the finite W-algebra $\mathcal{W}({\mathfrak{g}},e)$ .

Moreover, as stated in [ Reference Goodwin and Topley6 , lemma 4·4] (see also [ 18 , remark 2·1]) we can identify $\mathcal{W}(\mathfrak{g},e)$ with $U(\mathfrak{g},e)^\mathcal{M}$ , i.e., we have

(2·1) \begin{align}&\mathcal{W}(\mathfrak{g},e) =U(\mathfrak{g},e)^{\mathcal{M}}\nonumber \\&\text{and}\;\varphi(Z_p(\tilde{\mathfrak{p}}))^{\mathcal{M}}\cong Z_p(\tilde{\mathfrak{p}})^{\mathcal{M}}=Z_p(\tilde{\mathfrak{p}})\cap \mathcal{W}(\mathfrak{g},e).\end{align}

Under the weakened restriction on p, the results in [ Reference Shu and Zeng20 , proposition 2·1] as below still hold.

3·1·1. As we identify $S(\tilde{\mathfrak{p}})$ with ${\unicode{x1D55C}}[\chi+\mathfrak{m}^\perp]$ , we similarly regard the $\mathcal{M}$ -algebra $Z_p(\tilde{\mathfrak{p}})$ as the coordinate algebra of the Frobenius twist $(\chi+\mathfrak{m}^\perp)^{(1)}\subseteq(\mathfrak{g}^*)^{(1)}$ of $\chi+\mathfrak{m}^\perp$ , where the natural action of $\mathcal{M}$ on $(\chi+\mathfrak{m}^\perp)^{(1)}$ is a Frobenius twist of the coadjoint action of $\mathcal{M}$ on $\chi+\mathfrak{m}^\perp$ .

3·1·2. Further description of $Z_0(\mathcal{W})$ via good transverse slices

Define the Killing isomorphism $\kappa$ from $\mathfrak{g}$ to $\mathfrak{g}^*$ via sending $x\in\mathfrak{g}$ to $(x,\cdot)$ . As ${\unicode{x1D55C}}[\chi+\mathfrak{m}^\perp]^{\mathcal{M}}\cong{\unicode{x1D55C}}[\chi+\kappa(\mathfrak{v})]$ by [ Reference Goodwin and Topley6 , lemma 5·1] or [ 17 , lemma 3·2], write

\begin{align*} Z_p(\tilde{\mathfrak{p}})^{\mathcal{M}}\;:\!=\;{\unicode{x1D55C}}[(\chi+\mathfrak{m}^\perp)^{(1)}]^{\mathcal{M}}={\unicode{x1D55C}}[(\chi+\kappa(\mathfrak{v}))^{(1)}], \end{align*}

which is, under the Killing isomorphism, the function algebra on the Frobenius twist of $\kappa({\mathcal{S}})$ for

(3·1) \begin{align}{\mathcal{S}}\;:\!=\;e+\mathfrak{v}.\end{align}

Note that ${\mathcal{S}}$ and $\mathfrak{v}$ are isomorphic as affine varieties. We can regard ${\mathcal{S}}$ as a good transverse slice, instead of $\mathfrak{v}$ , for the description of $Z_0(\mathcal{W})$ .

So there is an isomorphism $\phi$ of affine schemes between $\text{Specm}(Z_0(\mathcal{W}))$ and the variety ${{\mathcal{S}}}$ arising from $\mathfrak{g}^{(1)*}$ (this $\phi$ is in the same sense as in section 2·3·2). More precisely,

\begin{align*}\textrm{Specm}(Z_0(\mathcal{W}))={\tilde{\mathfrak{p}}}^{(1)*}/ \mathcal{M}= \kappa({\mathcal{S}})^{(1)}.\end{align*}

3·2·1. Let us first recall some basic material on G-invariants of $S(\mathfrak{g}^*)$ and regular elements of $\mathfrak{g}$ (see for example, [ Reference Jantzen9 , section 7·13] or [ Reference Varadarajan28 , chapter 4]).

Recall that an element $X\in\mathfrak{g}$ is said to be regular if the adjoint orbit $\text{Ad}(G).X$ has the maximal dimension among the adjoint orbits, which is equal to 2N, where $N=\#\Phi^+$ denotes the number of positive roots of G. Equivalently, the dimension of $\mathfrak{g}_X$ is the smallest one among all centralisers, which is equal to r. Denote by $\mathfrak{g}_{\text{reg}}$ the set of regular elements of $\mathfrak{g}$ .

Denote by

$$P_X(t) = \text{det}(t.1 -\text{ad} X)$$

the characteristic polynomial of $\text{ad} X$ in the variable t. Then we have

\begin{align*}P_X(t) =\sum_{j=0}^{n} (-1)^{n-j} F_j(X)t^{j},\end{align*}

where $n=\dim\mathfrak{g}$ and $F_n\equiv 1$ , and all $F_j$ are polynomial function on $\mathfrak{g}$ . All $F_j$ are $\text{Ad}(G)$ -invariants. Furthermore, $S(\mathfrak{g}^*)^G$ are generated by those $F_j$ . Note that $\text{det}(\text{ad} X)=0$ , and then $F_0\equiv 0$ . The smallest integer $t\gt 0$ such that $F_t\not\equiv0$ is equal to r, the rank of $\mathfrak{g}$ . Then an element $X\in\mathfrak{g}$ is regular semisimple if and only if $F_r(X)\ne 0$ . The regular semisimple elements constitute an open subset of $\mathfrak{g}$ , which is denoted by $\mathfrak{g}_{\text{rss}}$ .

3·2·2. Recall that $r=\text{rank}\,\mathfrak{g}$ , and we already assume the hypotheses (H1)-(H3). By Chevalley restriction theorem (see for example, [ Reference Jantzen9 , section 7·12]), there are an isomorphism $\mathsf{Res}\;:\; S(\mathfrak{g}^*)^G \cong S(\mathfrak{h}^*)^{W}$ as ${{\unicode{x1D55C}}}$ -algebras for any given Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ and the abstract Weyl group W of $\mathfrak{g}$ . The map $\mathsf{Res}$ is actually the restriction of the natural map $\Upsilon\;:\; S(\mathfrak{g}^*)\rightarrow S(\mathfrak{h}^*)$ , sending $f\in S(\mathfrak{g}^*)$ to $f|_{S(\mathfrak{h})}$ . Consequently, there is an isomorphism $\mathsf{res}$ of varieties:

(3·2) \begin{align}\mathfrak{g}/\hskip-3pt/ G{\overset{\cong}{\longrightarrow}}\mathfrak{h}/ W.\end{align}

Then we have the morphism $\zeta\;:\;\mathfrak{g}\rightarrow \mathfrak{h}/ W$ . Set $\varpi_{\mathcal{S}}\;:\!=\;\zeta|_{\mathcal{S}}$ . Note that $U(\mathfrak{g})^G\cong U(\mathfrak{h})^{W}$ which are isomorphic to a polynomial ring (see, for example [ Reference Brown and Gordon3 , lemma 3·3] or [ Reference Jantzen8 , section 9·6]). There exists a set of algebraically independent generators $\{g_1,\ldots,g_r\}$ in $U(\mathfrak{g})^G$ ( $\cong U(\mathfrak{h})^{W}$ ). Let $\text{gr}_S(g_i)=\tilde g_i$ be the images of $g_i$ associated to the standard grading of $U(\mathfrak{g})$ . We can further choose such $g_i$ with $i=1,\ldots,{r}$ such that the elements $\kappa(\tilde g_1),\ldots,\kappa(\tilde g_r)$ form a free generating set of $S(\mathfrak{g}^*)^G$ .

Note that under above assumption, we have the adjoint quotient map:

\begin{equation*}\nu\;:\;\mathfrak{g}\rightarrow \mathbb{A}^r\end{equation*}

which sends $x\in\mathfrak{g}$ to $(\kappa(\tilde g_1)(x),\ldots,\kappa(\tilde g_r)(x))$ . Consider the composite of $\nu$ and the canonical isomorphism $\eta\;:\; \mathbb{A}^r\rightarrow \mathfrak{h}/ W$ . Then we have $\varpi_{\mathcal{S}}=\eta\circ\nu|_{\mathcal{S}}$ . In virtue of [ Reference Slodowy21 , section 7·3-7·4] and [ 16 , theorem 5·4 and remark 5·4], we have the following result.

Lemma 3·1. Under the hypothesis (H3) in section 1·3·1, we consider the restriction of $\zeta$ to ${\mathcal{S}}$ , i.e.,

$$\varpi_{\mathcal{S}}\;:\; {\mathcal{S}}\rightarrow \mathfrak{h}/ {W}.$$

The morphism $\varpi_{\mathcal{S}}$ is faithfully flat.

Consequently, $\varpi_{\mathcal{S}}$ is surjective and all fibers of $\varpi_{\mathcal{S}}$ have the same dimension $\mathsf{d}-r$ with $\mathsf{d}=\dim \mathfrak{g}_e$ (see [ 16, Reference Slodowy21 ] or [ Reference Shu and Zeng20 , proposition 4·11(i)]).

Set ${\mathcal{S}}_{\text{reg}}\;:\!=\;{\mathcal{S}}\cap \mathfrak{g}_{\text{reg}}$ . Then [ Reference Shu and Zeng20 , proposition 4·13] can be proved in the same way under the hypotheses (H1)-(H3).

3·2·3. Let $\mathsf{t}$ be the translation

\begin{equation*}\begin{array}{llll}\mathsf{t}\;:\;&\mathfrak{v}&{\overset{\cong}\rightarrow} &{\mathcal{S}}\\ &x&\mapsto&e+x,\end{array}\end{equation*}

which is an isomorphism of affine varieties. Then one obtains a morphism

\begin{align*}\psi\;:\!=\;\varpi_{\mathcal{S}}\circ \mathsf{t}\;:\;\mathfrak{v}\rightarrow\mathbb{A}^r,\qquad x\mapsto(\psi_1(x),\ldots,\psi_r(x)).\end{align*}

The morphism $\psi$ is still faithfully flat with $\psi^{-1}(\xi)\cong\varpi_{\mathcal{S}}^{-1}(\xi)$ for any $\xi\in\mathbb{A}^r$ , and the fiber $\psi^{-1}(\xi)$ is normal. We choose a basis $\{x_1, \ldots, {x_\mathsf{d}}\}$ of $\mathfrak{g}_e$ . Since $\mathfrak{v}$ is dual to $\mathfrak{g}_e$ via $(\cdot,\cdot)$ (see section 1·3·2), we let $x^*_1, \ldots, {x_\mathsf{d}^*}$ denote the linear functions on ${{\unicode{x1D55C}}}$ defined via $x^*_i (y)=(x_i , y)$ for $y\in\mathfrak{v}$ .

The following lemma can be obtained from [ Reference Brown and Gordon3 , corollary 3·4] and its proof.

Lemma 3·3. Under the hypotheses (H1)-(H3), the Jacobian matrix

\begin{equation*}\left(\partial \psi_i\over\partial x^*_j\right)_{\tiny\begin{array}{l}1\leqslant i\leqslant r\\1\leqslant j\leqslant {\mathsf{d}}\end{array}}\end{equation*}

has rank r in $\mathfrak{v}$ .

3·3·1. Let us first return to the p-center $Z_0(\mathcal{W})=Z_p(\tilde{\mathfrak{p}})^{\mathcal{M}} $ . Recall that $Z_0(\mathcal{W})=\mathcal{W}({\mathfrak{g}},e)\cap \varphi(Z_p(\mathfrak{g}))\cong\varphi(Z_p(\tilde{\mathfrak{p}}))^{\mathcal{M}} $ by (2·1). By the discussion in section 3·1·2, we further have

(3·3) \begin{align} \text{Specm}(Z_0(\mathcal{W}))= \kappa({\mathcal{S}})^{(1)}. \end{align}

In the meantime, one can describe ${\mathcal{S}}$ via $\mathfrak{v}$ (see section 3·2·3). So we have an isomorphism of algebras

\begin{align*}Z_0(\mathcal{W})\cong \ {{\unicode{x1D55C}}}[\kappa(\mathfrak{v})^{(1)}]. \end{align*}

3·3·2. Recall that $Z_1(\mathcal{W})=\varphi(U(\mathfrak{g})^G)$ . By the same arguments as in [ Reference Shu and Zeng20 ], under the present condition (H1)-(H3), we still have the following property for finite W-algebras as in [ Reference Shu and Zeng20 , lemma 3·2].

Lemma 3·4. There exists an isomorphism between ${\unicode{x1D55C}}$ -algebras

$$\varphi|_{U(\mathfrak{g})^{G}}\;:\; U(\mathfrak{g})^G\xrightarrow\sim Z_1(\mathcal{W}),$$

where $\varphi|_{U(\mathfrak{g})^{G}}$ denotes the restriction of the map $\varphi$ as in section 2·3 to the subalgebra $U(\mathfrak{g})^{G}$ of $Z(\mathfrak{g})$ .

To simplify notation, in the following discussions we still write $\varphi$ for the map $\varphi|_{U(\mathfrak{g})^{G}}$ as in Lemma 3·4. By the Harish–Chandra theorem, we have $S(\mathfrak{h})^{W}\cong U(\mathfrak{g})^G$ . So arising from $\varphi$ there is an isomorphism of varieties

(3·4) \begin{align}\text{Specm}(Z_1(\mathcal{W}))\cong \mathfrak{h}^*/ {W}.\end{align}

3·4·1. Proof for Part (i)

We still set $n=\dim\mathfrak{g}$ and $r=\text{rank}\,\mathfrak{g}$ . By Lemma 2·1(iii), we can consider the fraction field of $Z(\mathcal{W})$ . Set $\mathbb{F}_0\;:\!=\;\text{Frac}(Z_0(\mathcal{W}))$ and $\mathbb{F}\;:\!=\;\text{Frac}(Z(\mathcal{W}))$ . Then $\mathbb{F}_0$ is a subfield of $\mathbb{F}$ . Denote by $\mathcal{Q}$ the fractional field of $\tilde{\mathcal{Z}}$ . By the same arguments as in (step 1) of [ Reference Shu and Zeng20 , section 5·2], we can show the following result.

Recall that in section 2·1 we have introduced the Kazhdan filtration on $U(\mathfrak{g})$ and its sub-quotients, and also their corresponding graded algebras. Write $\text{gr}(\tilde{\mathcal{Z}})$ and $\text{gr}(Z(\mathcal{W}))$ for the graded algebras of $\tilde{\mathcal{Z}}$ and $Z(\mathcal{W})$ respectively. Note that $\tilde{\mathcal{Z}}\subseteq Z(\mathcal{W})$ and ${{\unicode{x1D55C}}}$ is contained in $\tilde{\mathcal{Z}}$ , by inductive arguments on the degrees we can finally deduce the claim

$$\tilde{\mathcal{Z}}=Z(\mathcal{W})$$

amounts to the claim that

(3·5) \begin{equation}\text{gr}(\tilde{\mathcal{Z}})=\text{gr}(Z(\mathcal{W})).\end{equation}

From Lemma 3·5 one easily derives that their graded algebras $\text{gr}(Z(\mathcal{W}))$ and $\text{gr}(\tilde{\mathcal{Z}})$ also share the same fraction field. It follows from Proposition 2·2(iii) that $\text{gr}(Z(\mathcal{W}))$ is integral over $\text{gr}(Z_0(\mathcal{W}))$ , so, a fortiori, $\text{gr}(Z(\mathcal{W}))$ is integral over $\text{gr}(\tilde{\mathcal{Z}})$ . Therefore, to prove (3·5) it is sufficient to show that $\text{gr}(\tilde{\mathcal{Z}})$ is integrally closed.

Due to Proposition 3·2, by the same arguments as in (step 3) of [ Reference Shu and Zeng20 , section 5·2] we can show that $\text{Specm}(\text{gr}(\tilde{\mathcal{Z}}))$ is smooth outside of a subvariety of codimension 2. On the other hand, as a result of Lemma 3·3 we can prove the following result, by the same arguments as in [ Reference Shu and Zeng20 , lemma 5·1].

According to Serre’s theorem [ Reference Serre23 , chapter 4], a complete intersection variety is normal if it is smooth outside of a subvariety of codimension 2. Hence by the above analysis, $\text{Specm}(\text{gr}(\tilde{\mathcal{Z}}))$ is normal, i.e., $\text{gr}(\tilde{\mathcal{Z}})$ is integral closed. This is desired for the proof of the equality $\tilde{\mathcal{Z}}=Z(\mathcal{W})$ . As to the second statement, it follows from the arguments in [ Reference Shu and Zeng20 , lemma 3·6] which works in our current situation. The proof of (i) is completed.

3·4·2. The proof for parts (ii) and (iii)

The proof of part (ii) is based on the arguments in section 3·4·1 and the classical result that $Z_1(\mathfrak{g})$ is a complete intersection over $Z_0(\mathfrak{g})\cap Z_1(\mathfrak{g})$ (see for example, [ Reference Brown and Gordon3 , theorem 3·5(3)]). In the same way as in [ Reference Shu and Zeng20 , section 5·3], we can show that $Z_1(\mathcal{W})$ is a complete intersection over $Z_0(\mathcal{W})\cap Z_1(\mathcal{W})$ . Furthermore, the multiplication map $\mu\;:\;Z_0(\mathcal{W})\otimes_{Z_0(\mathcal{W})\cap Z_1(\mathcal{W})} Z_1(\mathcal{W})\rightarrow Z(\mathcal{W})$ is an isomorphism of ${\unicode{x1D55C}}$ -algebras. This completes the proof of (ii).

As for (iii), based on Proposition 3·2 the proof can be done using the same arguments as in [ Reference Shu and Zeng20 , section 4·8].

5·2·1. Note that $\mathfrak{h}$ is a toral subalgebra. There is a basis consisting of toral elements which means all of them satisfying $h^{[p]}=h$ (see for example, [ Reference Strade and Farnsteiner25 , theorem 3·6 of chapter 2]). Thus we can choose a toral ${{\unicode{x1D55C}}}$ -basis $\{h_1,\ldots, h_r\}$ in $\mathfrak{h}$ whose $\mathbb{F}_p$ -linear span is denoted by $\mathfrak{h}_{\mathbb{F}_p}$ (we can take them as the same as we did in section 4·3). Then $\mathfrak{h}_{\mathbb{F}_p}$ is an $\mathbb{F}_p$ -form of $\mathfrak{h}$ .

By Remark 4·5(1), we define a variety

(5·3) \begin{align}\mathcal{C}_{{\mathcal{S}}}\;:\!=\;\{&(\vartheta^p,\lambda, g)\in \kappa({\mathcal{S}})^{(1)}_{\text{rss}}\times \mathfrak{h}^* \times G \nonumber\\ &\mid (\vartheta')^p(h)=\lambda(h)^p-\lambda(h^{[p]}) \text{ for all } h\in\mathfrak{h} \nonumber\\ &\qquad \text{ with }\vartheta'=\text{Ad}^*(g^{-1})\vartheta\in \mathfrak{h}^{*}\}.\end{align}

Then $\mathcal{C}_{{\mathcal{S}}}$ is a closed subvariety of $\kappa({\mathcal{S}})^{(1)}_{\text{rss}} \times \mathfrak{h}^*\times G$ , which can be described in more detail below. The morphism $\mathfrak{F}$ from $\mathfrak{h}^*$ to $\mathfrak{h}^{(1)*}$ sends $\lambda$ to $\mathfrak{F}(\lambda)$ which takes value $\lambda(h)^p-\lambda(h^{[p]})$ at $h\in \mathfrak{h}$ . Then $\mathfrak{F}(\lambda)=0$ for all $\lambda\in (\mathfrak{h}_{\mathbb{F}_p})^*$ . So $\mathcal{C}_{{\mathcal{S}}}$ is just the solution space of the equation $(\vartheta')^p=\mathfrak{F}(\lambda)$ in the variety $\kappa({\mathcal{S}})_{\text{rss}} \times \mathfrak{h}^*\times G$ . More precisely, write $\vartheta_i$ for the functional $h_i\mapsto \vartheta'(h_i)$ , and $\lambda_i$ for the functional $h_i\mapsto \lambda(h_i)$ . Thus, $\mathcal{C}_{{\mathcal{S}}}$ can be regarded a closed subset of $\kappa({\mathcal{S}})^{(1)}_{\text{rss}} \times \mathfrak{h}^*\times G$ defined by the equations

(5·4) \begin{align}&\vartheta_i^p=\lambda_i^p-\lambda_i, \;\; i=1,\ldots,r, \nonumber\\&\vartheta'=\text{Ad}^*(g^{-1})\vartheta\in \mathfrak{h}^{*}.\end{align}

5·2·2. Note that the action of $n\in \mathsf{N}$ on $\mathfrak{h}^*$ amounts to an action of some $w\in W$ on $\mathfrak{h}^*$ . So we can define the action of $\mathsf{N}$ on $\mathcal{C}_{\mathcal{S}}$ as in the following lemma.

Lemma 5·3. There is an action of $\mathsf{N}$ on $\mathcal{C}_{\mathcal{S}}$ defined via for $(\vartheta^p,\lambda, g)\in \mathcal{C}_{\mathcal{S}}$ and $n\in\mathsf{N}$ ,

(5·5) \begin{align}n\maltese (\vartheta^p,\lambda, g)=(\vartheta^p,\text{Ad}^*(n)(\lambda), gn^{-1}).\end{align}

Proof. It suffices to show that for $(\vartheta^p, \lambda, g)\in \mathcal{C}_{\mathcal{S}}$ with

$$(\vartheta')^p(h)=\lambda(h)^p-\lambda(h^{[p]}) \text{ for all } h\in\mathfrak{h} \text{ with }\vartheta'=\text{Ad}^*(g^{-1})\vartheta\in \mathfrak{h}^{*}, $$

then

(5·6) \begin{align} \vartheta''=\text{Ad}^*((gn^{-1})^{-1})\vartheta\in \mathfrak{h}^* \end{align}

and furthermore, for all $h\in \mathfrak{h}$ , the following holds

(5·7) \begin{align}(\vartheta'')^p(h)=((\text{Ad}^*(n)(\lambda))h)^p-(\text{Ad}^*(n)\lambda)(h^{[p]}) \text{ for all } h\in\mathfrak{h}. \end{align}

The verification on (5·6) can be done as the same as in (5·2). In particular, the action of $n\in \mathsf{N}$ on $\mathfrak{h}^*$ amounts to an action of some $w\in W$ on $\mathfrak{h}^*$ . So we have

(5·8) \begin{align}\vartheta''=w(\vartheta')\in \mathfrak{h}^*.\end{align}

As to (5·7), we first note that the action of $\text{Ad}^*(n)$ on $\mathfrak{h}^*$ gives rise to an automorphism of the symmetric algebra $\text{Sym}(\mathfrak{h}^*)$ , and also the action preserves the p-mapping [p]. Then from (5·8), we have

(5·9) \begin{align}(\vartheta'')^p(h)&=(\text{Ad}^*(n)(\vartheta'))^p(h)\nonumber \\&=\vartheta'(\text{Ad}(n^{-1})h)^p\nonumber\\&=((\text{Ad}^*(n)\lambda)h)^p-(\text{Ad}^*(n)\lambda)(h^{[p]}) \end{align}

for all $h\in\mathfrak{h}$ . This completes the proof.

Consider the following variety

\begin{align*}\digamma\;:\!=\;&\{(\vartheta^p,\zeta^*(\lambda))\in \kappa({\mathcal{S}})^{(1)}_{\text{rss}}\times \mathfrak{h}^*/ W \mid (\vartheta')^p(h)=\lambda(h)^p-\lambda(h^{[p]}),\;\; \forall h\in\mathfrak{h} \\ &\hskip6cm\text{ with }\vartheta'=\text{Ad}^*(g^{-1})\vartheta\in \mathfrak{h}^{*} \text{ for some }g\in G\},\end{align*}

for which it is worthwhile reminding that for any $w\in W$ and the corresponding $n\in \mathsf{N}$ , $\vartheta''\;:\!=\;\text{Ad}^*((gn^{-1})^{-1})\vartheta=\text{Ad}^*(n)\vartheta'\in \mathfrak{h}^*$ , we have

(5·10) \begin{align}\vartheta''^{p}(h)=&w(\lambda)(h)^p-w(\lambda)(h^{[p]}) \text{ for all }h\in\mathfrak{h},\nonumber\\&\qquad \text{ with }\vartheta''=\text{Ad}^*((gn^{-1})^{-1})\vartheta.\end{align}

This shows that for any $w\in W$ , the equation (5·10) is well defined, which defines the point $(\vartheta^p, \zeta^*(w(\lambda)))$ (the same point as $(\vartheta^p, \zeta^*(\lambda))$ ). Furthermore, the equations in (5·10) show that $\digamma$ is an $\mathsf{N}$ -variety with $\mathsf{N}$ -action being the identify effect. Now the equations appearing in the Definition of (5·3) and the equations in (5·10) give rise to a natural $\mathsf{N}$ -equivariant morphism from $\mathcal{C}_{\mathcal{S}}$ to $\digamma$ , sending the point $(\vartheta^p, \lambda, g)\in \mathcal{C}_{\mathcal{S}}$ to the point $(\vartheta^p, \zeta^*(\lambda))\in \digamma $ . This morphism finally induces the $\mathsf{N}$ -quotient morphism $\mathcal{C}_{\mathcal{S}}/ \mathsf{N}$ to $\digamma$ , which is an isomorphism of varieties.

By definition, $\kappa({\mathcal{S}})_{\text{rss}}^{(1)}\times_{\mathcal{V}}\mathfrak{h}^*/ W$ can be identified with $\digamma$ . Hence the corollary is proved.

5·2·3. Now we establish a connection between the different varieties we introduced above.

Proposition 5·5. There exists an $\mathsf{N}$ -equivariant isomorphism of $\mathsf{N}$ -varieties

\begin{align*}\mathcal{C}_{{\mathcal{S}}} {\overset{\cong}{\longrightarrow}} \mathcal{U}_{{\mathcal{S}}}.\end{align*}

Proof. For $(\vartheta^p,\lambda,g)\in \mathcal{C}_{\mathcal{S}}$ , by definition we have $\vartheta'\;:\!=\;\text{Ad}^*(g^{-1})\vartheta\in \mathfrak{h}^{*}$ for $g\in G$ , and also

(5·11) \begin{equation}(\vartheta')^p(h)=\lambda(h)^p-\lambda(h^{[p]}) \text{ for all } h\in\mathfrak{h}.\end{equation}

By Lemma 4·6, $\text{Ad}^*(g)\lambda\in \kappa({\mathcal{S}})_{\text{rss}}$ . Define a map $\psi_1$ from $\mathcal{C}_{{\mathcal{S}}}$ to $\mathcal{U}_{{\mathcal{S}}}$ via sending $(\vartheta^p,\lambda, g)$ to $(\text{Ad}^*(g)\lambda, g)$ . By a direct check, it is $\mathsf{N}$ -equivariant.

For the inverse of $\psi_1$ , we define a map $\psi_2\;:\;\mathcal{U}_{{\mathcal{S}}}\longrightarrow \mathcal{C}_{{\mathcal{S}}}$ by sending any element $(\chi,g)\in \mathcal{U}_{\mathcal{S}}$ with $\lambda\;:\!=\;\text{Ad}^*(g^{-1})\chi\in \mathfrak{h}^*$ to $(\text{Ad}^*(g)\mathfrak{F}(\lambda), \lambda, g)\in\mathcal{C}_{{\mathcal{S}}}$ . Again we note that the action of $\text{Ad}^*(n)$ on $\mathfrak{h}^*$ for $n\in \mathsf{N}$ gives rise to an automorphism of the symmetric algebra $\text{Sym}(\mathfrak{h}^*)$ , and also the action preserves the p-mapping [p]. We still have that $\psi_2$ is $\mathsf{N}$ -equivariant.

Both $\psi_1$ and $\psi_2$ are clearly morphisms of varieties. Furthermore, it is easily confirmed by a straightforward computation that the compositions between $\psi_1$ and $\psi_2$ are identities. Hence, both $\mathcal{U}_{\mathcal{S}}$ and $\mathcal{C}_{\mathcal{S}}$ are $\mathsf{N}$ -equivariant isomorphic, as $\mathsf{N}$ -varieties. This completes the proof.

5·2·4. From now on, we identify $\mathscr{Z}$ with $\kappa({\mathcal{S}})^{(1)}\times_{\mathcal{V}}\mathfrak{h}^*/ W$ . Consider the projection on the first component for the decomposition of $\mathscr{Z}$ in (4·4):

$$\text{pr}_1\;:\; {\mathscr{Z}}\rightarrow \kappa({\mathcal{S}})^{(1)}.$$

The homomorphism $\text{pr}_1$ of varieties has the complete inverse image

$${\mathscr{Z}^{\ \flat}}\;:\!=\;\text{pr}_1^{-1}(\kappa({\mathcal{S}})^{(1)}_{\text{rss}}).$$

Then we have the following result.

Proposition 5·6. The following is true:

\begin{align*}\mathcal{C}_{{\mathcal{S}}}/ \mathsf{N}= {\mathscr{Z}^{\ \flat}}.\end{align*}