2.1 Local fields and extensions

Fix a prime number $p\gt0$ . Let k be a perfect field of characteristic p and fix an algebraic closure $\bar{k}$ of k. Let F be a complete discrete-valued field of characteristic 0 with valuation ring $\mathcal{O}$ and residue field k. Fix a uniformizer $\varpi\in\mathcal{O}$ . Let $\mathrm{val}:F\to\mathbb{Z}\cup\{\infty\}$ be the normalized valuation on F with $\mathrm{val}(\varpi)=1$ . Fix a (separable) algebraic closure $\overline{F}$ of F. For each positive integer m, let $\mu_m$ be the group of mth roots of unity in $\overline{F}$ . The valuation val on F extends uniquely to $\overline{F}$ , and the associated p-adic absolute value on $\overline{F}$ is defined by $|x|:=p^{-\mathrm{val}(x)}$ for any $x\in\overline{F}$ . Let $\overline{\mathcal{O}}\subset\overline{F}$ be the valuation ring, which is also the integral closure of $\mathcal{O}$ in $\overline{\mathcal{O}}$ . Let $F^{\mathrm{ur}}$ be the maximal unramified extension of F in $\overline{F}$ . Let $\breve{F}$ be the completion of $F^{\mathrm{ur}}$ and let $\breve{\mathcal{O}}\subset\breve{F}$ be its valuation ring. Let $\Gamma:=\mathrm{Gal}(\overline{F}/F)$ be the absolute Galois group of F and let $\Gamma_1=\mathrm{Gal}(\overline{F}/F^{\mathrm{ur}})$ be the inertia subgroup, which is canonically isomorphic to the absolute Galois group of $\breve{F}$ .

For any perfect k-algebra R, we let W(R) be its ring of p-typical Witt vectors. Let $\mathcal{O}_0=W(k)$ and let $F_0=W(k)[{1}/{p}]$ be the fraction field of $\mathcal{O}_0$ . Then, F is a finite totally ramified extension of $F_0$ . Let $W_{\mathcal{O}}(R):=W(R)\otimes_{W(k)}\mathcal{O}$ be the ring of $\mathcal{O}$ -ramified Witt vectors. In particular, we have $\breve{\mathcal{O}}=W_{\mathcal{O}}(\bar{k})$ . There is a unique group homomorphism

\begin{align*} [\cdot]:R^\times\to W_{\mathcal{O}}(R)^\times \end{align*}

that lifts the identity map modulo $\varpi$ . When $R=\bar{k}$ , the image $[\bar{k}^\times]\subset\breve{\mathcal{O}}$ is called the set of Teichmüller representatives of $\bar{k}^\times$ .

In the remainder of the article, except in § 5.3, we assume that $k=\bar{k}$ so that $F=\breve{F}$ , $\mathcal{O}=\breve{\mathcal{O}}$ .

2.2 Quasi-split groups and parahoric subgroups

Let G be a connected reductive group over F. Since F is complete and the residue field k is algebraically closed, G is automatically quasi-split by Steinberg’s theorem. We fix a pinning of G as follows. Let $F'/F$ be the minimal Galois extension such that G becomes split over F’. Let $\mathbb{G}$ be the split connected reductive group scheme over $\mathbb{Z}$ with Lie algebra $\mathbb{g}:=\mathrm{Lie}(\mathbb{G})$ , together with an isomorphism $\mathbb{G}_{F'}\cong G_{F'}$ . Fix a pinning $(\mathbb{T},\mathbb{B},(x_\alpha)_{\alpha\in\Delta})$ of $\mathbb{G}$ defined over $\mathbb{Z}$ , consisting of a split maximal torus $\mathbb{T}$ and a Borel subgroup $\mathbb{B}$ containing $\mathbb{T}$ and for each simple root $\alpha\in\Delta$ , a collection of $\mathbb{Z}$ -basis vectors $x_\alpha\in\mathbb{g}_\alpha(\mathbb{Z})$ in the root space $\mathbb{g}_\alpha\subset\mathbb{g}$ . Let $\mathbb{t}$ be the Lie algebra of $\mathbb{T}$ , viewed either as a $\mathbb{Z}$ -module or a scheme. Let $\mathsf{G}:=\mathbb{G}_k$ (respectively, $\mathsf{T}:=\mathbb{T}_k$ , $\mathsf{B}:=\mathbb{B}_k$ ) be the base change of $\mathbb{G}$ (respectively, $\mathbb{T},\mathbb{B}$ ) to k. In the following, we often simplify notation and still use $\mathbb{G}$ , $\mathbb{T}$ , etc. to denote their base change to other rings. We hope this will not cause any confusion since, in each case, it will be clear from the context what are the base rings.

The group $\mathrm{Out}(\mathbb{G})$ of outer automorphisms of $\mathbb{G}$ can be identified with the subgroup of the automorphism group scheme $\mathrm{Aut}(\mathbb{G})$ that stabilizes this pinning. The group G is the twisted form of $\mathbb{G}$ associated to a homomorphism

\begin{align*} \rho_G:\mathrm{Gal}(F'/F)\to\mathrm{Out}(\mathbb{G}). \end{align*}

We also get a Borel subgroup B and a maximal torus T of G, both defined over F. Let $\mathfrak{t}$ (respectively, $\mathfrak{b}$ ) be the Lie algebra of T (respectively, B).

Let $\mathcal{B}(G)$ be the (reduced) Bruhat–Tits building and let $\widetilde{\mathcal{B}}(G)$ be the extended Bruhat–Tits building of G(F). By definition, $\mathcal{B}(G)$ is also the Bruhat–Tits building of $G_{\mathrm{ad}}(F)$ or $G^{\mathrm{sc}}(F)$ . By Bruhat–Tits theory (cf. [Reference Kaletha and PrasadKP23] for a modern reference), to each point $x\in\widetilde{\mathcal{B}}(G)$ in the extended building is associated a sequence of open bounded (mod center) subgroups of G(F):

\begin{align*} \mathbf{P}_{x+}\subset\mathbf{P}_x\subset\mathbf{P}_x^\dagger\subset\widetilde{\mathbf{P}}_x, \end{align*}

where $\mathbf{P}_x$ is the parahoric subgroup, $\mathbf{P}_{x+}$ is the pro-unipotent radical of $\mathbf{P}_x$ , $\mathbf{P}_x^\dagger$ is the stabilizer of x in G(F), and $\widetilde{\mathbf{P}}_x$ is the normalizer of $\mathbf{P}_x$ in G(F) (which coincides with the stabilizer in G(F) of the image of x in the reduced building $\mathcal{B}(G)$ , see [Reference Kaletha and PrasadKP23, § 4.3]). We omit the subscript ‘x’ when it does not play any role and simply write $\mathbf{P}_+,\mathbf{P},\mathbf{P}^\dagger,\widetilde{\mathbf{P}}$ instead. For example, if $G=\mathrm{GL}_n$ and $\mathbf{P}=G(\mathcal{O})$ is the hyperspecial parahoric subgroup, then we have $G(\mathcal{O})_+=\ker(G(\mathcal{O})\to G(k))$ , $\mathbf{P}^\dagger=\mathbf{P}$ , and $\widetilde{\mathbf{P}}=\mathbf{P} Z_G(F)$ . We note that the group $\mathbf{P}^\dagger$ is bounded, while $\widetilde{\mathbf{P}}$ is only bounded mod center but not bounded in general.

For each parahoric subgroup $\mathbf{P}\subset G(F)$ , we let $\mathcal{G}_{\mathbf{P}}$ (respectively, $\mathcal{G}_{\mathbf{P}}^\dagger$ ) be the smooth group scheme over $\mathcal{O}$ with generic fiber G and group of $\mathcal{O}$ -points $\mathcal{G}_{\mathbf{P}}(\mathcal{O})=\mathbf{P}$ (respectively, $\mathcal{G}^\dagger_{\mathbf{P}}(\mathcal{O})=\mathbf{P}^\dagger$ ). Let $\mathsf{G}_{\mathbf{P}}$ (respectively, $\mathsf{G}^\dagger_{\mathbf{P}}$ ) be the reductive quotient of the special fiber of $\mathcal{G}_{\mathbf{P}}$ (respectively, $\mathcal{G}^\dagger_{\mathbf{P}}$ ). Then, $\mathcal{G}_{\mathbf{P}}$ is the fiberwise identity component of $\mathcal{G}^\dagger_{\mathbf{P}}$ and $\mathsf{G}_{\mathbf{P}}$ is the identity component of $\mathsf{G}^\dagger_{\mathbf{P}}$ .

2.3 Restriction of scalars and fixed-point loci

We explicitly construct some parahoric group schemes by restriction of scalars and taking fixed points under finite group actions. Here we review the related facts, following [Reference Bosch, Lütkebohmert and RaynaudBLR90] and [Reference EdixhovenEdi92].

Let $S'\to S$ be a morphism of schemes. For any scheme X’ over S’, recall that the restriction of scalars of X’ along $S'\to S$ , denoted $\mathrm{Res}_{S'/S}X'$ , is the functor that sends any S-scheme T to the set

\begin{align*} (\mathrm{Res}_{S'/S}X')(T):=\mathrm{Hom}_{S'}(T\times_SS',X'). \end{align*}

When $S=\mathrm{Spec} A$ and $S'=\mathrm{Spec} A'$ are affine schemes, we also write $\mathrm{Res}_{A'/A}X'$ instead of $\mathrm{Res}_{S'/S}X'$ .

1. (i) The restriction of scalars $\mathrm{Res}_{A'/A}X'$ is represented by an affine scheme.

2. (ii) If X’ is smooth over $\mathrm{Spec} A'$ , then $\mathrm{Res}_{A'/A}X'$ is smooth over $\mathrm{Spec} A$ .

The following result is a special case of [Reference EdixhovenEdi92, Proposition 3.4]. For the reader’s convenience, we provide a more straightforward proof in the current setting.

Since $X=\mathrm{Spec} A$ is smooth over S, there exists an $\mathcal{O}_S$ -algebra homomorphism $f:A\to R$ making the following diagram commutative.

In particular, we have $f(J)\subset I$ , so for any $\sigma\in\Sigma$ , there exists a map $i_\sigma:A\to I$ such that $f(\sigma(a))=f(a)+i_\sigma(a)$ for all $a\in A$ . For each $\sigma\in\Sigma$ , from the identity $f(\sigma(ab))=f(\sigma(a))f(\sigma(b))$ for any $a,b\in A$ , we get that

\begin{align*} i_\sigma(ab)=f(b)i_\sigma(a)+f(a)i_\sigma(b). \end{align*}

Define an $\mathcal{O}_S$ -linear map $\tilde{f}:A\to R$ by

\begin{align*} \tilde{f}(a)=\frac{1}{|\Sigma|}\sum_{\sigma\in\Sigma}f(\sigma(a)). \end{align*}

By the previous identity and the fact that $I^2=0$ , we get that for any $a,b\in A$ ,

\begin{equation*} \begin{split} \tilde{f}(ab)&=f(ab)+\frac{1}{|\Sigma|}\sum_{\sigma\in\Sigma}i_\sigma(ab)\\ &=f(a)f(b)+\frac{1}{|\Sigma|}f(b)\sum_{\sigma\in\Sigma}i_\sigma(a)+\frac{1}{|\Sigma|}f(a)\sum_{\sigma\in\Sigma}i_\sigma(b)\\ &=\bigg({\kern-1pt}f(a)+\frac{1}{|\Sigma|}\sum_{\sigma\in\Sigma}i_\sigma(a)\!\bigg)\bigg({\kern-1pt}f(b)+\frac{1}{|\Sigma|}\sum_{\sigma\in\Sigma}i_\sigma(b)\!\bigg)\\ &=\tilde{f}(a)\tilde{f}(b). \end{split} \end{equation*}

Thus, $\tilde{f}$ is an algebra homomorphism and by definition it factors through an $\mathcal{O}_S$ -algebra homomorphism $A/J\to R$ that lifts $\bar{f}$ .

2.4 Special integral model for tamely ramified groups

Let F be as in § 2.1 and assume that its residue field k is algebraically closed. Let G be a connected reductive group over F as in § 2.2. In this subsection, we assume moreover that the splitting extension $F'/F$ of G is tamely ramified of degree $e\geqslant1$ , $p\nmid e$ . We construct G together with a special integral model $\mathcal{G}_0$ and an Iwahori subgroup $\mathbf{I}\subset G(F)$ explicitly.

Let $\mathcal{O}'$ be the valuation ring of F’ and choose a uniformizer $\varpi'\in \mathcal{O}'$ such that $\varpi=(\varpi')^e$ is a uniformizer of $\mathcal{O}$ . Then, we have a canonical isomorphism $\mathrm{Gal}(F'/F)\cong\mu_e$ , where $\mu_e$ is the group of eth roots of unit in F.

Let $\mathcal{G}_0^\dagger:=(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{G})^{\mu_e}$ , where $\mu_e$ acts simultaneously on $\mathcal{O}'$ and $\mathbb{G}$ (through $\rho_G$ ). Let $\mathcal{G}_0$ be the fiberwise identity component of $\mathcal{G}_0^\dagger$ . Similarly, we let $\mathcal{T}$ be the fiberwise identity component of $(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{T})^{\mu_e}$ . By Lemmas 2.1 and 2.2, the group schemes $\mathcal{G}_0,\mathcal{G}_0^\dagger,\mathcal{T}$ are affine and smooth over $\mathcal{O}$ . The special fiber of $\mathcal{G}_0$ is $\mathsf{G}_0:=\mathsf{G}^{\mu_e,\circ}$ , the identity component of the fixed-point loci of the $\rho_G(\mu_e)$ action on $\mathsf{G}$ . Let $\mathsf{A}:=\mathbb{T}^{\mu_e,\circ}$ be the special fiber of $\mathcal{T}$ . Then, $\mathsf{G}_0$ is a reductive group over k and $\mathsf{A}$ is a maximal torus of $\mathsf{G}_0$ .

Let $\mathbf{P}_0^\dagger:=\mathcal{G}_0^\dagger(\mathcal{O})$ and $\mathbf{P}_0:=\mathcal{G}_0(\mathcal{O})$ . Then, $\mathbf{P}_0$ is a special parahoric subgroup of G(F) corresponding to a special point that we denote by ‘0’ in the Bruhat–Tits building $\mathcal{B}(G)$ . By construction, there is a natural reduction map $\mathrm{Red}_0:\mathcal{G}_0^\dagger\to\mathsf{G}$ whose image is $\mathsf{G}^{\mu_e}$ . It restricts to a surjective homomorphism (that we denote by the same symbol)

(2.1) \begin{equation} \mathrm{Red}_0:\mathcal{G}_0\to\mathsf{G}_0.\end{equation}

Let $\mathbf{I}\subset\mathcal{G}_0(\mathcal{O})$ be the inverse image of $\mathbb{B}(k)\cap\mathsf{G}_0(k)$ under $\mathrm{Red}_0$ . Then $\mathbf{I}$ is an Iwahori subgroup of G(F) which corresponds to an alcove in $\mathcal{B}(G)$ whose closure contains the special point 0.

In the following, for simplicity, we often denote the special $\mathcal{O}$ -model $\mathcal{G}_0$ also by G and denote $\mathfrak{g}=\mathrm{Lie}(\mathcal{G}_0)$ so that we can write $\mathbf{P}_0=G(\mathcal{O})$ , $\mathrm{Lie}(\mathbf{P}_0)=\mathfrak{g}(\mathcal{O})$ , etc. If G is split, then $F'=F$ , $\mathcal{G}_0=\mathcal{G}_0^\dagger=\mathbb{G}=G$ , and $\mathbf{P}_0=\mathbf{P}_0^\dagger=G(\mathcal{O})$ is a hyperspecial parahoric subgroup of G(F).

Our main theorems are valid under a condition that is slightly more general than being tamely ramified. We formulate it as follows.

2.5 Conditions on the residue characteristic

We review some standard conditions on the residue characteristic p of F determined by a reductive group.

First, let us recall the notion of bad primes and torsion primes for a root system, following [Reference Springer and SteinbergSS70, Chapter I, § 4].

A prime number p is bad (respectively, torsion) for a reductive group or a reductive Lie algebra over a field if it is bad (respectively, torsion) for its (absolute) root system. A prime number p is good for a root system (or reductive group, or reductive Lie algebra) if it is not bad for it.

The bad primes for an irreducible root system $\Sigma$ are listed according to the types of $\Sigma$ as follows.

• – Type $\mathrm{A}_n$ : none.

• – Type $\mathrm{B}_n(n\geqslant2),\mathrm{C}_n(n\geqslant3),\mathrm{D}_n(n\geqslant4)$ : $p=2$ .

• – Type $\mathrm{E}_6,\mathrm{E}_7,\mathrm{F}_4,\mathrm{G}_2$ : $p=2,3$ .

• – Type $\mathrm{E}_8$ : $p=2,3,5$ .

The torsion primes for an irreducible root system $\Sigma$ are listed according to the types of $\Sigma$ as follows.

• – Type $\mathrm{A}_n,\mathrm{C}_n(n\geqslant2)$ : none.

• – Type $\mathrm{B}_n(n\geqslant3), \mathrm{D}_n(n\geqslant4),\mathrm{G}_2$ : $p=2$ .

• – Type $\mathrm{E}_6,\mathrm{E}_7,\mathrm{F}_4$ : $p=2,3$ .

• – Type $\mathrm{E}_8$ : $p=2,3,5$ .

A prime number p is bad (respectively, torsion) for a general root system if it is so for some of its simple factors. We observe from the list above that the torsion primes for a root system are also bad primes (but not vice versa). Moreover, any bad prime of a root system divides the order of its Weyl group by [Reference Springer and SteinbergSS70, I.4.10].

Sometimes, we need conditions on p that depend not only on the root system but also the root datum of a reductive group. Let $\mathcal{R}=(X,\Sigma,X^\vee,\Sigma^\vee)$ be a root datum consisting of a pair of finite free $\mathbb{Z}$ -modules $X,X^\vee$ dual to each other, and a subset $\Sigma\subset X$ (respectively, $\Sigma^\vee\subset X^\vee$ ) of roots (respectively, coroots). Let $\mathcal{R}^{\mathrm{sc}}:=(X^{\mathrm{sc}},\Sigma,(X^{\mathrm{sc}})^\vee, \Sigma^\vee)$ be the associated simply connected root datum in which $(X^{\mathrm{sc}})^\vee=\mathbb{Z}\Sigma^\vee$ is the lattice generated by the coroots of $\mathcal{R}$ . The root datums $\mathcal{R}$ and $\mathcal{R}^{\mathrm{sc}}$ have the same underlying root system $\Sigma$ .

Note that

\begin{align*}|(X^\vee/\mathbb{Z}\Sigma_1^\vee)_{\mathrm{tor}}|=|(\mathbb{Z}\Sigma^\vee/\mathbb{Z}\Sigma_1^\vee)_{\mathrm{tor}}|\cdot|(X^\vee/\mathbb{Z}\Sigma^\vee)_{\mathrm{tor}}|. \end{align*}

The prime divisors of the first factor $|(\mathbb{Z}\Sigma^\vee/\mathbb{Z}\Sigma_1^\vee)_{\mathrm{tor}}|$ (when $\Sigma_1$ runs through all possible closed subsystems $\Sigma$ ) are precisely the torsion primes of the underlying root system $\Sigma$ . The second factor above has several equivalent interpretations. Let $\mathcal{R}^{\mathrm{der}}:=(X^{\mathrm{der}},\Sigma,(X^{\mathrm{der}})^\vee,\Sigma^\vee)$ be the ‘derived’ root datum of $\mathcal{R}$ , where $X^{\mathrm{der}}:=\mathrm{Im}(X\to X^{\mathrm{sc}})$ and $(X^{\mathrm{der}})^\vee:=X^\vee\cap\mathbb{Q}\Sigma^\vee$ (intersection inside $X^\vee\otimes_{\mathbb{Z}}\mathbb{Q}$ ) is the saturation of $\mathbb{Z}\Sigma^\vee$ in $X^\vee$ . Then, we have

\begin{align*} |(X^\vee/\mathbb{Z}\Sigma^\vee)_{\mathrm{tor}}|=|(X^{\mathrm{der}})^\vee/(X^{\mathrm{sc}})^\vee|=|X^{\mathrm{sc}}/X^{\mathrm{der}}|=|\mathrm{coker}(X\to X^{\mathrm{sc}})|. \end{align*}

In particular, p is torsion for $\mathcal{R}$ if either:

• – p is torsion for the underlying root system of $\mathcal{R}$ ; or

• – p divides the order of the cokernel of the natural homomorphism $X\to X^{\mathrm{sc}}$ .

So, our definition of torsion primes for a root datum is the same as the definition in [Reference DemazureDem73, § 5], by their Proposition 6 (see also [Reference Bouthier and ČesnavičiusBČ22, § 4.1.12]).

If $\mathcal{R}$ is the root datum of a split reductive group G, then $\mathcal{R}^{\mathrm{der}}$ (respectively, $\mathcal{R}^{\mathrm{sc}}$ ) is the root datum of the derived group $G^{\mathrm{der}}$ (respectively, its simply connected cover $G^{\mathrm{sc}}$ ). The finite abelian groups $\mathrm{coker}(X\to X^{\mathrm{sc}})$ and $\ker(G^{\mathrm{sc}}\to G)$ are dual to each other (in the sense of algebraic groups) and hence they have the same order. Here, order should be understood as the dimension of the coordinate ring of the algebraic group, e.g. $\mu_n$ has order n over any field. On the other hand, recall that the algebraic fundamental group of G is defined by $\pi_1(G)=X^\vee/\mathbb{Z}\Sigma^\vee$ and its torsion subgroup $\pi_1(G)_{\mathrm{tor}}=\pi_1(G^{\mathrm{der}})=(X^{\mathrm{der}})^\vee/(X^{\mathrm{sc}})^\vee$ is a quotient of $\pi_1(G_{\mathrm{ad}})$ . In the following table, we list the structure of $\pi_1(G_{\mathrm{ad}})$ for a split reductive group G according to the types.

For later reference, we recall some classical results below about centralizers of semisimple elements in Lie algebras. This also helps us have a glimpse of the relevance of the various conditions on p introduced above.

1. (i) The Lie algebra of $\mathsf{G}_\gamma$ coincides with the centralizer of $\gamma$ in $\mathrm{Lie}(\mathsf{G})$ . In other words, we have $\mathrm{Lie}(\mathsf{G}_\gamma)=\mathrm{Lie}(\mathsf{G})_\gamma$ .

2. (ii) Suppose that either $\mathrm{char}(\mathsf{k})=0$ or $\mathrm{char}(\mathsf{k})=p\gt0$ is a good prime for $\mathsf{G}$ , then $\mathsf{G}_\gamma^\circ$ is a Levi subgroup of $\mathsf{G}$ .

3. (iii) Suppose that either $\mathrm{char}(\mathsf{k})=0$ or $\mathrm{char}(\mathsf{k})=p\gt0$ is not a torsion prime for the root datum of $\mathsf{G}$ , then $\mathsf{G}_\gamma$ is connected.

3.1 Discriminant valuation

Recall that an element $\gamma\in G(F)$ is regular semisimple if it is semisimple and its centralizer $G_\gamma$ has minimal possible dimension. In fact, $\gamma$ is regular semisimple if and only if the identity component $G_\gamma^\circ$ is a maximal torus of G defined over F. There is a nonempty open subscheme $G^{\mathrm{rs}}\subset G$ such that $G^{\mathrm{rs}}(F)$ is the set of all regular semisimple elements.

Definition 3.1. The discriminant valuation of a regular semisimple element $\gamma\in G^{\mathrm{rs}}(F)$ is the integer defined by

\begin{align*} d_G(\gamma):=\mathrm{val}\det(\mathrm{Id}-\mathrm{ad}(\gamma) \mid \mathfrak{g}(F)/\mathfrak{g}_\gamma(F)). \end{align*}

We can compute $d_G(\gamma)$ as follows. After $G(\overline{F})$ -conjugation, we may assume that $\gamma\in T(\overline{F})$ for a maximal torus T in G. Let R be the set of roots of T on $\mathfrak{g}$ . Then, we have

(3.1) \begin{equation} d_G(\gamma)=\sum_{\alpha\in R}\mathrm{val}(1-\alpha(\gamma)).\end{equation}

3.2 Kottwitz homomorphism

The Kottwitz homomorphism is a generalization of the determinant valuation on $\mathrm{GL}_n(F)$ to all reductive groups. It gives an obstruction for an element to lie in a parahoric subgroup. Let $\pi_1(G)=\pi_1(G_{\overline{F}})$ be the algebraic fundamental group of G, which is a finitely generated abelian group with an action of the Galois group $\Gamma=\mathrm{Gal}(\overline{F}/F)$ . We have the Kottwitz homomorphism

\begin{align*} \kappa_G:G(F)\to\pi_1(G)_\Gamma, \end{align*}

whereFootnote ² $\pi_1(G)_\Gamma$ is the group of $\Gamma$ -coinvariants of $\pi_1(G)$ .

Let $G(F)^0:=\ker(\kappa_G)$ and let $G(F)^\dagger$ be the inverse image of the torsion subgroup of $\pi_1(G)_\Gamma$ under $\kappa_G$ . Any parahoric subgroup $\mathbf{P}\subset G(F)$ is contained in $G(F)^0$ and the Kottwitz homomorphism $\kappa_G$ induces an injective homomorphism $\widetilde{\mathbf{P}}/\mathbf{P}\hookrightarrow\pi_1(G)_\Gamma$ , which is bijective if $\mathbf{P}$ is an Iwahori subgroup (see [Reference Kaletha and PrasadKP23, Proposition 11.6.1]).

Let S be a maximal F-split torus in G and let N (respectively, T) be the normalizer (respectively, centralizer) of S in G. By Steinberg’s theorem, the assumption on F implies that G is quasi-split and T is a maximal torus of G defined over F. Let $\mathcal{T}$ be the fiberwise identity component of the finite-type Néron model of T. Then, $\mathcal{T}(\mathcal{O})$ is the unique parahoric subgroup of T(F) and the Kottwitz homomorphism $\kappa_T$ for T induces an isomorphism $X_*(T)_\Gamma\cong T(F)/\mathcal{T}(\mathcal{O})$ . Let $W=N(F)/T(F)$ be the (relative) Weyl group of G and let $\widetilde{W}:=N(F)/\mathcal{T}(\mathcal{O})$ be the Iwahori–Weyl group of G. Let $W_{\mathrm{aff}}\subset\widetilde{W}$ be the affine Weyl group of the root system of G.

We choose a special point, denoted ‘0’, and an alcove $\mathfrak{a}$ whose closure contains 0, both in the apartment for S in the Bruhat–Tits building of G. They determine decompositions

\begin{align*} \widetilde{W}=X_*(T)_\Gamma\rtimes W,\quad \widetilde{W}=W_{\mathrm{aff}}\rtimes\widetilde{W}_{\mathfrak{a}}, \end{align*}

where $\widetilde{W}_{\mathfrak{a}}$ is the stabilizer of $\mathfrak{a}$ in $\widetilde{W}$ . Moreover, the Kottwitz homomorphism induces an isomorphism

\begin{align*} \iota_{\mathfrak{a}}:\widetilde{W}_{\mathfrak{a}}\cong\pi_1(G)_\Gamma. \end{align*}

See, for example, [Reference Kaletha and PrasadKP23, §§ 7.8, 11.5] (note that, in [Reference Kaletha and PrasadKP23], the Iwahori–Weyl group is denoted by $\widetilde{W}^0$ ).

Definition 3.2. Define the homomorphism $w^G_{\mathfrak{a}}:\pi_1(G)_\Gamma\to W$ to be the composition of the following natural homomorphisms:

\begin{align*} w^G_{\mathfrak{a}}:\pi_1(G)_\Gamma\cong\widetilde{W}_{\mathfrak{a}}\hookrightarrow\widetilde{W}\to W, \end{align*}

where the first map is the inverse of $\iota_{\mathfrak{a}}$ .

The defect of an element $x\in\pi_1(G)_\Gamma$ is the non-negative integer

\begin{align*} \mathrm{def}(x):=\dim_{\mathbb{Q}} X_*(S)_{\mathbb{Q}}-\dim_{\mathbb{Q}} X_*(S)_{\mathbb{Q}}^{w^G_{\mathfrak{a}}(x)}, \end{align*}

where $X_*(S)_{\mathbb{Q}}:=X_*(S)\otimes_{\mathbb{Z}}\mathbb{Q}$ and $X_*(S)_{\mathbb{Q}}^{w^G_{\mathfrak{a}}(x)}$ is the subspace of $w^G_{\mathfrak{a}}(x)$ -invariants.

The invariant $\mathrm{def}(x)$ is independent of the choice of the alcove $\mathfrak{a}$ . Indeed, the group N(F) acts transitively on the set of alcoves in the apartment of S and, for any $n\in N(F)$ , the maps $w_{\mathfrak{a}}$ and $w_{n(\mathfrak{a})}$ are conjugate by n. Therefore, $\dim_{\mathbb{Q}} X_*(S)_{\mathbb{Q}}^{w_{\mathfrak{a}}(x)}=\dim_{\mathbb{Q}} X_*(S)_{\mathbb{Q}}^{w_{n(\mathfrak{a})}(x)}$ . Moreover, $\mathrm{def}(x)$ is also independent of the choice of the maximal F-split torus S, since any two such tori are conjugate by an element of G(F), which induces canonical isomorphisms between the associated (Iwahori) Weyl groups. It will be convenient to have a description of $\mathrm{def}(x)$ that does not depend on the torus S. For this, we have the following simple lemma which follows directly from the definitions.

3.3 Topological Jordan decomposition for the group

Recall from [Reference Kaletha and PrasadKP23, Definition 2.2.1] that, for an affine F-scheme S of finite type, a subset $B\subset S(F)$ is bounded if, for any regular function $f\in F[S]$ , the set $\{|f(b)|,b\in B\}$ is bounded above. We say that an element $\gamma\in G(F)$ is bounded if the subgroup $\langle\gamma\rangle\subset G(F)$ generated by $\gamma$ is bounded.

1. (i) The element $\gamma$ is bounded.

2. (ii) For any algebraic representation $\rho$ of G (defined over $\overline{F}$ ), any generalized eigenvalue $\lambda$ of $\rho(\gamma)$ in $\overline{F}$ satisfies $|\lambda|=1$ .

3. (iii) There exists a parahoric subgroup $\mathbf{P}\subset G(F)$ such that $\gamma\in\mathbf{P}^\dagger$ . In other words, $\gamma$ has a fixed point in the extended Bruhat–Tits building of G(F).

4. (iv) There exists an Iwahori subgroup $\mathbf{I}\subset G(F)$ such that $\gamma\in\mathbf{I}^\dagger$ . In other words, $\gamma$ fixes an alcove in the extended Bruhat–Tits building of G(F).

Suppose that $\gamma\in G(F)$ is unbounded. Let H be the Zariski closure of $\langle\gamma\rangle$ in G. Then, $\gamma$ is also unbounded in H. Let $\rho:G\to\mathrm{GL}(V)$ be an algebraic representation with finite kernel. By [Reference Kaletha and PrasadKP23, Lemma 2.2.11], there is an eigenvalue $\lambda$ of $\rho(\gamma)$ such that $|\lambda|\gt1$ .

1. (i) $\Leftrightarrow$ (iii). By [Reference Kaletha and PrasadKP23, Proposition 2.2.13], any bounded subgroup of G(F) is contained in a maximal bounded open subgroup. Therefore, the equivalence follows from [Reference Kaletha and PrasadKP23, Theorem 4.2.15].

2. (iii) $\Leftrightarrow$ (iv). Let $\mathbf{I}\subset G(F)$ be an Iwahori subgroup contained in $\mathbf{P}$ and suppose $\gamma\in\mathbf{P}^\dagger$ . Let $\mathcal{G}_{\mathbf{P}}^\dagger$ be the smooth $\mathcal{O}$ -model of G such that $\mathcal{G}_{\mathbf{P}}^\dagger(\mathcal{O})=\mathbf{P}^\dagger$ and let $\mathsf{G}_{\mathbf{P}}^\dagger$ be its special fiber. Let $\bar{\gamma}\in\mathsf{G}_{\mathbf{P}}^\dagger(k)$ be the image of $\gamma\in\mathbf{P}^\dagger$ under the natural reduction map. The image of $\mathbf{I}$ in $\mathsf{G}_{\mathbf{P}}$ is a Borel subgroup that we denote by $\mathsf{B}_{\mathbf{P}}$ . By [Reference SteinbergSte68, Theorem 7.2 on p. 49], the flag variety $\mathsf{G}_{\mathbf{P}}/\mathsf{B}_{\mathbf{P}}$ has a fixed point under the conjugation action of $\bar{\gamma}$ . Therefore, $\gamma$ is $\mathbf{P}^\dagger$ conjugate to an element in $\mathbf{I}^\dagger$ and we are done.

• – Strongly semisimple if it is semisimple and for any algebraic representation $\rho$ of G (defined over $\overline{F}$ ), each eigenvalue of $\rho(\gamma)$ lies in the set of Teichmüller representatives $[\bar{k}^\times]$ .

• – Topologically unipotent if, for any algebraic representation $\rho$ of G (defined over $\overline{F}$ ) and any generalized eigenvalue $\alpha$ of $\rho(\gamma)$ in $\overline{F}$ (i.e. $\alpha$ is a root of the characteristic polynomial of $\rho(\gamma)$ ), we have $|\alpha-1|\lt1$ .

• – Strongly topologically unipotent if there exists a parahoric subgroup $\mathbf{P}\subset G(F)$ with pro-unipotent radical $\mathbf{P}_+$ such that $\gamma\in\mathbf{P}_+$ .

An element $\gamma\in G(F)$ is (strongly) topologically unipotent (respectively, strongly semisimple, bounded) mod center if its image in $G_{\mathrm{ad}}(F)$ has this property.

1. (i) An element $\gamma\in G(F)$ is topologically unipotent if and only if there exists a finite extension $E/F$ such that $\gamma$ is strongly topologically unipotent in G(E). In particular, any strongly topologically unipotent element in G(F) is topologically unipotent.

2. (ii) An element $\gamma\in G(F)$ is strongly topologically unipotent if and only if it is topologically unipotent and $\kappa_G(\gamma)=1$ .

3. (iii) If the residue characteristic p does not divide the order of the torsion subgroup of $\pi_1(G)_\Gamma$ , then an element $\gamma\in G(F)$ is topologically unipotent if and only if it is strongly topologically unipotent.

Finally, we state the following result of Spice [Reference SpiceSpi08, Theorem 2.38] on the existence and uniqueness of topological Jordan decompositions.

3.4 Arithmetic invariants of tori

In this subsection, we introduce a numerical invariant of a conjugacy class defined by the Artin conductor of its centralizer. Then, we review a result of Chai and Yu [Reference Chai and YuCY01] on the Artin conductor of tori. This result is used later in Theorem 7.9 to compute the dimension of the regular locus of the Witt-vector affine Springer fiber for Lie algebras.

Let T be an F-algebraic group of multiplicative type. We are only interested in the case of tori but the slightly more general setting would make no difference in most of the following discussion. Let $\Lambda:=X_*(T_{\overline{F}})$ be the lattice of coweights of T defined over $\overline{F}$ , equipped with an action of $\mathrm{Gal}(\overline{F}/F)$ through a homomorphism $\rho_T:\mathrm{Gal}(\overline{F}/F)\to\mathrm{GL}(\Lambda):=\mathrm{Aut}_{\mathbb{Z}}(\Lambda)$ . We define some isogeny invariants of T associated to the representation

\begin{align*}\rho_{T,\mathbb{Q}}=\rho_T\otimes_{\mathbb{Z}}\mathbb{Q}:\mathrm{Gal}(\overline{F}/F)\to\mathrm{GL}(\Lambda_{\mathbb{Q}}),\end{align*}

where $\Lambda_{\mathbb{Q}}:=\Lambda\otimes_{\mathbb{Z}}\mathbb{Q}$ .

Definition 3.10. The Artin conductor of T, denoted $\mathrm{Art}(T)$ , is defined to be the Artin conductor of the representation $\rho_{T,\mathbb{Q}}$ . The defect of T is the non-negative integer

\begin{align*} \mathrm{def}(T):=\dim_{\mathbb{Q}}\Lambda_{\mathbb{Q}}-\dim_{\mathbb{Q}}\Lambda_{\mathbb{Q}}^{\mathrm{Gal}(\overline{F}/F)}, \end{align*}

which is the difference between the $\overline{F}$ -rank of T and the F-rank of T.

The Artin conductor $\mathrm{Art}(T)$ equals $\mathrm{def}(T)$ if and only if T splits over a tamely ramified extension of F. In general, $\mathrm{Art}(T)$ is always a non-negative integer by the Hasse–Arf theorem (see [Reference SerreSer68, Chapter VI. § 2, Theorem 1]) and $\mathrm{Art}(T)\geqslant\mathrm{def}(T)$ . The difference $\mathrm{Art}(T)-\mathrm{def}(T)$ is the Swan conductor of the representation $\rho_{T,\mathbb{Q}}$ and measures the wild ramification of T.

We extend this definition to general reductive groups over F as follows.

Definition 3.11. The Artin conductor (respectively, defect) of a reductive group G over F are the integers

\begin{align*} \mathrm{Art}(G):=\mathrm{Art}(T),\quad\mathrm{def}(G):=\mathrm{def}(T), \end{align*}

where $T\subset G$ is a maximally split maximal torus defined over F, i.e. T is a maximal torus of G defined over F and contains a maximal F-split torus $S\subset G$ . When G is quasi-split, then $T=Z_G(S)$ is the centralizer of S.

Now, let T be a torus over F and let $\mathfrak{t}=\mathrm{Lie}(T)$ be the Lie algebra of T, viewed either as an F-scheme or an F-vector space. We specialize the discussion in § 2.2 to the case where $G=T$ . Then, we get a unique smooth $\mathcal{O}$ -model $T^\dagger$ of T such that $T^\dagger(\mathcal{O})$ is the maximal bounded subgroup of T(F) and the fiberwise identity component $T^{\dagger,0}$ is the unique parahoric group scheme of T. In fact, $T^\dagger$ is the finite-type Néron model of T in the terminology of [Reference Chai and YuCY01, § 3] and [Reference NgôNgô10].Footnote ⁴ The Lie algebra $\mathfrak{t}^\dagger:=\mathrm{Lie}(T^\dagger)$ is an $\mathcal{O}$ -lattice in $\mathfrak{t}(F)$ . Let $L/F$ be a finite Galois extension contained in $\overline{F}$ such that $\rho_T$ factors through $\mathrm{Gal}(L/F)$ and let $\mathcal{O}_L$ be the valuation ring of L. Then, T is split over L and there is a natural homomorphism $(T^\dagger)_{\mathcal{O}_L}\to(T_L)^\dagger$ that induces an embedding of $\mathcal{O}_L$ -modules $\mathfrak{t}^\dagger\otimes_{\mathcal{O}}\mathcal{O}_L\subset\mathfrak{t}_L^\dagger$ .

Theorem 3.12 [Reference Chai and YuCY01]. Let $e(L|F)$ be the ramification index of the extension $L/F$ . Then, we have

\begin{align*} \frac{1}{e(L|F)}\mathrm{length}_{\mathcal{O}_L} \bigg(\frac{\mathfrak{t}_L^\dagger}{\mathfrak{t}^\dagger\otimes_{\mathcal{O}}\mathcal{O}_L}\bigg)=\frac{1}{2}\mathrm{Art}(T). \end{align*}

Now, suppose that there exists a nondegenerate symmetric F-bilinear form $(\cdot,\cdot)$ on $\Lambda\otimes_{\mathbb{Z}} F$ that is invariant under $\mathrm{Gal}(L/F)$ (which acts on $\Lambda$ via $\rho_T$ ). Recall that the dual of an $\mathcal{O}$ -lattice $M\subset\Lambda\otimes_{\mathbb{Z}} F$ is defined to be

\begin{align*} M^\vee:=\{x\in \Lambda\otimes_{\mathbb{Z}} F\mid (x,M)\subset\mathcal{O}\}. \end{align*}

We would also like to define the dual of $\mathcal{O}$ -lattices in $\mathfrak{t}(F)$ . For this, we first extend $(\cdot,\cdot)$ to an L-bilinear form $(\cdot,\cdot)_L$ on $\Lambda\otimes_{\mathbb{Z}} L$ . Since T splits over L, there are canonical identifications $\mathfrak{t}(L)=\Lambda\otimes_{\mathbb{Z}} L$ and

\begin{align*} \mathfrak{t}(F)=(\Lambda\otimes_{\mathbb{Z}} L)^{\mathrm{Gal}(L/F)}, \end{align*}

where $\mathrm{Gal}(L/F)$ acts diagonally. By $\mathrm{Gal}(L/F)$ -invariance of $(\cdot,\cdot)$ , the extended form $(\cdot,\cdot)_L$ restricts to an F-bilinear form on $\mathfrak{t}(F)$ . In other words, for any $x,y\in\mathfrak{t}(F)$ , we have $(x,y)_L\in F$ . Then, for any $\mathcal{O}$ -lattice $M\subset\mathfrak{t}(F)$ , we define its dual to be

\begin{align*} M^\vee:=\{x\in\mathfrak{t}(F)\mid (x,M)_L\subset\mathcal{O}\}=\{x\in\mathfrak{t}(F)\mid (x,M)_L\subset\mathcal{O}_L\}, \end{align*}

where the equality follows from the equality $\mathcal{O}_L\cap F=\mathcal{O}$ .

Corollary 3.13. With notation as above. Assume, moreover, that $\Lambda\otimes_{\mathbb{Z}}\mathcal{O}$ is self-dual under $(\cdot,\cdot)$ . Then, we have

\begin{align*} \mathrm{length}_{\mathcal{O}}(\mathfrak{t}^{\dagger,\vee}/\mathfrak{t}^\dagger)=\mathrm{Art}(T). \end{align*}

Proof. First, we claim that for any $\mathcal{O}$ -lattice $M\subset\mathfrak{t}(F)$ , we have

\begin{align*} M^\vee\otimes_{\mathcal{O}}\mathcal{O}_L=(M\otimes_{\mathcal{O}}\mathcal{O}_L)^\vee, \end{align*}

where the right-hand side means the dual of the $\mathcal{O}_L$ -lattice under the pairing $(\cdot,\cdot)_L$ . Indeed, after choosing an $\mathcal{O}$ -basis $\{m_1,\dotsc,m_r\}$ for M and letting $m_1^\vee,\dotsc,m_r^\vee$ be the dual basis satisfying $(m_i,m_j^\vee)_L=\delta_{ij}$ , both sides are identified with the $\mathcal{O}_L$ submodule of $\Lambda\otimes_{\mathbb{Z}} L$ spanned by $\{m_i^\vee,1\leqslant i\leqslant r\}$ .

Applying this identity to $M=\mathfrak{t}^\dagger$ , we get that

\begin{align*} \mathrm{length}_{\mathcal{O}}(\mathfrak{t}^{\dagger,\vee}/\mathfrak{t}^\dagger)=\frac{1}{e(L|F)}\mathrm{length}_{\mathcal{O}_L}(\mathfrak{t}^{\dagger,\vee}/\mathfrak{t}^\dagger)\otimes_{\mathcal{O}}\mathcal{O}_L =\frac{1}{e(L|F)}\mathrm{length}_{\mathcal{O}_L}\frac{(\mathfrak{t}^\dagger\otimes_{\mathcal{O}}\mathcal{O}_L)^\vee}{\mathfrak{t}^\dagger\otimes_{\mathcal{O}}\mathcal{O}_L}. \end{align*}

Since T splits over L, we have $\mathfrak{t}_L^\dagger=\Lambda\otimes_{\mathbb{Z}}\mathcal{O}_L$ , which is self-dual by assumption. So, we have

\begin{align*} \mathfrak{t}^\dagger\otimes_{\mathcal{O}}\mathcal{O}_L\subset\mathfrak{t}_L^\dagger=\mathfrak{t}_L^{\dagger,\vee}\subset(\mathfrak{t}^\dagger\otimes_{\mathcal{O}}\mathcal{O}_L)^\vee \end{align*}

and then, by Theorem 3.12, we conclude that

\begin{align*}\mathrm{length}_{\mathcal{O}}(\mathfrak{t}^{\dagger,\vee}/\mathfrak{t}^\dagger)=\frac{2}{e(L|F)}\mathrm{length}_{\mathcal{O}_L}\bigg(\frac{\mathfrak{t}_L^\dagger}{\mathfrak{t}^\dagger\otimes_{\mathcal{O}}\mathcal{O}_L}\bigg)=\mathrm{Art}(T). \end{align*}

We end this section with a result on the behavior of the Artin conductor under restrictions of scalars. Later, this is used in several reduction steps (see § 7.1 and Proposition 8.4) in the proof of the main theorems.

Proposition 3.14. Let $E/F$ be a finite extension field and let $\mathrm{Disc}_{E/F}\subset\mathcal{O}$ be its discriminant ideal. Let H be a reductive group over E with absolute rank r. Then, we have

\begin{align*} \mathrm{Art}(\mathrm{Res}_{E/F}H)=\mathrm{val}(\mathrm{Disc}_{E/F})r+\mathrm{Art}(H). \end{align*}

4.1 Chevalley quotient and regular semisimple elements

The Chevalley base for the Lie algebra $\mathfrak{g}$ is the invariant quotient of $\mathfrak{g}$ (viewed as an F-scheme) by the adjoint action of G, denoted $\mathfrak{c}=\mathfrak{g}//G:=\mathrm{Spec}(F[\mathfrak{g}]^{G})$ . It comes with a natural morphism

\begin{align*} \chi_G:\mathfrak{g}\to\mathfrak{c} \end{align*}

called the Chevalley morphism. Similarly, for the split form $\mathbb{G}$ of G, we have its Chevalley morphism $\chi_{\mathbb{G}}:\mathbb{g}\to\mathbb{c}$ that is actually defined over $\mathbb{Z}$ .

The $\mathrm{Out}(\mathbb{G})$ -action and the adjoint action of $\mathbb{G}$ on $\mathbb{g}$ combine into an action of the semi-direct product $\mathbb{G}\rtimes\mathrm{Out}(\mathbb{G})$ on $\mathbb{g}$ , which then induces an action of $\mathrm{Out}(\mathbb{G})$ on $\mathbb{c}$ making the morphism $\chi_{\mathbb{G}}$ equivariant under $\mathrm{Out}(\mathbb{G})$ . Consequently, $\chi_G$ is the $\mathrm{Out}(\mathbb{G})$ -twist of $\chi_{\mathbb{G}}$ over F’ defined by the homomorphism $\rho_G:\mathrm{Gal}(F'/F)\to\mathrm{Out}(\mathbb{G})$ .

Recall that an element in $\mathfrak{g}$ (respectively, $\mathbb{g}$ ) is regular if its centralizer in G (respectively, $\mathbb{G}$ ) has minimal possible dimension (or equivalently, its adjoint orbit has maximal possible dimension). The regular elements in $\mathbb{g}$ form an open subscheme $\mathbb{g}^{\mathrm{reg}}\subset\mathbb{g}$ that is $\mathbb{G}\rtimes\mathrm{Out}(\mathbb{G})$ -stable and descends to the open subscheme $\mathfrak{g}^{\mathrm{reg}}$ of regular elements in $\mathfrak{g}$ . Then, the Chevalley morphism $\chi_G$ is flat and its restriction to the regular open subscheme $\chi_G^{\mathrm{reg}}:\mathfrak{g}^{\mathrm{reg}}\to\mathfrak{c}$ is smooth. This follows by Galois descent from the corresponding statements for the split group $\mathbb{G}$ .

Let $\mathbb{g}^{\mathrm{rs}}\subset\mathbb{g}^{\mathrm{reg}}$ be the open subscheme of regular semisimple elements. There is an open subscheme $\mathbb{c}^{\mathrm{rs}}\subset\mathbb{c}$ such that $\mathbb{g}^{\mathrm{rs}}=\chi_{\mathbb{G}}^{-1}(\mathbb{c}^{\mathrm{rs}})$ . The complement of $\mathbb{c}^{\mathrm{rs}}$ in $\mathbb{c}$ is the vanishing loci of the discriminant function $\Delta_{\mathbb{g}}$ . Let us recall its definition below. For this, we view the split group $\mathbb{G}$ and related objects $\mathbb{g},\mathbb{c}$ , etc. to be defined over $\mathbb{Z}$ .

Let $P(X)\in \mathbb{Z}[\mathbb{g}][X]$ be the universal characteristic polynomial for adjoint actions of elements in $\mathbb{g}$ . In other words, for any ring R and any $\gamma\in\mathbb{g}(R)$ , viewed as a morphism $\gamma:\mathrm{Spec} R\to\mathbb{g}$ , the pullback $\gamma^*P(X)$ is the characteristic polynomial of the endomorphism $\mathrm{ad}(\gamma)$ on the free R-module $\mathbb{g}(R)$ . The characteristic polynomials are invariant under the action of $\mathbb{G}\rtimes\mathrm{Out}(\mathbb{G})$ and therefore we have $P(X)\in\mathbb{Z}[\mathbb{c}]^{\mathrm{Out}(\mathbb{G})}$ , the set of $\mathrm{Out}(\mathbb{G})$ -invariant polynomials on $\mathbb{c}$ with $\mathbb{Z}$ -coefficients. Moreover, P(X) is divisible by $X^{\mathbb{r}}$ , where $\mathbb{r}$ is the rank of $\mathbb{G}$ . The coefficient of the $X^{\mathbb{r}}$ term is the discriminant $\Delta_{\mathbb{g}}\in\mathbb{Z}[\mathbb{c}]^{\mathrm{Out}(\mathbb{G})}$ that we also view as an element in $\mathbb{Z}[\mathbb{g}]$ . Then, the regular semisimple locus $\mathbb{c}^{\mathrm{rs}}$ is the complement of the principal divisor $\Delta_{\mathbb{g}}=0$ .

The open subschemes $\mathbb{g}^{\mathrm{rs}}$ and $\mathbb{c}^{\mathrm{rs}}$ are stable under the $\mathrm{Out}(\mathbb{G})$ -action and descend to open subschemes $\mathfrak{g}^{\mathrm{rs}}\subset\mathfrak{g}$ and $\mathfrak{c}^{\mathrm{rs}}\subset\mathfrak{c}$ of regular semisimple elements. Let $\Delta_{\mathfrak{g}}\in F[\mathfrak{c}]=(F'[\mathbb{c}])^{\mathrm{Gal}(F'/F)}$ be the image of $\Delta_{\mathbb{g}}$ . Then, $\mathfrak{c}^{\mathrm{rs}}$ is the complement of the principal divisor $\Delta_{\mathfrak{g}}=0$ on $\mathfrak{c}$ .

Lemma 4.1. For any regular semisimple element $\gamma\in\mathfrak{g}^{\mathrm{rs}}(F)$ , we have

\begin{align*} \Delta_{\mathfrak{g}}(\gamma)=\det(\mathrm{ad}(\gamma)\mid\mathfrak{g}(F)/\mathfrak{g}_\gamma(F)), \end{align*}

where $\mathfrak{g}_\gamma=\ker(\mathrm{ad}(\gamma))=\mathrm{Lie}(G_\gamma)$ is the centralizer of $\gamma$ in the Lie algebra. More generally, for any semisimple element $\gamma\in\mathfrak{t}(\overline{F})=\mathbb{t}(\overline{F})$ , we have

\begin{align*} \Delta_{\mathfrak{g}}(\gamma)=\prod_{\alpha\in R}d\alpha(\gamma), \end{align*}

where R is the set of roots of $\mathbb{T}$ in $\mathbb{g}$ and $d\alpha:\mathbb{t}(\overline{F})\to\overline{F}$ is the differential of a root $\alpha\in R$ .

Definition 4.2. The discriminant valuation of a regular semisimple element $\gamma\in\mathfrak{g}^{\mathrm{rs}}(F)$ is the integer defined by

\begin{align*} d_{\mathfrak{g}}(\gamma):=\mathrm{val}(\Delta_{\mathfrak{g}}(\gamma))=\mathrm{val}\det(\mathrm{ad}(\gamma)\mid\mathfrak{g}(F)/\mathfrak{g}_\gamma(F)). \end{align*}

Assume, in the rest of this subsection, that the extension $F'/F$ is tamely ramified of degree $e\geqslant1$ and identify $\mathrm{Gal}(F'/F)\cong\mu_e$ . The construction in § 2.4 provides natural finite-type $\mathcal{O}$ -models for the F-schemes $\mathfrak{g}$ , $\mathfrak{g}^{\mathrm{reg}}$ , $\mathfrak{c}$ , etc. We use the same symbol to denote the resulting $\mathcal{O}$ -schemes of finite type. Then, we have

\begin{align*} \mathfrak{g}=(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{g})^{\mu_e}=\mathrm{Lie}(\mathcal{G}_0),\quad\mathfrak{g}^{\mathrm{reg}}=(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{g}^{\mathrm{reg}})^{\mu_e},\quad\mathfrak{c}=(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{c})^{\mu_e}. \end{align*}

We can talk about the $\mathcal{O}$ -points $\mathfrak{g}(\mathcal{O}),\mathfrak{c}(\mathcal{O})$ , etc. In particular, from the definition we see that $\mathfrak{g}^{\mathrm{reg}}(\mathcal{O}) = \mathfrak{g}(\mathcal{O})\cap\mathbb{g}^{\mathrm{reg}}(\mathcal{O}')$ .

For any closed point $\gamma\in\mathbb{g}$ , we have $\dim\mathbb{G}_\gamma\geqslant\mathbb{r}$ , the absolute rank of $\mathbb{G}$ . The locus where equality holds is the open subscheme $\mathbb{g}^{\mathrm{reg}}\subset\mathbb{g}$ .

Then, $I^{\mathrm{reg}}$ is an open subscheme of the fixed-point subscheme of $\mu_e$ on $(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{I}^{\mathrm{reg}})|_{\mathfrak{g}^{\mathrm{reg}}}$ , the base change of $\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{I}^{\mathrm{reg}}$ along the closed embedding $\mathfrak{g}^{\mathrm{reg}}\to\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{g}^{\mathrm{reg}}$ .

The group scheme J in the proposition is called the universal regular centralizer for $\mathfrak{g}$ (in G). In general, we need more assumptions on either p or the group G to guarantee its smoothness.

In general, we deduce the smoothness of $\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}(\chi_{\mathbb{G}}^{\mathrm{reg}}):\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{g}^{\mathrm{reg}}\to\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{c}$ from the smoothness of $\chi_{\mathbb{G}}^{\mathrm{reg}}$ by the infinitesimal lifting criterion. Then, we apply Lemma 2.2 to the base change of $\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}(\chi_{\mathbb{G}}^{\mathrm{reg}})$ along the closed subscheme $\mathfrak{c}\subset\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{c}$ to get the smoothness of $\chi_G^{\mathrm{reg}}$ . The group scheme I is an open subgroup scheme of $(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{I})^{\mu_e}$ and hence descends to an open subgroup scheme J of $(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{J})^{\mu_e}$ . The existence of the isomorphism $(\chi^{\mathrm{reg}})^*J\cong I^{\mathrm{reg}}$ that extends uniquely to a homomorphism $\chi^*J\to I$ follows from the corresponding statements for $\mathbb{J}$ .

4.2 Topological Jordan decomposition for the Lie algebra

• – bounded if $\chi_G(\gamma)\in\mathbb{c}(\overline{\mathcal{O}})$ ;

• – topologically nilpotent if it is bounded and the image of $\chi_G(\gamma)$ under the natural reduction map $\mathbb{c}(\overline{\mathcal{O}})\to\mathbb{c}(k)$ equals zero; and

• – strongly semisimple if it is semisimple, bounded, and, for any root $\alpha\in\Sigma$ and any $\gamma'\in\mathfrak{t}(\overline{F})$ that is $G(\overline{F})$ -conjugate to $\gamma$ , we have either $d\alpha(\gamma')=0$ or $|d\alpha(\gamma')|=1$ .

• – We have that $\gamma_0$ is strongly semisimple in $\mathfrak{g}(F)$ and G(F)-conjugate to an element in $\mathfrak{t}(\mathcal{O})$ .

• – The centralizer $G_{\gamma_0}$ is an F-Levi subgroup of G and $\gamma_1$ is a topologically nilpotent element in $\mathrm{Lie}(G_{\gamma_0})$ .

If, moreover, $\gamma$ is regular semisimple in $\mathfrak{g}(F)$ , then $\gamma_1$ is regular semisimple in $\mathrm{Lie}(G_{\gamma_0})$ .

Proof. Let $F'\subset\overline{F}$ be the minimal Galois extension of F over which G becomes split. Let $\Gamma:=\mathrm{Gal}(\overline{F}/F)$ , $\Gamma':=\mathrm{Gal}(\overline{F}/F')$ , and $\Theta:=\mathrm{Gal}(F'/F)\cong\Gamma/\Gamma'$ . Let $\rho_G:\Gamma\to\mathrm{Out}(\mathbb{G})$ be the twisting homomorphism of G as in § 2.2. Then, $\rho_G$ is the composition of the projection $\Gamma\twoheadrightarrow\Theta$ and an embedding $\Theta\hookrightarrow\mathrm{Out}(\mathbb{G})$ . For each $\sigma\in\Gamma$ , we let $\sigma_G$ denote the Galois action on $G(\overline{F})$ or $\mathfrak{g}(\overline{F})$ and simply use $\sigma(\cdot)$ to denote the Galois action with respect to the split form $\mathbb{G}$ . We have

\begin{align*} \sigma_G(g)=\rho_G(\sigma)\sigma(g),\quad\sigma_G(x)=\rho_G(\sigma)\sigma(x)\quad\mbox{for all }g\in G(\overline{F}), x\in\mathfrak{g}(\overline{F}). \end{align*}

Let $T_\gamma\subset G_\gamma$ be a maximal F-torus. Then, $T_\gamma$ is also a maximal F-torus of G since $\gamma$ is semisimple. Let $g\in G(\overline{F})$ be an element such that $\mathrm{Ad}(g)^{-1}T_{\overline{F}}=T_{\gamma,\overline{F}}$ (where T is the maximal F-torus of G, which is part of the pinning of G fixed in § 2.2). Let $\gamma'=\mathrm{ad}(g)\gamma$ . Since $\gamma$ lies in the center of $\mathfrak{g}_\gamma$ , we have $\gamma\in\mathrm{Lie}(T_\gamma)$ . Moreover, since $\gamma'$ is bounded, we get that $\gamma'\in\mathbb{t}(\overline{\mathcal{O}})$ . For any $\sigma\in\Gamma$ , we have $g\cdot\sigma_G(g)^{-1}\in N_G(T)(\overline{F})$ and we let $w_\sigma$ denote its image in $\mathbb{W}$ . Then, we have

(4.1) \begin{equation} \gamma'=w_\sigma\cdot\sigma_G(\gamma')=w_\sigma\cdot\rho_G(\sigma)\sigma(\gamma')\quad\mbox{for all }\sigma\in\Gamma. \end{equation}

Recall that $\mathbb{t}$ is the split $\mathcal{O}$ -model of $\mathfrak{t}$ . Let $\bar{\gamma}'\in\mathbb{t}(k)$ be the image of $\gamma'$ under the reduction map $\mathfrak{t}(\overline{\mathcal{O}})\to\mathbb{t}(k)$ . Since $\chi_G(\gamma)=\chi_G(\gamma')\in\mathfrak{c}(\mathcal{O})$ , we have $\chi_{\mathbb{G}}(\bar{\gamma}')\in\mathbb{c}^{\Theta}(k)$ .

We claim that the natural map $\mathbb{t}^{\Theta}(k)\to\mathbb{c}^{\Theta}(k)$ is surjective. By the assumption that G is tamely ramified, the order of $\Theta$ is invertible in $\mathcal{O}$ . So $\mathbb{t}^\Theta$ and $\mathbb{c}^\Theta$ are both smooth $\mathcal{O}$ -schemes by Lemma 2.2. In particular, any element in $\mathbb{c}^{\Theta}(k)$ lifts to an element in $\mathbb{c}^{\Theta}(\mathcal{O})$ by [Reference Kaletha and PrasadKP23, Lemma 8.1.3]. Since the morphism $\mathbb{t}^\Theta\to\mathbb{c}^\Theta$ is finite (hence proper), it suffices to show surjectivity of the map $\mathbb{t}^\Theta(\overline{F})\to\mathbb{c}^\Theta(\overline{F})$ . Thus, we are reduced to the characteristic-0 situation and the claim follows from [Reference LevyLev25]. Since $\mathbb{W}$ acts transitively on the fiber $\chi_{\mathbb{G}}^{-1}(\chi_{\mathbb{G}}(\bar{\gamma}'))$ , after conjugation by $N_G(T)(\overline{F})$ we may and do assume that $\bar{\gamma}'\in\mathbb{t}^{\Theta}(k)$ , so that $\rho_G(\sigma)(\bar{\gamma}')=\bar{\gamma}'$ for all $\sigma\in\Gamma$ .

Let $\Sigma_1\subset\Sigma$ be the subset consisting of roots $\alpha$ such that $\mathrm{val}(d\alpha(\gamma'))\gt0$ . Then, $\Sigma_1$ is the closed subsystem of $\Sigma$ consisting of the roots whose root hyperplane contains $\bar{\gamma}'$ . Moreover, $\Sigma_1$ is $\Theta$ -stable since $\bar{\gamma}'$ is $\Theta$ -invariant. Let $\mathbb{W}_1$ be the Weyl group of the root system $\Sigma_1$ and let $\mathbb{H}\subset \mathbb{G}$ be the split reductive subgroup containing $\mathbb{T}$ and having root system $\Sigma_1$ . By assumption, p is good for $\mathbb{G}$ and does not divide the order of $\pi_1(\mathbb{G}^{\mathrm{der}})$ , so p is not a torsion prime for the root datum of $\mathbb{G}$ . This implies that the centralizer k-group $\mathsf{G}_{\bar{\gamma}'}=\mathbb{G}_{k,\bar{\gamma}'}$ is connected by Proposition 2.6. Then, by [Reference SteinbergSte75, Lemma 3.7], we get that $\mathbb{W}_1=\mathbb{W}_{\bar{\gamma}'}$ , the stabilizer of $\bar{\gamma}'$ in $\mathbb{W}$ . From equation (4.1), we deduce

\begin{align*} \bar{\gamma}'=w_\sigma\cdot\rho_G(\sigma)(\bar{\gamma}')=w_\sigma\cdot\bar{\gamma}'\quad\mbox{for all }\sigma\in\Gamma. \end{align*}

Therefore, the assignment $\sigma\mapsto w_\sigma$ defines a homomorphism

\begin{align*} \rho_{\bar{\gamma}'}:\Gamma\to\mathbb{W}_{\bar{\gamma}'}=\mathbb{W}_1. \end{align*}

Let $\mathbb{t}_1\subset\mathbb{t}$ be the vanishing loci of $\{d\alpha,\alpha\in\Sigma_1\}$ defined by

\begin{align*} \mathbb{t}_1:=\mathrm{Spec}\frac{\mathcal{O}[\mathbb{t}]}{\langle d\alpha;\alpha\in\Sigma_1\rangle}=\mathrm{Spec}\frac{\mathrm{Sym}_{\mathcal{O}}(X^*(T)\otimes_{\mathbb{Z}}\mathcal{O})}{\langle \alpha,\alpha\in\Sigma_1\rangle}. \end{align*}

Then, we have $\Omega^1_{\mathbb{t}_1/\mathcal{O}}=(X^*(T)/\mathbb{Z}\Sigma_1)\otimes_{\mathbb{Z}}\mathcal{O}$ . Our assumptions on p imply that $X^*(T)/\mathbb{Z}\Sigma_1$ does not have p-torsion. Indeed, $\mathbb{Z}\Sigma/\mathbb{Z}\Sigma_1$ is p-torsion-free since p is good for G, while the order of the torsion subgroup of $(X^*(T)/\mathbb{Z}\Sigma)$ equals the order of $\pi_0(Z_{\mathbb{G}})$ , which is prime to p by assumption. So, $\Omega^1_{\mathbb{t}_1/\mathcal{O}}$ is a free $\mathcal{O}$ -module of finite rank and hence $\mathbb{t}_1$ is a smooth $\mathcal{O}$ -scheme. Moreover, $\mathbb{t}_1$ is preserved by the $\Theta$ -action and we have $\mathbb{t}_1\subset\mathbb{t}^{\mathbb{W}_1}$ . The fixed-point locus $\mathbb{t}_1^{\Theta}$ is a smooth $\mathcal{O}$ -scheme by Lemma 2.2. By construction, we have $\bar{\gamma}'\in\mathbb{t}_1^{\Theta}(k)$ and then by [Reference Kaletha and PrasadKP23, Lemma 8.1.3] there exists an element

\begin{align*} \gamma_0'\in\mathbb{t}_1^{\Theta}(\mathcal{O})\subset\mathfrak{t}(\mathcal{O}) \end{align*}

whose image under the natural reduction map $\mathbb{t}_1(\mathcal{O})\to\mathbb{t}_1(k)$ equals $\bar{\gamma}'$ . Let $\gamma_1':=\gamma'-\gamma_0'$ . Then, for any $\sigma\in\Gamma$ , we have $w_\sigma\cdot\sigma_G(\gamma_0')=\gamma_0'$ and by (4.1) we get

\begin{align*} w_\sigma\cdot\sigma_G(\gamma_1')=\gamma_1'. \end{align*}

Let $\gamma_0:=\mathrm{ad}(g)^{-1}(\gamma_0')$ and $\gamma_1:=\mathrm{ad}(g)^{-1}(\gamma_1')$ . Then, we have $\gamma=\gamma_0+\gamma_1$ and the above equations imply that $\gamma_0,\gamma_1\in\mathfrak{g}_\gamma(F)$ .

For any root $\alpha\in\Sigma_1$ we have $d\alpha(\gamma_0')=0$ and for any $\alpha\in\Sigma\backslash\Sigma_1$ we have $d\alpha(\bar{\gamma}')\in k^\times$ , so that $|d\alpha(\gamma_0')|=1$ . This implies that $\gamma_0'\in\mathfrak{t}(\mathcal{O})\subset\mathfrak{g}(F)$ is strongly semisimple and hence $\gamma_0\in\mathfrak{g}(F)$ is strongly semisimple. On the other hand, since the image of $\gamma_1'$ under the reduction map $\mathbb{t}(\overline{\mathcal{O}})\to\mathbb{t}(k)$ equals 0, it is a topologically nilpotent element in $\mathfrak{g}_{\gamma_0'}(\overline{F})$ . So $\gamma_1$ is a topologically nilpotent element in $\mathfrak{g}_{\gamma_0}(F)=\mathrm{Lie}(G_{\gamma_0})$ .

By our construction, $\gamma_0$ is $G(\overline{F})$ conjugate to the element $\gamma_0'\in\mathfrak{t}(\mathcal{O})$ . Let $G_{\gamma_0,\gamma_0'}$ be the F-scheme classifying elements $g\in G$ such that $\mathrm{ad}(g)(\gamma_0)=\gamma_0'$ . Then, $G_{\gamma_0,\gamma_0'}$ is nonempty, hence a $G_{\gamma_0}$ -torsor over F. Since $G_{\gamma_0}$ is connected reductive by Proposition 2.6, we have $H^1(F,G_{\gamma_0})=0$ by Steinberg’s theorem and the fact that F has cohomological dimension $\leqslant1$ . Thus, $G_{\gamma_0,\gamma_0'}$ is a trivial $G_{\gamma_0}$ -torsor, which means that $\gamma_0$ is G(F)-conjugate to $\gamma_0'\in\mathfrak{t}(\mathcal{O})$ . By Proposition 2.6, $G_{\gamma_0',\overline{F}}=\mathbb{H}_{\overline{F}}$ is a Levi subgroup of $G_{\overline{F}}$ . In addition, $G_{\gamma_0'}$ contains the maximally split maximal F-torus T of G since $\gamma_0'\in\mathfrak{t}(F)$ . Therefore, $G_{\gamma_0'}$ , and hence $G_{\gamma_0}$ , is an F-Levi subgroup of G.

Finally, suppose that $\gamma$ is regular semisimple. Then, $\gamma'$ is also regular semisimple and hence $d\alpha(\gamma_1')=d\alpha(\gamma')\ne0$ for all $\alpha\in\Sigma_1$ . So $\gamma_1$ is regular semisimple in $\mathrm{Lie}(G_{\gamma_0})$ .

4.3 Invariant pairing and quasi-logarithm

We review relevant facts from [Reference Kazhdan and VarshavskyKV06, § 1.8] and generalize some results there to the case of tamely ramified groups. Let G be an algebraic group over F and let $\rho:G\to\mathrm{GL}(V)$ be an algebraic representation on a finite-dimensional F-vector space V. Let $d\rho:\mathrm{Lie}(G)\to\mathrm{End}(V)$ be the induced representation of the Lie algebra. On $\mathrm{End}(V)$ we have the trace pairing defined by $(A,B):=\mathrm{Tr}(AB)$ . This induces a $G_{\mathrm{ad}}$ -invariant symmetric bilinear form on $\mathfrak{g}$ defined by $(X,Y)_\rho:=\mathrm{Tr}(d\rho(X)\circ d\rho(Y))$ . Suppose that $(\cdot,\cdot)_\rho$ is nondegenerate. Then, we can define the orthogonal projection $\mathrm{pr}_{\mathfrak{g}}:\mathrm{End}(V)\to\mathfrak{g}$ such that $\mathrm{pr}_{\mathfrak{g}}(A)\in\mathfrak{g}$ is the unique element satisfying $(\mathrm{pr}_{\mathfrak{g}}(A),X)_\rho=\mathrm{Tr}(A\circ d\rho(X))$ for all $X\in\mathfrak{g}$ . Define a morphism $\Phi_\rho:G\to\mathfrak{g}$ by $\Phi_\rho(g):=\mathrm{pr}_{\mathfrak{g}}(\rho(g)-\mathrm{Id})$ . Then, $\Phi_\rho$ is a quasi-logarithm map in the sense of [Reference Kazhdan and VarshavskyKV06], whose definition is as follows.

1. (i) For any parahoric subgroup $\mathbf{P}\subset G(F)$ , $\Phi$ extends to a morphism of $\mathcal{O}$ -schemes $\mathcal{G}_{\mathbf{P}}\to\mathrm{Lie}(\mathbf{P})$ .

2. (ii) There exists a parahoric subgroup $\mathbf{P}\subset G(F)$ such that $\Phi$ extends to a morphism of $\mathcal{O}$ -schemes $\mathcal{G}_{\mathbf{P}}\to\mathrm{Lie}(\mathbf{P})$ .

Suppose that $\Phi$ extends to a morphism of $\mathcal{O}$ -group schemes $\mathcal{G}_{\mathbf{P}}\to\mathrm{Lie}(\mathbf{P})$ for a parahoric subgroup $\mathbf{P}\subset G(F)$ . Let $\mathbf{I}\subset G(F)$ be an Iwahori subgroup contained in $\mathbf{P}$ . Then, $\Phi$ induces a quasi-logarithm map on the special fiber $\overline{\mathcal{G}}_{\mathbf{P}}$ and the image of $\mathbf{I}$ in the special fiber $\overline{\mathcal{G}}_{\mathbf{P}}$ is a Borel subgroup. Then, by [Reference Kazhdan and VarshavskyKV06, Lemma 1.8.3(a)] (which holds also for non-reductive groups), we deduce that $\Phi(\mathbf{I})\subset\mathrm{Lie}(\mathbf{I})$ . For any parahoric subgroup $\mathbf{P}'$ containing $\mathbf{I}$ , we have $\mathbf{P}'=\bigcup_{x\in\mathbf{P}'}x\mathbf{I} x^{-1}$ and $\mathrm{Lie}(\mathbf{P}')=\bigcup_{x\in\mathbf{P}'}\mathrm{ad}(x)\mathbf{P}'$ . By the G-equivariance of $\Phi$ , we get that $\Phi(\mathbf{P}')\subset\mathrm{Lie}(\mathbf{P}')$ and that this holds for any parahoric subgroup $\mathbf{P}'$ . Thus, $\Phi$ extends to a morphism of $\mathcal{O}$ -schemes $\mathcal{G}_{\mathbf{P}'}\to\mathrm{Lie}(\mathcal{G}_{\mathbf{P}'})$ for any parahoric subgroup $\mathbf{P}'\subset G(F)$ by [Reference Bruhat and TitsBT84, Proposition 1.7.6].

We are interested in extending the pairing $(\cdot,\cdot)_\rho$ and the map $\Phi_\rho$ to the parahoric group schemes of G. First, we consider the case where G is split.

1. (i) For any parahoric subgroup $\mathbf{P}\subset G(F)$ , we have

   \begin{align*} \mathrm{Lie}(\mathbf{P}_+)=\mathrm{Lie}(\mathbf{P})^\perp:=\{Y\in\mathfrak{g}(F),(Y,\mathrm{Lie}(\mathbf{P}))_\rho\subset\varpi\mathcal{O}\}. \end{align*}

2. (ii) Then, there exists a parahoric subgroup $\mathbf{P}\subset G(F)$ such that $\mathrm{Lie}(\mathbf{P}_+)=\mathrm{Lie}(\mathbf{P})^\perp$ .

In this case, the associated quasi-logarithm map $\Phi_\rho$ is defined over $\mathcal{O}$ .

This is [Reference Kazhdan and VarshavskyKV06, Lemma 1.8.7(a)] and [Reference Kazhdan and VarshavskyKV06, 1.8.9].

1. (i) The trace pairing $(\cdot,\cdot)_\rho$ defined by $\rho$ is a perfect pairing on the $\mathcal{O}$ -module $\mathfrak{g}^\natural=\mathrm{Lie}(G^\natural)$ .

2. (ii) The natural homomorphism $\mathrm{Out}(G^\natural)\to\mathrm{Out}(G_{\mathrm{ad}})$ between outer automorphism groups is an isomorphismFootnote ⁵ and the trace pairing $(\cdot,\cdot)_\rho$ is $\mathrm{Out}(G_{\mathrm{ad}})$ -invariant. In other words, for any $x,y\in\mathfrak{g}^\natural(F)$ and any $\theta\in\mathrm{Out}(G_{\mathrm{ad}})$ , we have $(x,y)_\rho=(\theta(x),\theta(y))_\rho$ .

3. (iii) The residue characteristic p divides neither the order of $\pi_1(G^{\natural,\mathrm{der}})$ Footnote ⁶ nor the order of the component group of the center of $G^\natural$ .

Next, we show that $(\cdot,\cdot)_\rho$ is $\mathrm{Out}(G_{\mathrm{ad}})$ -invariant. If G is exceptional, then $(\cdot,\cdot)_\rho$ is the Killing form and is easily checked to be invariant under any automorphism of the Lie algebra. If G is of type $\mathrm{A}_{n-1}$ , then any nontrivial outer automorphism of $\mathfrak{g}^\natural=\mathrm{Mat}_n$ is conjugate to $x\mapsto-{}^tx$ , which clearly fixes the usual trace pairing. If G is of type B or C, then $\mathrm{Out}(G_{\mathrm{ad}})$ is trivial and there is nothing to prove. If G is of type $\mathrm{D}_n$ for $n\geqslant5$ , then any nontrivial outer automorphism of $\mathfrak{g}^\natural=\mathfrak{so}(2n)$ has order 2 and is induced by conjugation by an element in $\mathrm{O}_{2n}$ (which contains $G^\natural=\mathrm{SO}_{2n}$ as a subgroup of index 2; see e.g. [Reference SpringerSpr09, § 17.3.2]), hence it fixes the trace pairing $(\cdot,\cdot)_\rho$ for the standard representation $\rho$ of $G^\natural=\mathrm{SO}_{2n}$ . Thus, we are left with the case where G is of type $\mathrm{D}_4$ . In this case, $\mathrm{Out}(G_{\mathrm{ad}})=S_3$ and the action of a generator of order 2 on $\mathfrak{g}^\natural=\mathfrak{so}(8)$ fixes the trace pairing for the same reason as in the previous case. Finally, we refer to [Reference Mikosz and WeberMW15, Appendix] for an explicit description of the action of an order-3 outer automorphism on $\mathfrak{g}^\natural=\mathfrak{so}(8)$ , from which we easily check that it fixes the standard trace pairing.Footnote ⁷

Next, we generalize the statements on regular centralizers and Kostant sections to tamely ramified groups. Following the notation in § 2.4, we consider a connected reductive group G over F that splits over a degree-e tamely ramified extension $F'/F$ , where $p\nmid e$ and we assume that $G_{\mathrm{ad}}$ is absolutely simple and p is good for G. Let $\mathbb{G}$ be the split form of G and let $\rho_G:\mu_e\to\mathrm{Out}(\mathbb{G})$ be the twisting data of G. Then, we have the special integral model of G for which we use the same notation $G:=(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{G})^{\mu_e,\circ}$ . Let $\mathbb{G}^\natural$ be the split reductive group over $\mathcal{O}$ associated to $G^\natural$ from Lemma 4.12. Let $\rho_{G_{\mathrm{ad}}}:\mu_e\to\mathrm{Out}(\mathbb{G}_{\mathrm{ad}})$ be the composition of $\rho_G$ with the natural homomorphism $\mathrm{Out}(\mathbb{G})\to\mathrm{Out}(\mathbb{G}_{\mathrm{ad}})$ . By condition (ii) in Lemma 4.12, we can lift $\rho_{G_{\mathrm{ad}}}$ to a homomorphism $\mu_e\to\mathrm{Out}(\mathbb{G}^\natural)$ , from which we obtain an outer twist $G^\natural$ of $\mathbb{G}^\natural$ that comes with an isomorphism of F-groups $G^\natural_{\mathrm{ad}}\cong G_{\mathrm{ad}}$ . Moreover, we also get a special $\mathcal{O}$ -model $G^\natural:=(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{G}^\natural)^{\mu_e,\circ}$ . Let $\mathfrak{g}^\natural=\mathrm{Lie}(G^\natural)=(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{g}^\natural)^{\mu_e}$ be the Lie algebra and let $\mathfrak{c}^\natural=(\mathrm{Res}_{\mathcal{O}'/\mathcal{O}}\mathbb{c}^\natural)^{\mu_e}$ be the Chevalley base for $G^\natural$ , both defined over $\mathcal{O}$ . We have the Chevalley homomorphism $\chi_{G^\natural}:\mathfrak{g}^\natural\to\mathfrak{c}^\natural$ and the universal centralizer group scheme $I^\natural$ over $\mathfrak{g}^\natural$ . By condition (iii) of Lemma 4.12, the residue characteristic p is not a torsion prime for the root datum of G. Then, by Proposition 4.4, we have the universal regular centralizer $J^\natural$ over $\mathfrak{c}^\natural$ , together with an isomorphism $(\chi_{G^\natural}^{\mathrm{reg}})^*J^\natural\cong I^\natural|_{\mathfrak{g}^{\natural,\mathrm{reg}}}$ .