3.1 Integral representatives

The nilpotent orbits for $G$ over an algebraically closed field of good characteristic can be classified by combinatorial data ${\mathcal{C}}$ that is independent of the characteristic. For classical groups, nilpotent orbits can be classified by their Jordan canonical form in terms of partitions. For a partition $\unicode[STIX]{x1D70E}\in {\mathcal{C}}$ , let ${\mathcal{O}}_{F,\unicode[STIX]{x1D70E}}\subset \mathfrak{g}_{F}$ denote the corresponding orbit over the algebraically closed field $F$ . For $\unicode[STIX]{x1D70E}\in {\mathcal{C}}$ , we seek elements

(3.1) $$\begin{eqnarray}N_{\unicode[STIX]{x1D70E}}\in \mathfrak{g}\text{ such that }(N_{\unicode[STIX]{x1D70E}})_{k}\in O_{\overline{k},\unicode[STIX]{x1D70E}}\text{ and }(N_{\unicode[STIX]{x1D70E}})_{K}\in O_{\overline{K},\unicode[STIX]{x1D70E}}.\end{eqnarray}$$

This makes precise the statement that $N_{K}$ and $N_{k}$ ‘lie in the same nilpotent orbit’.

For symplectic and orthogonal groups, we can produce the desired $N_{\unicode[STIX]{x1D70E}}$ using a minor extension of the classical results known over algebraically closed fields [Reference JantzenJan04, § 1]. Let $G=\operatorname{Sp}_{m}$ with $m=2n$ , or $G=\operatorname{O}_{m}$ with $m=2n$ or $m=2n+1$ . We assume $n\geqslant 2$ . Recall that $\operatorname{Sp}_{m}$ and $\operatorname{O}_{m}$ are defined using standard pairings on a free ${\mathcal{O}}$ -module $M$ of rank $m$ . For $m=2n$ , the standard alternating pairing $\unicode[STIX]{x1D711}_{\text{std}}$ on ${\mathcal{O}}^{m}$ is the one given by the block matrix

$$\begin{eqnarray}\left(\begin{array}{@{}cc@{}}0 & I_{n}^{\prime }\\ -I_{n}^{\prime } & 0\end{array}\right),\end{eqnarray}$$

where $I_{n}^{\prime }$ denotes the anti-diagonal matrix with $1$ s on the diagonal. The standard symmetric pairing $\unicode[STIX]{x1D711}_{\text{std}}$ on ${\mathcal{O}}^{m}$ is the one given by the matrix $I_{m}^{\prime }$ .

The admissible partitions of $m$ are in bijection with nilpotent orbits of $\operatorname{Sp}_{m}$ or $\operatorname{O}_{m}$ over any algebraically closed field of good characteristic [Reference JantzenJan04, Theorem 1.6]. The corresponding orbit is the intersection of $\mathfrak{g}\subset \mathfrak{gl}_{m}$ with the $\operatorname{GL}_{m}$ -orbit corresponding to that partition of $m$ . Note that $\operatorname{GL}_{m}$ -orbit representatives in Jordan canonical form need not lie in $\mathfrak{g}$ .

We will construct nilpotents together with a pairing, and then show how to relate the constructed pairing to the standard pairings used to define $G$ . Let $\unicode[STIX]{x1D716}=1$ in the case of $\operatorname{O}_{m}$ , and $\unicode[STIX]{x1D716}=-1$ in the case of $\operatorname{Sp}_{m}$ .

Definition 3.5. Let $d\geqslant 2$ be an integer. Define $M(d)={\mathcal{O}}^{d}$ , with basis $v_{1},\ldots ,v_{d}$ and a perfect pairing $\unicode[STIX]{x1D711}_{d}$ such that

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{d}(v_{i},v_{j})=\left\{\begin{array}{@{}ll@{}}(-1)^{i}\quad & i+j=d+1,\\ 0\quad & \text{otherwise}\end{array}\right.\end{eqnarray}$$

(alternating for even $d$ , symmetric for odd $d$ ). Define a nilpotent $N_{d}\in \operatorname{End}(M(d))$ by $N_{d}v_{i}=v_{i-1}$ for $1<i\leqslant d$ and $N_{d}v_{1}=0$ .

Similarly, define $M(d,d)={\mathcal{O}}^{2d}$ with basis $v_{1},\ldots ,v_{d},v_{1}^{\prime },\ldots ,v_{d}^{\prime }$ and a perfect $\unicode[STIX]{x1D716}$ -symmetric pairing $\unicode[STIX]{x1D711}_{d,d}$ by extending

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{d,d}(v_{i},v_{j})=\unicode[STIX]{x1D711}_{d,d}(v_{i}^{\prime },v_{j}^{\prime })=0\quad \text{and}\quad \unicode[STIX]{x1D711}_{d,d}(v_{i},v_{j}^{\prime })=\left\{\begin{array}{@{}ll@{}}(-1)^{i}\quad & i+j=d+1,\\ 0\quad & \text{otherwise}.\end{array}\right.\end{eqnarray}$$

Define a nilpotent $N_{d,d}\in \operatorname{End}(M(d,d))$ by $N_{d,d}v_{i}=v_{i-1}$ and $N_{d,d}v_{i}^{\prime }=v_{i-1}^{\prime }$ for $1<i\leqslant d$ , and $N_{d,d}v_{1}=N_{d,d}v_{1}^{\prime }=0$ .

Note that the pairing $\unicode[STIX]{x1D711}_{d,d}$ can be symmetric or alternating depending on the parity of $d$ . It is straightforward to verify the pairings are perfect and that $N_{d}$ (respectively $N_{d,d}$ ) is skew with respect to $\unicode[STIX]{x1D711}_{d}$ (respectively $\unicode[STIX]{x1D711}d,d$ ) in the sense that for $v,w\in M(d)$ we have

$$\begin{eqnarray}\unicode[STIX]{x1D711}_{d}(N_{d}v,w)=-\unicode[STIX]{x1D711}_{d}(v,N_{d}w).\end{eqnarray}$$

Given an admissible partition $\unicode[STIX]{x1D70E}:m=m_{1}+m_{2}+\cdots +m_{r}$ , we will construct a free ${\mathcal{O}}$ -module of rank $m$ with an $\unicode[STIX]{x1D716}$ -symmetric perfect pairing and a nilpotent endomorphism that is skew with respect to the pairing such that the Jordan block structure of nilpotent endomorphism in geometric fibers is given by $\unicode[STIX]{x1D70E}$ . Let $n_{i}(\unicode[STIX]{x1D70E})=\text{\#}\{j:m_{j}=i\}$ .

• ∙ If $G=\operatorname{O}_{m}$ , then $n_{i}(\unicode[STIX]{x1D70E})$ is even for even $i$ , so we can define

  $$\begin{eqnarray}M_{\unicode[STIX]{x1D70E}}=\underset{i\,\text{odd}}{\bigoplus }M(i)^{\oplus n_{i}(\unicode[STIX]{x1D70E})}\oplus \underset{i\,\text{even}}{\bigoplus }M(i,i)^{\oplus n_{i}(\unicode[STIX]{x1D70E})/2}.\end{eqnarray}$$

• ∙ If $G=\operatorname{Sp}_{m}$ , then $n_{i}(\unicode[STIX]{x1D70E})$ is even for odd $i$ , so we can define

  $$\begin{eqnarray}M_{\unicode[STIX]{x1D70E}}=\underset{i\,\text{odd}}{\bigoplus }M(i,i)^{\oplus n_{i}(\unicode[STIX]{x1D70E})/2}\oplus \underset{i\,\text{even}}{\bigoplus }M(i)^{\oplus n_{i}(\unicode[STIX]{x1D70E})}.\end{eqnarray}$$

Let $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}$ and $N_{\unicode[STIX]{x1D70E}}$ denote the pairing and nilpotent endomorphism defined by the pairing and nilpotent endomorphism on each piece using Definition 3.5. In all cases, $M_{\unicode[STIX]{x1D70E}}$ is a free ${\mathcal{O}}$ -module of rank $m$ . For each $\unicode[STIX]{x1D70E},$ let $G_{\unicode[STIX]{x1D70E}}$ be the automorphism scheme $\text{}\underline{\operatorname{Aut}}(M_{\unicode[STIX]{x1D70E}},\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}})$ , so for an algebraically closed field $F$ over ${\mathcal{O}}$ we have an isomorphism $(G_{\unicode[STIX]{x1D70E}})_{F}\simeq G_{F}$ well defined up to $G(F)$ -conjugation by using $F$ -linear isomorphisms $(M_{\unicode[STIX]{x1D70E}},\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}})_{F}\simeq (F^{m},\unicode[STIX]{x1D711}_{\text{std}})$ .

Let $e_{1},e_{2},\ldots ,e_{m}$ be the standard basis for ${\mathcal{O}}^{m}$ . The elements $e_{i}$ and $e_{m+1-i}$ pair non-trivially under the standard pairing. When $m=2n+1$ , $e_{n+1}$ pairs non-trivially with itself under the standard pairing. We now relate the standard pairings to the pairings $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}$ .

Proof. The standard pairings are very similar to $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}$ . In the case of $\operatorname{Sp}_{m}$ , each basis vector pairs trivially against all but one other basis vector, with which it pairs as $\pm 1$ . So after reordering the basis, $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}$ is the standard pairing. The case of $\operatorname{O}_{m}$ is slightly more complicated. Let $\unicode[STIX]{x1D70E}:m=m_{1}+m_{2}+\cdots +m_{r}$ be an admissible partition. The construction of $M_{\unicode[STIX]{x1D70E}}$ and $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}$ gives a basis $\{v_{i,j}\}$ where $1\leqslant i\leqslant r$ and $1\leqslant j\leqslant m_{i}$ . From the construction of $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}$ , we see that $v_{i,j}$ pairs trivially against all basis vectors except for $v_{i,m_{i}+1-j}$ . So as long as $2j\neq m_{i}+1$ , we obtain a pair of basis vectors which are orthogonal to all others and which pair to $\pm 1$ . For each odd $m_{i}$ , the vector $v_{i,(m_{i}+1)/2}$ pairs non-trivially with itself. The standard pairing with respect to the basis $e_{i}$ has such a vector only when $m=2n+1$ and then only for one $e_{i}$ .

We must change the basis over ${\mathcal{O}}$ so that $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}$ becomes the standard symmetric pairing. Let $v=v_{i,(m_{i}+1)/2}$ and $v^{\prime }=v_{j,(m_{j}+1)/2}$ be two distinct vectors which pair non-trivially with themselves. In particular, $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}(v,v)=(-1)^{(m_{i}+1)/2}:=\unicode[STIX]{x1D702}$ and $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}(v^{\prime },v^{\prime })=(-1)^{(m_{j}+1)/2}:=\unicode[STIX]{x1D702}^{\prime }$ . Define

$$\begin{eqnarray}w=\frac{\sqrt{\unicode[STIX]{x1D702}}v-\sqrt{-\unicode[STIX]{x1D702}^{\prime }}v^{\prime }}{\sqrt{2}}\quad \text{and}\quad w^{\prime }=\frac{\sqrt{\unicode[STIX]{x1D702}}v+\sqrt{-\unicode[STIX]{x1D702}^{\prime }}v^{\prime }}{\sqrt{2}}.\end{eqnarray}$$

Then we see that $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}(w,w)=0=\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}(w^{\prime },w^{\prime })$ and $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}(w,w^{\prime })=1$ . Making this change of variable over ${\mathcal{O}}$ (which requires $\sqrt{-1},\sqrt{2}\in {\mathcal{O}}^{\times }$ ), we have reduced the number of basis vectors which pair non-trivially with themselves by two, and produced a new pair of basis vectors orthogonal to the others and which pair to $1$ . By induction, we may therefore pick a basis $v_{1}^{\prime },\ldots ,v_{m}^{\prime }$ for which at most one basis vector pairs non-trivially with itself under $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}$ . After re-ordering, we may further assume that $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}(v_{i}^{\prime },v_{j}^{\prime })=0$ unless $i+j=m+1$ , in which case $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}(v_{i}^{\prime },v_{j}^{\prime })=\pm 1$ . Suppose $j=m+1-i$ . If $i\neq j$ , by scaling $v_{i}^{\prime }$ we may assume that $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}(v_{i}^{\prime },v_{j}^{\prime })=1$ . If $i=j$ , we already know that $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}(v_{i}^{\prime },v_{j}^{\prime })=1$ . With respect to this basis, $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}$ is the standard pairing.

The last statement immediately follows from Lemma 3.6.◻

3.2 Pure nilpotent lifts

For a nilpotent element $\overline{N}\in \mathfrak{g}_{k}$ of type $\unicode[STIX]{x1D70E}\in {\mathcal{C}}$ , we will define the notion of a pure nilpotent lift of $\overline{N}$ in $\mathfrak{g}_{k}$ and study the space of such lifts, assuming there exists $N_{\unicode[STIX]{x1D70E}}\in \mathfrak{g}$ lifting $\overline{N}$ such that $(N_{\unicode[STIX]{x1D70E}})_{\overline{K}}\in O_{\overline{K},\unicode[STIX]{x1D70E}}$ and such that $Z_{G}(N_{\unicode[STIX]{x1D70E}})$ is smooth over ${\mathcal{O}}$ .

Definition 3.9. Fix an $N_{\unicode[STIX]{x1D70E}}\in \mathfrak{g}$ lifting $\overline{N}$ such that $(N_{\unicode[STIX]{x1D70E}})_{\overline{K}}\in O_{\overline{K},\unicode[STIX]{x1D70E}}$ and such that $Z_{G}(N_{\unicode[STIX]{x1D70E}})$ is smooth over ${\mathcal{O}}$ . Define the functor $\operatorname{Nil}_{\overline{N}}:{\mathcal{C}}_{{\mathcal{O}}}\rightarrow \operatorname{Sets}$ by

$$\begin{eqnarray}\operatorname{Nil}_{\overline{N}}(R)=\{N\in \mathfrak{g}_{R}:\operatorname{Ad}_{G}(g)(N_{\unicode[STIX]{x1D70E}})=N~\text{for some}~g\in \widehat{G}(R)\}.\end{eqnarray}$$

Call these $N\in \operatorname{Nil}_{\overline{N}}(R)$ the pure nilpotents lifting $\overline{N}$ .

This is obviously a subfunctor of the formal neighborhood of $\overline{N}$ in the affine space $\text{}\underline{\mathfrak{g}}$ over ${\mathcal{O}}$ attached to $\mathfrak{g}$ . The key to analyzing $\operatorname{Nil}_{\overline{N}}$ is that $Z_{G_{R}}(N)$ is smooth over $R$ since $Z_{G}(N_{\unicode[STIX]{x1D70E}})$ is ${\mathcal{O}}$ -smooth and $N$ is in the $G$ -orbit of $(N_{\unicode[STIX]{x1D70E}})_{R}$ . To ease notation below, we shall write $gNg^{-1}$ rather than $\operatorname{Ad}_{G}(g)(N)$ for $g\in \widehat{G}(R)$ .

Proof. We will use Schlessinger’s criterion to check pro-representability. As $\operatorname{Nil}_{\overline{N}}$ is a subfunctor of the formal neighborhood of the scheme $\text{}\underline{\mathfrak{g}}$ at $\overline{N}$ , the only condition to check is the analogue of Definition 2.3(ii): given a Cartesian diagram in ${\mathcal{C}}_{{\mathcal{O}}}$

and $N_{i}\in \operatorname{Nil}_{\overline{N}}(R_{i})$ such that $N_{1}$ and $N_{2}$ reduce to $N_{0}$ , we want to check that $N_{1}\times N_{2}\in \operatorname{Nil}_{\overline{N}}(R_{1}\times _{R_{0}}R_{2})$ . By definition, there exists $g_{1}\in \widehat{G}(R_{1})$ and $g_{2}\in \widehat{G}(R_{2})$ such that $N_{1}=g_{1}N_{\unicode[STIX]{x1D70E}}g_{1}^{-1}$ and $N_{2}=g_{2}N_{\unicode[STIX]{x1D70E}}g_{2}^{-1}$ . Consider the element $g_{1}g_{2}^{-1}\in \widehat{G}(R_{0})$ . Observe that

$$\begin{eqnarray}g_{1}g_{2}^{-1}N_{\unicode[STIX]{x1D70E}}g_{2}g_{1}^{-1}=g_{1}N_{0}g_{1}^{-1}=N_{\unicode[STIX]{x1D70E}}\in \mathfrak{g}_{R_{0}}.\end{eqnarray}$$

In particular, $g_{1}g_{2}^{-1}\in Z_{G}(N_{\unicode[STIX]{x1D70E}})(R_{0})$ . The extension $R_{2}\rightarrow R_{0}$ has nilpotent kernel, so as $Z_{G}(N_{\unicode[STIX]{x1D70E}})$ is smooth over ${\mathcal{O}}$ there exists $h\in Z_{G}(N_{\unicode[STIX]{x1D70E}})(R_{2})$ lifting $g_{1}g_{2}^{-1}$ . The element

$$\begin{eqnarray}(g_{1},hg_{2})\in R_{1}\times _{R_{0}}R_{2}\end{eqnarray}$$

conjugates $N_{1}\times N_{2}$ to $N_{\unicode[STIX]{x1D70E}}$ . Hence $N_{1}\times N_{2}\in \operatorname{Nil}_{\overline{N}}(R_{1}\times _{R_{0}}R_{2})$ .◻

Lemma 3.11. The functor $\operatorname{Nil}_{\overline{N}}$ is formally smooth, in the sense that for a small surjection $R_{2}\rightarrow R_{1}$ of coefficient ${\mathcal{O}}$ -algebras the map

$$\begin{eqnarray}\operatorname{Nil}_{\overline{N}}(R_{2})\rightarrow \operatorname{Nil}_{\overline{N}}(R_{1})\end{eqnarray}$$

is surjective. Moreover, when $Z_{G}(N_{\unicode[STIX]{x1D70E}})$ is ${\mathcal{O}}$ -smooth and $\operatorname{Nil}_{\overline{N}}$ is representable, it has relative dimension $\dim G_{k}-\dim Z_{G_{k}}(N_{k})$ over ${\mathcal{O}}$ .

4.1 Centralizers over fields

In this section, let $k$ be an algebraically closed field, $G$ be a connected reductive group over $k$ , and $N$ a nilpotent element of $\mathfrak{g}=\operatorname{Lie}G$ . As the formation of the scheme-theoretic centralizer commutes with base change, smoothness results for $Z_{G}(N)$ over $k$ will imply such results over general fields (not necessarily algebraically closed).

The group scheme $Z_{G}(N)$ is the fiber over $0\in \mathfrak{g}$ of the composition

$$\begin{eqnarray}G\overset{\operatorname{Ad}_{G}}{\longrightarrow }\operatorname{GL}(\mathfrak{g})\overset{T\mapsto TN-N}{\longrightarrow }\mathfrak{g}.\end{eqnarray}$$

Hence $\operatorname{Lie}Z_{G}(N)$ is the kernel of

$$\begin{eqnarray}\mathfrak{g}\overset{\operatorname{ad}_{\mathfrak{g}}}{\longrightarrow }\operatorname{End}(\mathfrak{g})\overset{T\mapsto TN}{\longrightarrow }\mathfrak{g}\end{eqnarray}$$

which is the Lie algebra centralizer $\mathfrak{z}_{\mathfrak{g}}(N)$ .

In a wide range of situations, all nilpotent centralizers are smooth. A direct calculation shows that this holds for $G=\operatorname{GL}_{n}$ (see [Reference JantzenJan04, § 2.3]), and a criterion of Richardson [Reference JantzenJan04, Theorem 2.5] can be applied to show to following.

4.2 Checking flatness over a Dedekind base

We want to analyze smoothness of centralizers in the relative setting (especially over $\operatorname{Spec}{\mathcal{O}}$ ). If $Z_{G}(N_{\unicode[STIX]{x1D70E}})\rightarrow \operatorname{Spec}{\mathcal{O}}$ is flat and the special and generic fibers are smooth, then $Z_{G}(N)$ is smooth over ${\mathcal{O}}$ . The following lemma gives a way to check that a morphism to a Dedekind scheme is flat.

Proof. It suffices to prove the result when $S=\operatorname{Spec}(A)$ for $A$ a discrete valuation ring with uniformizer $\unicode[STIX]{x1D70B}$ . Let $X_{\unicode[STIX]{x1D702}}$ be the generic fiber and $X_{s}$ the special fiber. Consider the schematic closure $\imath :X^{\prime }{\hookrightarrow}X$ of the generic fiber. The scheme $X^{\prime }$ is flat over $\operatorname{Spec}(A)$ since flatness is equivalent to being torsion-free over a discrete valuation ring, and there is an exact sequence

(4.1) $$\begin{eqnarray}0\rightarrow J\rightarrow {\mathcal{O}}_{X}\rightarrow \imath _{\ast }{\mathcal{O}}_{X^{\prime }}\rightarrow 0,\end{eqnarray}$$

where $J$ is a coherent sheaf killed by a power of $\unicode[STIX]{x1D70B}$ . We will show that $\imath$ is an isomorphism by analyzing the special fiber.

First, we claim that the dimension of each irreducible component on the special fiber of $X^{\prime }$ is the same as the dimension of the equidimensional $X_{\unicode[STIX]{x1D702}}$ . We will get this from flatness of $X^{\prime }$ . The generic fiber of $X^{\prime }$ is $X_{\unicode[STIX]{x1D702}}$ , which is equidimensional and non-empty by hypothesis. Furthermore, $X^{\prime }$ is the union of the closures $Z_{i}$ of the reduced irreducible components $X_{\unicode[STIX]{x1D702},i}$ of $X_{\unicode[STIX]{x1D702}}$ , and each $Z_{i}$ is $A$ -flat with integral $\unicode[STIX]{x1D702}$ -fiber, hence integral. We just need to analyze the dimension of irreducible components of $(Z_{i})_{s}$ when $(Z_{i})_{s}\neq \emptyset$ . Since $Z_{i}$ is integral, we can apply [Reference MatsumuraMat89, Theorem 15.1, 15.5] to such $Z_{i}$ to conclude that the dimension of each irreducible component of the special fiber of $X^{\prime }$ is the same as the dimension of the generic fiber.

Observe that the sections $\unicode[STIX]{x1D70E}_{i}$ factor through the closed subscheme $X^{\prime }\subset X$ , as we can check this on the generic fiber since $X^{\prime }$ is $A$ -flat. Thus $X^{\prime }$ meets every irreducible component of $X_{s}$ away from the other irreducible components of $X_{s}$ . We would have that $|X_{s}^{\prime }|=|X_{s}|$ if $X_{s}^{\prime }$ is equidimensional of the same dimension as the equidimensional $X_{s}$ . We have shown the dimension of any irreducible component in $X_{s}^{\prime }$ is the same dimension as the common dimension of irreducible components of the generic fiber $X_{\unicode[STIX]{x1D702}}$ of $X^{\prime }$ . By hypothesis, the dimension of any irreducible component of the generic fiber of $X$ is the same as the dimension of any irreducible component of the special fiber of $X$ . Thus the dimension of any irreducible component of $X_{s}^{\prime }$ is the same as the dimension of each irreducible component of $X_{s}$ , giving that $|X_{s}^{\prime }|=|X_{s}|$ . As $X_{s}$ is reduced, this forces $\imath _{s}:X_{s}^{\prime }{\hookrightarrow}X_{s}$ to be an isomorphism.

Now tensoring (4.1) with the residue field of $A$ gives an exact sequence

$$\begin{eqnarray}0\rightarrow J/\unicode[STIX]{x1D70B}J\rightarrow {\mathcal{O}}_{X,s}\rightarrow \imath _{\ast }{\mathcal{O}}_{X^{\prime },s}\rightarrow 0\end{eqnarray}$$

because ${\mathcal{O}}_{X^{\prime }}$ is $A$ -flat. But $J/\unicode[STIX]{x1D70B}J=0$ as $\imath _{s}$ is an isomorphism. Hence $J=\unicode[STIX]{x1D70B}J=\unicode[STIX]{x1D70B}^{2}J=\cdots =\unicode[STIX]{x1D70B}^{n}J=0$ for $n$ large, so $X=X^{\prime }$ is flat over $A$ .◻

4.3 Centralizers for orthogonal and symplectic groups

To apply Corollary 4.6, we need information about the component group of centralizers of nilpotents. For $\operatorname{GL}_{n}$ over a field, all such centralizers are connected. For symplectic and orthogonal groups, there is an explicit description of $Z(N_{\unicode[STIX]{x1D70E}})$ where $N_{\unicode[STIX]{x1D70E}}$ is the nilpotent constructed in § 3. We continue the notation of that section: $G$ is $\operatorname{Sp}_{m}$ or $\operatorname{O}_{m}$ (with $m\geqslant 4$ ) over a discrete valuation ring ${\mathcal{O}}$ with a residue field $k$ of good characteristic $p\neq 2$ .

Let $\unicode[STIX]{x1D70E}:m_{1}+\cdots +m_{r}$ be an admissible partition of $m$ . We assume that ${\mathcal{O}}$ is large enough so that Proposition 3.7 holds, and take $N:=N_{\unicode[STIX]{x1D70E}}$ . Then there exists elements $v_{1},\ldots ,v_{r}\in M:={\mathcal{O}}^{m}$ such that

$$\begin{eqnarray}v_{1},Nv_{1},\ldots ,N^{m_{1}-1}v_{1},v_{2},Nv_{2},\ldots ,N^{m_{r}-1}v_{r}\end{eqnarray}$$

is a basis for $M$ . Furthermore, $N^{m_{i}}v_{i}=0$ for $i=1,\ldots r$ , and the pairing between basis elements is given by $\unicode[STIX]{x1D711}:=\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D70E}}$ . In particular, each $v_{i}$ pairs non-trivially with only one other basis element $X^{d_{i}-1}v_{i^{\ast }}$ , for some $i^{\ast }\in \{1,\ldots ,r\}$ .

To understand the $G$ -centralizer of $N$ , we construct an associated grading of $M$ as in [Reference JantzenJan04, § 3.3,3.4]. This is motivated by the Jacobson–Morosov theory of $\mathfrak{sl}_{2}$ -triples over a field of sufficiently large characteristic, but for symplectic and orthogonal groups it is constructed by hand in characteristic $p\neq 2$ below.

We now record some elementary properties of the preceding construction; all are routine to check, and the analogous proofs over a field may be found in [Reference JantzenJan04, § 3.4]. We have that $M=\bigoplus _{s}M(s)$ , and

$$\begin{eqnarray}v_{i}\in M(-(d_{i}-1)),Nv_{i}\in M(-(d_{i}-1)+2),\ldots ,N^{d_{i}-1}v_{r}\in M(d_{r}-1).\end{eqnarray}$$

Furthermore, we know $N\cdot M(s)\subset M(s+2)$ and $M(s)=N\cdot M(s-2)\oplus L(s)$ for $s\leqslant 0$ .

The dimension of $M(s)$ is $m_{s}(\unicode[STIX]{x1D70E}):=\text{\#}\{j:d_{j}-1\geqslant |s|\}$ . The dimension of $L(s)$ equals $l_{s}(\unicode[STIX]{x1D70E}):=m_{s+1}(\unicode[STIX]{x1D70E})-m_{s}(\unicode[STIX]{x1D70E})$ . Furthermore, the pairing $\unicode[STIX]{x1D711}$ interacts well with the grading: a computation with basis elements gives that $\unicode[STIX]{x1D711}(M(s),M(t))\neq 0$ implies $s+t=0$ .

The above grading on $M$ corresponds to the one-parameter subgroup $\unicode[STIX]{x1D706}:\mathbf{G}_{m}\rightarrow G$ for which the action of $t\in \mathbf{G}_{m}$ on $M(s)$ is given by scaling by $t^{s}$ . The dynamic method (see [Reference Conrad, Gabber and PrasadCGP15, § 2.1]) associates to $\unicode[STIX]{x1D706}$ a parabolic subgroup $P_{G}(\unicode[STIX]{x1D706})$ with Levi $Z_{G}(\unicode[STIX]{x1D706})$ . Define $C_{N}$ and $U_{N}$ to be the scheme-theoretic intersections

$$\begin{eqnarray}\displaystyle C_{N} & = & \displaystyle Z_{G}(N)\cap Z_{G}(\unicode[STIX]{x1D706})=\{g\in Z_{G}(N):gM(i)=M(i)\text{ for all }i\},\nonumber\\ \displaystyle U_{N} & = & \displaystyle Z_{G}(N)\cap U_{G}(\unicode[STIX]{x1D706})=\{g\in Z_{G}(N):(g-1)M^{(i)}\subset M^{(i+1)}\text{ for all }i\}.\nonumber\end{eqnarray}$$

We finally give a concrete description of $C_{N}$ . We first define a pairing on $L(s)$ . Recall that the space $L(s)$ of ‘lowest weight vectors’ in $M(s)$ has basis $\{v_{i}:1-d_{i}=s\}$ . We define a pairing on $L(s)$ by

$$\begin{eqnarray}\unicode[STIX]{x1D713}_{s}(v,w)=\unicode[STIX]{x1D711}(v,N^{-s}w).\end{eqnarray}$$

A direct calculation shows that $\unicode[STIX]{x1D713}_{s}$ is non-degenerate and that $\unicode[STIX]{x1D713}_{s}$ is symmetric if $(-1)^{s}=\unicode[STIX]{x1D716}$ and is alternating if $(-1)^{s}=-\unicode[STIX]{x1D716}$  [Reference JantzenJan04, § 3.7].

A point of $C_{N}$ preserves the grading on $M$ , and since it commutes with the ‘raising operator’ $N$ its action on $M$ is determined by its action on the space $L(s)$ of ‘lowest weight vectors’ in $M(s)$ . So the following fact is no surprise.

Proposition 4.11. There is an isomorphism of algebraic groups

$$\begin{eqnarray}C_{N}\simeq \mathop{\prod }_{s\leqslant 0}\text{}\underline{\operatorname{Aut}}(L(s),\unicode[STIX]{x1D713}_{s}).\end{eqnarray}$$

The corresponding statement over a field is [Reference JantzenJan04, § 3.8 Propositions 2, 3]: the proof is the same.

Example 4.12. Let $G=\operatorname{Sp}_{m}$ . Unraveling when $\unicode[STIX]{x1D713}_{s}$ is symmetric or alternating, we see that

$$\begin{eqnarray}C_{N}\simeq \mathop{\prod }_{s\leqslant 0;s\,\text{odd}}\operatorname{O}(L(s),\unicode[STIX]{x1D713}_{s})\times \mathop{\prod }_{s\leqslant 0;s\,\text{even}}\operatorname{Sp}(L(s),\unicode[STIX]{x1D713}_{s}).\end{eqnarray}$$

The special fibers of the symplectic factors are connected, while the orthogonal factors have two connected components in the special fiber. Thus there are $2^{t}$ connected components, where $t$ is the number of odd $s$ for which $L(s)\neq 0$ . For each component, there is a section $g\in C_{N}({\mathcal{O}})$ meeting that component corresponding to a choice of $\pm \operatorname{Id}\in \operatorname{O}(L(s),\unicode[STIX]{x1D713}_{s})$ for each odd $s$ with $L(s)\neq 0$ . The connected components of $Z_{G}(N)$ are the same as those for $C_{N}$ by Fact 4.9.

Example 4.13. Let $G=\operatorname{O}_{m}$ . We likewise see that

$$\begin{eqnarray}C_{N}\simeq \mathop{\prod }_{s\leqslant 0;s\,\text{even}}\operatorname{O}(L(s),\unicode[STIX]{x1D713}_{s})\times \mathop{\prod }_{s\leqslant 0;s\,\text{odd}}\operatorname{Sp}(L(s),\unicode[STIX]{x1D713}_{s}).\end{eqnarray}$$

Again there are $2^{t}$ connected components of $Z_{G}(N)$ , where $t$ is the number of even $s$ for which $L(s)\neq 0$ .

Now suppose that $G=\operatorname{SO}_{m}$ . The elements $N$ we considered in this section are representatives for some of the nilpotent orbits of $\operatorname{SO}_{m}$ . The group $C_{N}$ has the same structure as for $G=\operatorname{O}_{m}$ , except we require that the overall determinant be $1$ ; this has $2^{t-1}$ connected components. Though $\operatorname{SO}_{m}$ has more nilpotent orbits than $\operatorname{O}_{m}$ , according to Remark 3.3 their representatives are conjugate by an element $\operatorname{O}_{m}(k)$ with determinant $-1$ to the representatives constructed in Proposition 3.7. This shows that there are sections $g\in C_{N}({\mathcal{O}})$ meeting each component.

4.4 Smoothness of centralizers

We now return the case when $G$ is an almost-reductive group over a discrete valuation ring ${\mathcal{O}}$ with residue field $k$ of very good characteristic $p>0$ . Suppose we are given an integral representative $N=N_{\unicode[STIX]{x1D70E}}\in \mathfrak{g}:=\operatorname{Lie}G$ for the nilpotent orbit on geometric fibers corresponding to $\unicode[STIX]{x1D70E}\in {\mathcal{C}}$ as in (3.1): that is, an element such that

$$\begin{eqnarray}N_{k}\in O_{\overline{k},\unicode[STIX]{x1D70E}}\quad \text{and}\quad N_{K}\in O_{\overline{K},\unicode[STIX]{x1D70E}}.\end{eqnarray}$$

Proposition 3.7 provides such $N$ in symplectic and orthogonal cases when $\sqrt{-1},\sqrt{2}\in {\mathcal{O}}^{\times }$ . We wish to check that the $Z_{G}(N)$ is smooth over ${\mathcal{O}}$ . This $N$ satisfies

(4.2) $$\begin{eqnarray}Z_{G_{K}}(N_{K})\text{ and }Z_{G_{k}}(N_{k})\text{ are smooth of the same dimension.}\end{eqnarray}$$

Now we wish to check the conditions necessary to apply Corollary 4.6. We define

$$\begin{eqnarray}A(N)=Z_{G_{k}}(N_{k})(k)/Z_{G_{k}}(N_{k})^{\circ }(k),\end{eqnarray}$$

and study when the following holds:

(4.3) $$\begin{eqnarray}\text{each element of }A(N)\text{ arises from some }s\in Z_{G}(N)({\mathcal{O}}).\end{eqnarray}$$

Note that this checks the criterion in Corollary 4.6 as

$$\begin{eqnarray}Z_{G_{k}}(N_{k})(k)/Z_{G_{k}}(N_{k})^{\circ }(k)=(Z_{G_{k}}(N_{k})/Z_{G_{k}}(N_{k})^{\circ })(k)\end{eqnarray}$$

by Lang’s theorem, and since the irreducible components of $Z_{G_{k}}(N_{K})$ are the same as connected components by smoothness.

We are free to make a local flat extension of ${\mathcal{O}}$ , as it suffices to check flatness after such an extension. In particular, it suffices to check (4.3) when ${\mathcal{O}}$ is Henselian and $k$ is algebraically closed, and $\sqrt{-1},\sqrt{2}\in {\mathcal{O}}$ . Examples 4.12 and 4.13 give such sections when $G=\operatorname{Sp}_{m}$ or $G=\operatorname{SO}_{m}$ . We will use these cases to get a result for similitude groups.

Let $\unicode[STIX]{x1D70B}:\widetilde{G^{\prime }}\rightarrow G^{\prime }$ be the simply connected central cover of the derived group $G^{\prime }$ over ${\mathcal{O}}$ . As $p$ is very good, $\widetilde{G^{\prime }}$ and $G^{\prime }$ have isomorphic Lie algebras via $\unicode[STIX]{x1D70B}$ and $\operatorname{Lie}G^{\prime }$ is a direct factor of $\operatorname{Lie}G$ with complement $\operatorname{Lie}(Z_{G})$ , so we may abuse notation and view $N$ as an element of all of these Lie algebras over ${\mathcal{O}}$ .

Let $S$ be a (split) maximal central torus in $G$ . Consider the isogeny $S\times \widetilde{G^{\prime }}\rightarrow G$ . As $S$ acts trivially on $N$ , we see that $S\times Z_{\widetilde{G^{\prime }}}(N)$ is the preimage of $Z_{G^{\prime }}(N)$ under this isogeny. As $p$ is very good for $G$ , we obtain finite étale surjections

$$\begin{eqnarray}Z_{\widetilde{G^{\prime }}}(N)\rightarrow Z_{G^{\prime }}(N)\quad \text{and}\quad S\times Z_{\widetilde{G^{\prime }}}(N)\rightarrow Z_{G}(N)\end{eqnarray}$$

over ${\mathcal{O}}$ .

For example, this lets us pass between $\operatorname{Sp}_{2n}$ and $\operatorname{GSp}_{2n}$ by way of the projective symplectic group.

5.1 Passing between unipotents and nilpotents

As before, $G$ is either $\operatorname{GSp}_{m}$ or $\operatorname{GO}_{m}$ (or $\operatorname{GL}_{m}$ to recover the results of [Reference Clozel, Harris and TaylorCHT08, § 2.4.4]) over the ring of integers ${\mathcal{O}}$ in a $p$ -adic field with residue field $k$ of characteristic $p>0$ . As always, we assume that $p$ is very good for $G_{k}$ (i.e. $p\neq 2$ ). Let $\mathfrak{g}=\operatorname{Lie}(G)$ .

As in § 3.2, we work with a pure nilpotent $N_{\unicode[STIX]{x1D70E}}\in \mathfrak{g}$ for which $Z_{G}(N_{\unicode[STIX]{x1D70E}})$ is ${\mathcal{O}}$ -smooth, $(N_{\unicode[STIX]{x1D70E}})_{\overline{K}}\in O_{\overline{K},\unicode[STIX]{x1D70E}}$ , and $(N_{\unicode[STIX]{x1D70E}})_{\overline{k}}\in O_{\overline{k},\unicode[STIX]{x1D70E}}$ . Define $\overline{N}:=(N_{\unicode[STIX]{x1D70E}})_{k}$ . We studied deformations of $\overline{N}$ in § 3.2, but will ultimately want to analyze deformations of Galois representations which take on unipotent values at certain elements of a local Galois group. Thus, we need a way to pass between unipotent and nilpotent elements. For classical groups, we can use a truncated version of the exponential and logarithm maps as follows.

Fact 5.1. Suppose that $p\geqslant m$ and that $R$ is an ${\mathcal{O}}$ -algebra. If $A\in \operatorname{Mat}_{m}(R)$ has characteristic polynomial $x^{m}$ , then

$$\begin{eqnarray}\exp (A):=1+A+A^{2}/2+\cdots +A^{m-1}/(m-1)!\end{eqnarray}$$

has characteristic polynomial $(x-1)^{m}$ . If $B\in \operatorname{Mat}_{m}(R)$ has characteristic polynomial $(x-1)^{m}$ , then

$$\begin{eqnarray}\log (B):=(B-1)-(B-1)^{2}/2+\cdots +(-1)^{m}(B-1)^{m-1}/(m-1)\end{eqnarray}$$

has characteristic polynomial $x^{m}$ . Furthermore for $C\in \operatorname{GL}_{m}(R)$ and an integer $q$ , we have

$$\begin{eqnarray}\displaystyle \begin{array}{@{}ll@{}}\bullet ~\exp (CAC^{-1})=C\exp (A)C^{-1}\quad & \bullet ~\exp (\log (B))=B\\ \bullet ~\log (CBC^{-1})=C\log (B)C^{-1}\quad & \bullet ~\exp (qA)=\exp (A)^{q}\\ \bullet ~\log (\exp (A))=A\quad & \bullet ~\log (B^{q})=q\log (B).\end{array} & & \displaystyle \nonumber\end{eqnarray}$$

This is [Reference TaylorTay08, Lemma 2.4]. The key idea is that because all the higher powers of $A$ and $B-1$ vanish and all of the denominators appearing are invertible as $p\geqslant m$ , we can deduce these facts from results about the exponential and logarithm in characteristic zero.

Suppose $J$ is the matrix for a perfect symmetric or alternating pairing over $R$ .

Proof. Directly from the definitions we see that $\exp (A^{T})=\exp (A)^{T}$ . Observe that

$$\begin{eqnarray}\exp (JAJ^{-1})=JBJ^{-1}\quad \text{and}\quad \exp (-A^{T})=(B^{T})^{-1}.\end{eqnarray}$$

Thus $JAJ^{-1}=-A^{T}$ if and only if $(B^{T})^{-1}=JBJ^{-1}$ .◻

We shall use this exponential map to convert pure nilpotents into unipotent elements. Let $R$ be a coefficient ring over ${\mathcal{O}}$ . By Definition 3.9, any pure nilpotent $N\in \operatorname{Nil}_{\overline{N}}(R)$ is $G(R)$ -conjugate to $N_{\unicode[STIX]{x1D70E}}$ , so it has characteristic polynomial $x^{m}$ . Denoting the derived group of $G$ by $G^{\prime }$ , any nilpotent element of $\mathfrak{g}$ lies in $(\mathfrak{g}^{\prime })=(\operatorname{Lie}G^{\prime })$ , so $NJ+JN=0$ (and not just $NJ+JN=\unicode[STIX]{x1D706}J$ for some $\unicode[STIX]{x1D706}\in {\mathcal{O}}$ ). Thus, Corollary 5.2 shows that $\exp (N)\in G(R)$ . This gives an exponential map

(5.1) $$\begin{eqnarray}\exp :\operatorname{Nil}_{\overline{N}}\rightarrow G\end{eqnarray}$$

such that for $g\in \widehat{G}(R)$ , $N\in \operatorname{Nil}_{\overline{N}}(R)$ , and $q\in \mathbf{Z}$ we have $\exp (qN)=\exp (N)^{q}$ and $g\exp (N)g^{-1}=\exp (\operatorname{Ad}_{G}(g)N)$ .

5.2 Minimally ramified deformations

As before, $G$ is $\operatorname{GSp}_{m}$ , $\operatorname{GO}_{m}$ or $\operatorname{GL}_{m}$ over the ring of integers ${\mathcal{O}}$ of a $p$ -adic field with residue field $k$ with $p>m$ . Let $L$ be a finite extension of $\mathbf{Q}_{\ell }$ (with $\ell \neq p$ ), and denote its absolute Galois group by $\unicode[STIX]{x1D6E4}_{L}$ . Consider a representation $\overline{\unicode[STIX]{x1D70C}}:\unicode[STIX]{x1D6E4}_{L}\rightarrow G(k)$ . We wish to define a (large) smooth deformation condition for $\overline{\unicode[STIX]{x1D70C}}$ generalizing the minimally ramified deformation condition for $\operatorname{GL}_{n}$ defined in [Reference Clozel, Harris and TaylorCHT08, § 2.4.4]. In this section we do so for a special class of tamely ramified representations. This requires making an étale local extension of ${\mathcal{O}}$ , which will be harmless for our purposes.

Recall that $\unicode[STIX]{x1D6E4}_{L}^{\operatorname{t}}$ , the Galois group of the maximal tamely ramified extension of $L$ , is isomorphic to the semidirect product

$$\begin{eqnarray}\widehat{\mathbf{Z}}\ltimes \mathop{\prod }_{p^{\prime }\neq \ell }\mathbf{Z}_{p^{\prime }},\end{eqnarray}$$

where $\widehat{\mathbf{Z}}$ is generated by a Frobenius $\unicode[STIX]{x1D719}$ and the conjugation action by $\unicode[STIX]{x1D719}$ on the direct product is given by the cyclotomic character. We consider representations of $\unicode[STIX]{x1D6E4}_{L}^{\operatorname{t}}$ which factor through the quotient $\widehat{\mathbf{Z}}\ltimes \mathbf{Z}_{p}$ . Picking a topological generator $\unicode[STIX]{x1D70F}$ for $\mathbf{Z}_{p}$ , the action is explicitly given by

$$\begin{eqnarray}\unicode[STIX]{x1D719}\unicode[STIX]{x1D70F}\unicode[STIX]{x1D719}^{-1}=q\unicode[STIX]{x1D70F},\end{eqnarray}$$

where $q$ is the size of the residue field of $L$ . Note $q$ is a power of $\ell$ , so it is relatively prime to $p$ . This leads us to study representations of the group $T_{q}:=\widehat{\mathbf{Z}}\ltimes \mathbf{Z}_{p}$ .

Let $\overline{\unicode[STIX]{x1D70C}}:T_{q}\rightarrow G(k)$ be such a representation. We first claim that $\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F})\in G(k)$ is unipotent. This element decomposes as a commuting product of semisimple and unipotent elements of $G(k)$ . The order of a semisimple element in $G(k)$ is prime to $p$ , while by continuity there is an $r\geqslant 0$ such that $\unicode[STIX]{x1D70F}^{p^{r}}\in \ker (\overline{\unicode[STIX]{x1D70C}})$ . Thus $\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F})$ is unipotent.

Informally, a deformation $\unicode[STIX]{x1D70C}:T_{q}\rightarrow G(R)$ will be minimally ramified if $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70F})$ lies in the ‘same’ unipotent orbit as $\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F})$ . To make this meaningful over an infinitesimal thickening of $k$ , we shall use the notion of pure nilpotents as in Definition 3.9 since unipotence and unipotent orbits are not good notions when not over a field. As $\overline{N}:=\log (\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F}))$ is nilpotent, by Remark 3.8 after making an étale local extension of ${\mathcal{O}}$ we may assume that there exists a pure nilpotent $N_{\unicode[STIX]{x1D70E}}\in \mathfrak{g}$ lifting $\overline{N}$ for which $Z_{G}(N_{\unicode[STIX]{x1D70E}})$ is smooth. Making a further extension if necessary, we may also assume that the unit $q\in {\mathcal{O}}^{\times }$ is a square. We obtain an exponential map $\exp :\operatorname{Nil}_{\overline{N}}\rightarrow G$ as in (5.1).

Example 5.5. Take $G=\operatorname{GL}_{m}$ . Then $X\mapsto 1_{m}+X$ gives an identification of nilpotents and unipotents. Up to conjugacy, over algebraically closed fields parabolic subgroups correspond to partitions of $m$ and every nilpotent orbit is the Richardson orbit of such a parabolic. Let $\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F})-1_{m}=:\overline{N}$ correspond to the partition $\unicode[STIX]{x1D70E}=n_{1}+n_{2}+\cdots +n_{r}$ . By Example 3.2, the lift $N_{\unicode[STIX]{x1D70E}}$ of $\overline{N}$ is conjugate to a block nilpotent matrix with blocks of size $n_{1},n_{2},\ldots ,n_{r}$ . The points $N\in \operatorname{Nil}_{\overline{N}}(R)$ are the $\widehat{G}(R)$ -conjugates of $N_{\unicode[STIX]{x1D70E}}$ . It is clear (since $p>m$ ) that if $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70F})\in \operatorname{Nil}_{\overline{N}}(R)$ , then

(5.2) $$\begin{eqnarray}\displaystyle \ker (\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70F})-1_{m})^{i}\otimes _{R}k\rightarrow \ker (\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F})-1_{m})^{i} & & \displaystyle\end{eqnarray}$$

is an isomorphism for all $i$ . Conversely, repeated applications of [Reference Clozel, Harris and TaylorCHT08, Lemma 2.4.15] show that any $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70F})$ satisfying this collection of isomorphism conditions is $\widehat{G}(R)$ -conjugate to $N_{\unicode[STIX]{x1D70E}}$ . Thus the minimally ramified deformation condition for $\operatorname{GL}_{m}$ defined in [Reference Clozel, Harris and TaylorCHT08] agrees with our definition. Note that the identification $X\mapsto 1_{m}+X$ does not satisfy $qX\rightarrow (1+X)^{q}$ , so it will not work in our argument. The proof of [Reference Clozel, Harris and TaylorCHT08, Lemma 2.4.19] uses a different method for which this non-homomorphic identification suffices.

The functor of minimally ramified lifts is pro-representable by a ring $R_{\overline{\unicode[STIX]{x1D70C}}}^{\operatorname{m.r.}\Box }$ as it suffices to specify images of $\unicode[STIX]{x1D70F}$ and $\unicode[STIX]{x1D719}$ subject to constraints that $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70F})=\exp (N)$ and $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D719})\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70F})\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D719})^{-1}=\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70F})^{q}$ .

Proof. Let $\overline{\unicode[STIX]{x1D6F7}}=\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D719})\in G(k)$ and let $\widehat{G}_{\overline{\unicode[STIX]{x1D6F7}}}$ be the formal completion of $G$ at $\overline{\unicode[STIX]{x1D6F7}}$ . Using the relation

$$\begin{eqnarray}\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D719})\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F})\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D719})^{-1}=\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F})^{q},\end{eqnarray}$$

we deduce that $\overline{\unicode[STIX]{x1D6F7}}\,\overline{N}\,\overline{\unicode[STIX]{x1D6F7}}^{-1}=q\overline{N}$ . Therefore we study the functor $M_{\overline{N}}$ on $\widehat{{\mathcal{C}}}_{{\mathcal{O}}}$ defined by

$$\begin{eqnarray}M_{\overline{N}}(R)=\{(\unicode[STIX]{x1D6F7},N):N\in \operatorname{Nil}_{\overline{N}}(R),\,\unicode[STIX]{x1D6F7}\in \widehat{G}_{\overline{\unicode[STIX]{x1D6F7}}}(R),\,\unicode[STIX]{x1D6F7}N\unicode[STIX]{x1D6F7}^{-1}=qN\}\subset \operatorname{Nil}_{\overline{N}}(R)\times \widehat{G}_{\overline{\unicode[STIX]{x1D6F7}}}(R).\end{eqnarray}$$

Any such lift $(\unicode[STIX]{x1D6F7},N)$ to a coefficient ring $R$ determines a homomorphism $T_{q}\rightarrow G(R)$ lifting $\overline{\unicode[STIX]{x1D70C}}$ via $\unicode[STIX]{x1D719}\mapsto \unicode[STIX]{x1D6F7}$ and $\unicode[STIX]{x1D70F}\mapsto \exp (N)$ : it is continuous because $\exp (\overline{N})$ is unipotent. We will analyze $M_{\overline{N}}$ through the composition

$$\begin{eqnarray}M_{\overline{N}}\rightarrow \operatorname{Nil}_{\overline{N}}\rightarrow \operatorname{Spf}{\mathcal{O}}.\end{eqnarray}$$

First, observe that $M_{\overline{N}}\rightarrow \operatorname{Nil}_{\overline{N}}$ is relatively representable as ‘ $\unicode[STIX]{x1D6F7}N=qN\unicode[STIX]{x1D6F7}$ ’ is a formal closed condition on points $\unicode[STIX]{x1D6F7}$ of $(\widehat{G}_{\overline{\unicode[STIX]{x1D6F7}}})_{R}$ for each $N\in \operatorname{Nil}_{\overline{N}}(R)$ .

From Lemma 3.11, we know that $\operatorname{Nil}_{\overline{N}}$ is formally smooth over ${\mathcal{O}}$ , and the universal nilpotent is $gN_{\unicode[STIX]{x1D70E}}g^{-1}$ for some $g\in \widehat{G}(\operatorname{Nil}_{\overline{N}})$ . To check formal smoothness of the map $M_{\overline{N}}\rightarrow \operatorname{Nil}_{\overline{N}}$ , it therefore suffices to check the formal smoothness of the fiber of $M_{\overline{N}}$ over the ${\mathcal{O}}$ -point $N_{\unicode[STIX]{x1D70E}}$ of $\operatorname{Nil}_{\overline{N}}$ .

We have written down $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70E}}\in G({\mathcal{O}})$ satisfying $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70E}}N_{\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70E}}^{-1}=qN_{\unicode[STIX]{x1D70E}}$ in Remark 4.14. Observe that $\overline{\unicode[STIX]{x1D6F7}}\,\overline{\unicode[STIX]{x1D6F7}}_{\unicode[STIX]{x1D70E}}^{-1}\in Z_{G}(N_{\unicode[STIX]{x1D70E}})(k)$ . By smoothness, we may lift $\overline{\unicode[STIX]{x1D6F7}}\,\overline{\unicode[STIX]{x1D6F7}}_{\unicode[STIX]{x1D70E}}^{-1}$ to an element $s\in Z_{G}(N_{\unicode[STIX]{x1D70E}})({\mathcal{O}})$ . Then $s\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70E}}$ reduces to $\overline{\unicode[STIX]{x1D6F7}}$ and satisfies $(s\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70E}})N_{\unicode[STIX]{x1D70E}}(s\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D70E}})^{-1}=qN_{\unicode[STIX]{x1D70E}}$ , so the fiber of $M_{\overline{N}}$ over $N_{\unicode[STIX]{x1D70E}}$ has an ${\mathcal{O}}$ -point. The relative dimension of the formally smooth $\operatorname{Nil}_{\overline{N}}$ is $\dim G_{k}-\dim Z_{G_{k}}(\overline{N})$ by Lemma 3.11, and $M_{\overline{N}}\rightarrow \operatorname{Nil}_{\overline{N}}$ is a $\widehat{Z_{G}(N_{\unicode[STIX]{x1D70E}})}$ -torsor since it has an ${\mathcal{O}}$ -point over $N_{\unicode[STIX]{x1D70E}}$ . As $Z_{G}(N_{\unicode[STIX]{x1D70E}})$ is smooth it follows that $M_{\overline{N}}$ is formally smooth over $\operatorname{Spf}{\mathcal{O}}$ of relative dimension $\dim G_{k}$ .◻

Let $S$ be the (torus) quotient of $G$ by its derived group $G^{\prime }$ , and $\unicode[STIX]{x1D707}:G\rightarrow S$ the quotient map. For use later, we now study a variant where we fix a lift $\unicode[STIX]{x1D708}:T_{q}\rightarrow S({\mathcal{O}})$ of $\unicode[STIX]{x1D707}\circ \overline{\unicode[STIX]{x1D70C}}:T_{q}\rightarrow S(k)$ , as follows.

5.3 Deformation conditions based on parabolic subgroups

The use of nilpotent orbits is not the only approach to defining a deformation condition at ramified places not above $p$ . As discussed in the introduction, the method used to prove [Reference Clozel, Harris and TaylorCHT08, Lemma 2.4.19] suggests a generalization from $\operatorname{GL}_{n}$ to other groups $G$ based on deformations lying in certain parabolic subgroups of $G$ . This deformation condition is not smooth for algebraic groups beyond $\operatorname{GL}_{n}$ , so it does not work in Ramakrishna’s method. In this section we give a conceptual explanation for this phenomenon.

Let $P\subset G$ be a parabolic ${\mathcal{O}}$ -subgroup. The Richardson orbit for $P_{k}$ intersects $(\operatorname{Lie}R_{u}P)_{k}$ in a dense open set which is a single geometric orbit under $P_{k}$ . Suppose that $\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F})$ is the exponential of a $k$ -point $\overline{N}$ in the Richardson orbit, and consider deformations $\unicode[STIX]{x1D70C}:T_{q}\rightarrow G({\mathcal{O}})$ of $\overline{\unicode[STIX]{x1D70C}}$ ramified with respect to $P$ in the sense that $\unicode[STIX]{x1D70C}(\unicode[STIX]{x1D70F})\in P$ (compare with Definition 1.2). This requires specifying a lift of $\overline{N}$ that lies in $\operatorname{Lie}P$ . One could hope that such lifts would automatically be $G({\mathcal{O}})$ -conjugate to the fixed lift $N_{\unicode[STIX]{x1D70E}}$ defined in Proposition 3.7, reminiscent of the definition we gave for $\operatorname{Nil}_{\overline{N}}$ , a situation in which the associated deformation (or lifting) ring is smooth.

We now show that often smoothness fails if $\overline{N}$ does not lie in the Richardson orbit of $P_{k}$ . Lifts of $\overline{N}$ can ‘change nilpotent type’ yet still lie in a parabolic lifting $P_{k}$ , such as the example of the standard Borel subgroup in $\operatorname{GL}_{3}$ with

$$\begin{eqnarray}N=\left(\begin{array}{@{}ccc@{}}0 & 1 & 0\\ 0 & 0 & p\\ 0 & 0 & 0\end{array}\right)\quad \text{lifting}\quad \overline{N}=\left(\begin{array}{@{}ccc@{}}0 & 1 & 0\\ 0 & 0 & 0\\ 0 & 0 & 0\end{array}\right).\end{eqnarray}$$

In particular, we easily obtain non-pure nilpotents. This is very bad: the nilpotent orbits over a field are smooth but the nilpotent cone is not smooth, so the deformation problem of deforming with respect to $P$ should not be smooth because ‘it sees multiple orbits’. Furthermore, even if we could lift $\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F})$ appropriately, there would still be problems lifting $\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D719})$ because the centralizer of a non-pure nilpotent is not smooth over ${\mathcal{O}}$ (the special and generic fiber typically have different dimensions). So it is crucial to choose a parabolic such that $\overline{N}$ lies in the Richardson orbit of $P_{k}$ .

For $\operatorname{GL}_{n}$ , all nilpotent orbits are Richardson orbits. This is not true in general. In particular, we should not expect the deformation condition of being ramified with respect to a parabolic to be liftable. Example 1.3 illustrates this phenomenon for $\operatorname{GSp}_{4}$ , which we now revisit in a more conceptual manner.

Example 5.9. Take $G=\operatorname{GSp}_{4}$ . Parabolic subgroups correspond to isotropic flags. Up to conjugacy, these subgroups are stabilizers of the flags

$$\begin{eqnarray}\displaystyle & \displaystyle 0\subset \operatorname{Span}(v_{1})\subset \operatorname{Span}(v_{1},v_{2})\subset \operatorname{Span}(v_{1},v_{2},v_{3})\subset k^{4},\quad 0\subset k^{4} & \displaystyle \nonumber\\ \displaystyle & \displaystyle 0\subset \operatorname{Span}(v_{1})\subset \operatorname{Span}(v_{1},v_{2},v_{3})\subset k^{4},\quad 0\subset \operatorname{Span}(v_{1},v_{2})\subset k^{4}, & \displaystyle \nonumber\end{eqnarray}$$

where $\{v_{1},v_{2},v_{2},v_{4}\}$ is a basis of $k^{4}$ . Their Richardson orbits correspond to the nilpotent orbits indexed respectively by the partitions $1+1+1+1$ , $4$ , $4$ , and $2+2$ . In particular, the same nilpotent orbit is associated with two flags, and the nilpotent orbit corresponding to the partition $2+1+1$ does not appear. This corresponds to a nilpotent orbit that is not a Richardson orbit; for the representation in Example 1.3, $\log (\overline{\unicode[STIX]{x1D70C}}(\unicode[STIX]{x1D70F}))$ is in this nilpotent orbit.