On the geometric connected components of moduli spaces of $p$-adic shtukas and local Shimura varieties.

In [Vieh-Rap], Rapoport and Viehmann propose that there should be a theory of $p$-adic local Shimura varieties. They conjectured the existence of towers of rigid-analytic spaces whose cohomology groups “understand” the local Langlands correspondence for general $p$-adic reductive groups. In this way, these towers of rigid-analytic varieties would “interact” with the local Langlands correspondence in a similar fashion to how Shimura varieties “interact” with the global Langlands correspondence. Moreover, they conjectured many properties and compatibilities that these towers should satisfy.

In the last decade, the theory of local Shimura varieties went through a drastic transformation with Scholze’s introduction of perfectoid spaces and the theory of diamonds. In [Ber2], Scholze and Weinstein construct the sought for towers of rigid analytic spaces and generalized them to what are now known as moduli spaces of $p$-adic shtukas. Moreover, since then, many of the expected properties and compatibilities for local Shimura varieties have been verified and generalized to moduli spaces of $p$-adic shtukas. The study of the geometry and cohomology of local Shimura varieties and moduli spaces of $p$-adic shtukas is still a very active area of research due to their connection to the local Langlands correspondence. The main aim of this article is to study the locally profinite space of connected components, and describe explicitly the continuous right action of the group $G({\mathbb{Q}_{p}})\times J_{b}({\mathbb{Q}_{p}})\times W_{E}$ on this space. In particular, we prove and generalize [Vieh-Rap, Conjecture 4.26] for the case of unramified groups.

Let us recall the formalism of local Shimura varieties and moduli spaces of $p$-adic shtukas. Local $p$-adic shtuka datum over ${\mathbb{Q}_{p}}$ is a triple $(G,[b],[\mu])$ where $G$ is a reductive group over ${\mathbb{Q}_{p}}$, $[\mu]$ is a conjugacy class of geometric cocharacters $\mu:\mathbb{G}_{m}\to G$ and $[b]$ is an element of Kottwitz set $B(G,[\mu])$. Whenever $[\mu]$ is minuscule we say that $(G,[b],[\mu])$ is local Shimura datum. We let $E/{\mathbb{Q}_{p}}$ denote the reflex field of $[\mu]$ and $\breve{E}=E\cdot\breve{\mathbb{Q}}_{p}$. Associated to $(G,[b],[\mu])$ there is a tower of diamonds over ${\mathrm{Spd}}(\breve{E},O_{\breve{E}})$, denoted $({\rm{Sht}}_{G,[b],[\mu],{{\mathcal{K}}}})_{{\mathcal{K}}}$, where ${{\mathcal{K}}}\subseteq G({\mathbb{Q}_{p}})$ ranges over compact subgroup of $G({\mathbb{Q}_{p}})$. Moreover, whenever $[\mu]$ is minuscule and ${{\mathcal{K}}}$ is a compact open subgroup, then $({\rm{Sht}}_{G,[b],[\mu],{{\mathcal{K}}}})_{{\mathcal{K}}}$ is represented by the diamond associated to a unique smooth rigid-analytic space $\mathbb{M}_{{\mathcal{K}}}$ over $\breve{E}$. The tower $(\mathbb{M}_{{\mathcal{K}}})_{{\mathcal{K}}}$ is the local Shimura variety.

Associated to $[b]\in B(G,\mu)$ there is a reductive group $J_{b}$ over ${\mathbb{Q}_{p}}$. After basechange to a completed algebraic closure, each individual space $({\rm{Sht}}_{G,[b],[\mu],{{\mathcal{K}}}}\times\mathbb{C}_{p})_{{\mathcal{K}}}$ comes equipped with continuous and commuting right actions by $J_{b}({\mathbb{Q}_{p}})$ and the Weil group $W_{E}$. Moreover, the tower receives a right action by the group $G({\mathbb{Q}_{p}})$ by using correspondences. When we let ${{\mathcal{K}}}=\{e\}$ we obtain the space at infinite level, denoted ${\rm{Sht}}_{G,[b],[\mu],\infty}\times\mathbb{C}_{p}$, which overall comes equipped with a continuous right action by $G({\mathbb{Q}_{p}})\times J_{b}({\mathbb{Q}_{p}})\times W_{E}$.

Since the actions are continuous the groups $G({\mathbb{Q}_{p}})\times J_{b}({\mathbb{Q}_{p}})\times W_{E}$ act continuously on $\pi_{0}({\rm{Sht}}_{G,[b],[\mu],\infty}\times\mathbb{C}_{p})$ and our main theorem describes explicitly this action whenever $G$ is an unramified reductive group over ${\mathbb{Q}_{p}}$ and $([b],[\mu])$ is HN-irreducible. It is natural to expect that the methods of this paper combined with those of [Hans] and [gaisin] could be used to remove the HN-irreducible condition. We do not pursue this generality.

Before stating our main theorem we set some notation. Let $(G,[b],[\mu])$ be local $p$-adic shtuka datum with $G$ an unramified reductive group over ${\mathbb{Q}_{p}}$. Let $G^{{\mathrm{der}}}$ denote the derived subgroup of $G$ and $G^{{\mathrm{sc}}}$ denote the simply connected cover of $G^{{\mathrm{der}}}$, let $N$ denote the image of $G^{{\mathrm{sc}}}({\mathbb{Q}_{p}})$ in $G({\mathbb{Q}_{p}})$ and let $G^{\circ}=G({\mathbb{Q}_{p}})/N$. This is a locally profinite topological group and it is the maximal abelian quotient of $G({\mathbb{Q}_{p}})$ when this later is considered as an abstract group. Let $E\subseteq\mathbb{C}_{p}$ be the field of definition of $[\mu]$, let ${{\mathrm{Art}}}_{E}:W_{E}\to E^{\times}$ be Artin’s reciprocity character from local class field theory. In §4 we associate to $[\mu]$ a continuous map of topological groups ${{\mathrm{Nm}}}_{[\mu]}^{\circ}:E^{\times}\to G^{\circ}$ and we associate to $[b]$ a map ${\mathrm{det}}^{\circ}:J_{b}({\mathbb{Q}_{p}})\to G^{\circ}$.

The general construction of ${{\mathrm{Nm}}}_{[\mu]}^{\circ}$ and ${\mathrm{det}}^{\circ}$ uses z-extensions and we do not review it in this introduction. Nevertheless, whenever $G^{{\mathrm{sc}}}=G^{{\mathrm{der}}}$ we can construct them as follows. In this case, $G^{\circ}=G^{{\mathrm{ab}}}({\mathbb{Q}_{p}})$ with $G^{{\mathrm{ab}}}$ is the co-center of $G$. If we let ${\mathrm{det}}:G\to G^{{\mathrm{ab}}}$ be the quotient map we can consider the induced data $\mu^{{\mathrm{ab}}}={\mathrm{det}}\circ[\mu]$ and $[b^{{\mathrm{ab}}}]=[{\mathrm{det}}(b)]$. Then ${{\mathrm{Nm}}}^{\circ}_{[\mu]}$ can be defined as:

$$E^{\times}\xrightarrow{\mu^{{\mathrm{ab}}}}G^{{\mathrm{ab}}}(E)\xrightarrow{{{\mathrm{Nm}}}^{G^{{\mathrm{ab}}}}_{E/{\mathbb{Q}_{p}}}}G^{{\mathrm{ab}}}({\mathbb{Q}_{p}})=G^{\circ}.$$

Here for a torus $T$ over ${\mathbb{Q}_{p}}$, like $G^{{\mathrm{ab}}}$, we are letting ${{\mathrm{Nm}}}^{T}_{E/{\mathbb{Q}_{p}}}:T^{{\mathrm{ab}}}(E)\to T^{{\mathrm{ab}}}({\mathbb{Q}_{p}})$ denote the usual norm map

$$t\mapsto\prod_{\gamma\in{{\mathrm{Gal}}}(E/\mathbb{Q}_{p})}\gamma(t).$$

On the other hand, ${\mathrm{det}}^{\circ}:J_{b}({\mathbb{Q}_{p}})\to G^{{\mathrm{ab}}}({\mathbb{Q}_{p}})$ is ${\mathrm{det}}=j_{b^{{\mathrm{ab}}}}\circ{\mathrm{det}}_{b}$ where ${\mathrm{det}}_{b}:J_{b}({\mathbb{Q}_{p}})\to J_{b^{{\mathrm{ab}}}}({\mathbb{Q}_{p}})$ is obtained from functoriality of the formation of $J_{b}$, and $j_{b^{{\mathrm{ab}}}}:J_{b^{{\mathrm{ab}}}}({\mathbb{Q}_{p}})\xrightarrow{\cong}G^{{\mathrm{ab}}}({\mathbb{Q}_{p}})$ is obtained from regarding $J_{b^{{\mathrm{ab}}}}({\mathbb{Q}_{p}})$ and $G^{{\mathrm{ab}}}({\mathbb{Q}_{p}})$ as subgroups of $G^{{\mathrm{ab}}}(K_{0})$ and exploiting that $G^{{\mathrm{ab}}}$ is commutative. Our first main theorem is:

Let us comment on previous results in the literature. Before a full theory of local Shimura varieties was available the main examples of local Shimura varieties one could work with were the ones obtained as the generic fiber of a Rapoport–Zink space ([RZ]). The most celebrated examples of Rapoport–Zink spaces are of course the Lubin–Tate tower and the tower of covers of Drinfeld’s upper half space. In [deJong] de Jong, as an application of his theory of fundamental groups, computes the connected components of the Lubin–Tate tower for ${\mathrm{GL}}_{n}({\mathbb{Q}_{p}})$. In [Strauch], Strauch computes by a different method the connected components of the Lubin–Tate tower for ${\mathrm{GL}}_{n}(F)$ and an arbitrary finite extension $F$ of ${\mathbb{Q}_{p}}$ (including ramification). In [ChenDet], M. Chen constructs $0$-dimensional local Shimura varieties and studies their geometry. In a later paper [Chen], she constructs her “determinant” map and uses these $0$-dimensional local Shimura varieties to describe connected components of Rapoport–Zink spaces of EL and PEL type associated to more general unramified reductive groups. Our result goes beyond the previous ones in that the only condition imposed on $G$ is unramifiedness. In this way, our result is the first to cover very general families of local Shimura varieties that can not be constructed from a Rapoport–Zink space. In particular, our result is new for local Shimura varieties associated to reductive groups of exceptional types.

The central strategy of Chen’s result builds on and heavily generalizes the central strategy used by de Jong. Two key inputs of Chen’s work to the strategy are the use of her “generic” crystalline representations and her collaboration with Kisin and Viehmann on computing the connected components of affine Deligne–Lusztig varieties [CKV]. Our strategy takes these two inputs as given.

We build on the central strategy employed by de Jong and Chen, but the versatility of Scholze’s theory of diamonds and the functorial construction of local Shimura varieties allow us to make simplifications and streamline the proof. Since our arguments take place in Scholze’s category of diamonds rather than the category of rigid analytic spaces, our argument works even for moduli spaces of $p$-adic shtukas that are not a local Shimura variety. In these (non-representable) cases, the result is new even for $G={\mathrm{GL}}_{n}$.

Our new main contribution to the central strategy is the use of specialization maps. To use these specialization maps in a rigorous way, we developed a formalism whose details were worked out in the separate paper [Specializ].

Let us sketch the central strategy to prove 1. Once one knows that $\pi_{0}({\rm{Sht}}_{G,b,[\mu],\infty}\times\mathbb{C}_{p})$ is a right $G^{\circ}$-torsor, computing the actions by $W_{E}$ and $J_{b}({\mathbb{Q}_{p}})$ in terms of the $G^{\circ}$ action can be reduced to the tori case using functoriality, z-extensions and the determinant map. These uses mainly group theoretic methods and down to earth diagram chases. In the tori case, the $J_{b}({\mathbb{Q}_{p}})$ action is easy to compute and the $W_{E}$ action can be bootstrapped to an easier case as follows. For tori $T$, by the work of Kottwitz, we know that the set $B(T,\mu)$ has a unique element so that the data of $b$ is redundant. We can consider the category of pairs $(T,\mu)$ where $T$ is a torus over ${\mathbb{Q}_{p}}$ and $\mu$ is a geometric cocharacter whose field of definition is $E$. The construction of moduli spaces of shtukas is functorial with respect to this category. Moreover, this category has an initial object given by $({\mathrm{Res}}_{E/{\mathbb{Q}_{p}}}(\mathbb{G}_{m}),\mu_{u})$ where

$$\mu_{u}:\mathbb{G}_{m}\to{\mathrm{Res}}_{E/{\mathbb{Q}_{p}}}(\mathbb{G}_{m})_{E}$$

is the unique map of tori that on $E$-points is given by the formula

$$f\mapsto f\otimes_{\mathbb{Q}_{p}}f.$$

After more diagram chasing one can again reduce the tori case to this “universal” case. Finally, this case can be done explicitly using the theory of Lubin–Tate groups and their relation to local class field theory. As we have mentioned, the tori case was already handled by M. Chen in [ChenDet], but for the convenience of the readers we recall part of the story in a different language.

Let us sketch how to prove that $\pi_{0}({\rm{Sht}}_{G,b,[\mu],\infty}\times\mathbb{C}_{p})$ is a $G^{\circ}$ torsor in the simplest case. For this, let $G$ be semisimple and simply connected. Our theorem then says that ${\rm{Sht}}_{G,b,[\mu],\infty}\times\mathbb{C}_{p}$ is connected.

The first step is to prove that $G({\mathbb{Q}_{p}})$ acts transitively on $\pi_{0}({\rm{Sht}}_{G,b,[\mu],\infty}\times\mathbb{C}_{p})$. Using the Grothendieck–Messing period map one realizes that this is equivalent to proving that the $b$-admissible locus of Scholze’s $B_{\mathrm{dR}}$-Grassmannian is connected. This fact is a result of Hansen and Weinstein to which we give an alternative proof.

For the next step, let $x\in\pi_{0}({\rm{Sht}}_{G,b,[\mu],\infty}\times\mathbb{C}_{p})$ and let $G_{x}\subseteq G({\mathbb{Q}_{p}})$ denote the stabilizer of $x$. Let ${{\mathcal{K}}}\subseteq G({\mathbb{Q}_{p}})$ be a hyperspecial subgroup of $G$. We claim that it is enough to prove that $G_{x}$ is open and that $G({\mathbb{Q}_{p}})={{\mathcal{K}}}\cdot G_{x}$. Indeed, ${{\mathcal{K}}}$ surjects onto $G({\mathbb{Q}_{p}})/G_{x}$ so that this space is discrete and compact therefore finite. By a theorem of Margulis [Marg], since we assumed $G$ to be simply connected, the only open subgroup of finite index is the whole group so that $G_{x}=G({\mathbb{Q}_{p}})$. The proof that $G_{x}$ is open relies heavily on M. Chen’s main result of [Chen] on “generic” crystalline representations. To be able to apply her result in our context one uses that for suitable $p$-adic fields $K$, every crystalline representation is realized as a ${\mathrm{Spd}}(K,O_{K})$-valued point in Scholze’s $B_{\mathrm{dR}}$-Grassmannian. For the convenience of the reader, we include a discussion on how to think of crystalline representations as ${\mathrm{Spd}}(K,O_{K})$-valued points.

Finally, proving that $G({\mathbb{Q}_{p}})={{\mathcal{K}}}\cdot G_{x}$ is equivalent to proving that ${\rm{Sht}}_{G,b,[\mu],{{\mathcal{K}}}}\times\mathbb{C}_{p}$, the ${{\mathcal{K}}}$-level moduli space of shtukas, is connected. This is where our theory of specialization maps gets used, which leads to our second main theorem. Suppose $G$ general reductive group over ${\mathbb{Q}_{p}}$ (no longer assumed to be unramified) and assume that ${{\mathcal{K}}}\subseteq G({\mathbb{Q}_{p}})$ can be realized as the ${\mathbb{Z}_{p}}$-points of a parahoric group scheme $\mathscr{G}$ over ${\mathbb{Z}_{p}}$. In this circumstance, Scholze and Weinstein, construct a v-sheaf ${{\mathrm{Sht}}^{\mathscr{G}_{b},\leq\mu}_{{O_{\breve{E}}}}}$ defined over ${\mathrm{Spd}}(O_{\breve{E}})$ and whose generic fiber is ${\rm{Sht}}_{G,[b],[\mu],{{\mathcal{K}}}}$ ([Ber2, §25]).

Fortunately for us, the study of connected components of affine Deligne–Lusztig varieties has enough literature [CKV], [Nie] [Hu-Zhou]. In the HN-irreducible case, and $G$ unramified, they can be identified with certain subsets of $\pi_{1}(G)$. If we go back to the assumptions of 1 and assume again that $G$ is semi-simple and simply connected, we get $\pi_{1}(G)=\{e\}$, which finishes the (sketch of) the proof of 1 for this case. The central strategy used for general unramified groups $G$ is not very different in spirit and only requires more patience.

The proof of 2 uses the machinery from integral $p$-adic Hodge theory as discussed in [Ber2], the formalism developed in [Specializ], and for general parahoric our recent collaboration [AGLR22]. The key inputs to prove that ${{\mathrm{Sht}}^{\mathscr{G}_{b},\leq\mu}_{{O_{\breve{E}}}}}$ has a specialization map are Kedlaya’s work [KedAinf] and Anschütz’ work [Ans2, Theorem 1.1] on extending vector bundles and $\mathscr{G}$-torsors over the punctured spectrum of $A_{\mathrm{inf}}$. Recall that ${{\mathrm{Sht}}^{\mathscr{G}_{b},\leq\mu}_{{O_{\breve{E}}}}}$ parametrizes triples $({{\mathcal{T}}},\Phi,\rho)$ where $({{\mathcal{T}}},\Phi)$ is a shtuka with $\mathscr{G}$ structure and $\rho:{{\mathcal{T}}}\to\mathscr{G}_{b}$ is $\varphi$-equivariant trivialization over ${\mathcal{Y}}_{[r,\infty]}$ for large enough $r$. A key observation is that $({{\mathrm{Sht}}^{\mathscr{G}_{b},\leq\mu}_{{O_{\breve{E}}}}})^{{\mathrm{red}}}$ is roughly speaking the locus in which $\rho$ is meromorphic. With this in mind we prove $({{\mathrm{Sht}}^{\mathscr{G}_{b},\leq\mu}_{{O_{\breve{E}}}}})^{\mathrm{red}}={X_{\mathscr{G}}^{\leq\mu}(b)}$. The finiteness properties (being rich), are known facts coming from the Grothendieck–Messing period morphism and general results on affine Deligne–Lusztig varieties. Finally, to prove surjectivity of the specialization map and relate the connected components of the generic fiber with the connected components of the reduced special fiber, one is led to study the tubular neighborhoods of $({{\mathrm{Sht}}^{\mathscr{G}_{b},\leq\mu}_{{O_{\breve{E}}}}},{\rm{Sht}}_{G,b,[\mu],{{\mathcal{K}}}})$ (as in [Specializ, Definition 4.18, Definition 4.38]). To do this, we construct a “local model diagram” for tubular neighborhoods. We clarify below.

Before stating our last main theorem we setup some terminology and formulate a conjectural statement that is philosophically aligned with Grothendieck–Messing theory. Let ${{\mathcal{M}}^{\mathscr{G},\leq\mu}_{O_{E}}}$ denote the local model studied in [AGLR22] and let ${{\mathcal{A}}_{\mathscr{G},\mu}}=({{\mathcal{M}}^{\mathscr{G},\leq\mu}_{O_{E}}})^{\mathrm{red}}$ denote its reduced special fiber. This is the $\mu$-admissible locus in the Witt vector affine flag variety. We let $F\supseteq\breve{E}$ be a nonarchimedean field extension with ring of integers $O_{F}$ and algebraically closed residue field $k_{F}$.

The weaker version that we are able to prove at the moment is as follows.

Let us mention that this version of the local model diagram, although not completely satisfactory, has already found some applications in the recent representability results of Pappas and Rapoport [pappas2021padic].

Finally, to establish the identity $\pi_{0}({\rm{Sht}}_{G,b,[\mu],{{\mathcal{K}}}}\times\mathbb{C}_{p})\cong\pi_{0}({X_{\mathscr{G}}^{\leq\mu}(b)})$ one is reduced to proving that all the tubular neighborhoods of the local model ${{\mathcal{M}}^{\mathscr{G},\leq\mu}_{O_{\mathbb{C}_{p}}}}$ have connected generic fiber. As was observed in [AGLR22], the condition that these tubular neighborhoods are generically connected is a “kimberlite analogue” of normality. When $\mathscr{G}$ is hyperspecial, we prove this normality in [Specializ] using a Demazure resolution. Unfortunately, this part of the argument doesn’t generalize directly for general parahoric groups $\mathscr{G}$, and the proof of normality will require more sophisticated tools.

Let us comment on the organization of this paper. The goal of the first chapter is to prove 1 using mainly generic fiber methods and taking as a black box some “integral method inputs”, which we justify in the second chapter. In the first two sections, we recall the relation between crystalline representations, Scholze’s theory of diamonds, Chen’s “generic” cyrstalline representations, and other geometric constructions that appear in modern rational $p$-adic Hodge theory. This part of the paper is purely expository, but it is important for the rest of the argument to have these relations in mind. In the third section we discuss local Shimura varieties associated to tori and we review M. Chen’s results on this objects. In section four, the details of the proof of 1 are provided.

The goal of the second chapter is to prove 2 and 3. In the first section we collect some facts from integral $p$-adic Hodge theory required for our argument to go through. In the second section, we recall the kimberlite structure of the local model. In the third section, we establish the main properties we need to construct a specialization map for moduli spaces of shtukas. In the final section, we prove 3 and finish the proof of 2.

We thank the author’s PhD advisor, Sug Woo Shin, for his interest, his insightful questions and suggestions at every stage of the project, and for his generous constant encouragement and support during the PhD program. Laurent Fargues, David Hansen, Peter Scholze and Jared Weinstein for answering questions the author had on $p$-adic Hodge theory and the theory of diamonds. Georgios Pappas and Michael Rapoport for bringing to our attention a serious flaw on an attempt we had to prove 1.

The author would also like to thank João Lourenço, Alexander Bertoloni, Rahul Dalal, Gabriel Dorfsman-Hopkins, Zixin Jiang, Dong Gyu Lim, Sander Mack-Crane, Gal Porat, Koji Shimizu for various degrees of help during the preparation of the manuscript.

This work was supported by the Doctoral Fellowship from the “University of California Institute for Mexico and the United States” (UC MEXUS) by the “Consejo Nacional de Ciencia y Tecnología” (CONACyT), and by Peter Scholze’s Leibniz price.

When $R$ is a characteristic $p$ ring we let $W(R)$ denote the ring of $p$-typical Witt vectors of $R$ and we denote by $\varphi:W(R)\to W(R)$ the canonical lift of arithmetic Frobenius. For a Huber pair $(R,R^{+})$ we use the abbreviations ${\mathrm{Spd}}(R)$ and ${\mathrm{Spa}}(R)$ when the entry $R^{+}$ is understood from the context. Whenever $f:R\to R^{\prime}$ is a ring homomorphism (respectively morphism of Huber pairs), we let $f^{{\mathrm{op}}}:{\mathrm{Spec}}(R^{\prime})\to{\mathrm{Spec}}(R)$ (respectively $f^{{\mathrm{op}}}:{\mathrm{Spa}}({R^{\prime}})\to{\mathrm{Spa}}({R})$ or $f^{{\mathrm{op}}}:{{\mathrm{Spd}}({R^{\prime}})}\to{{\mathrm{Spd}}({R})}$) the morphism of spaces induced by $f$.

We let $k$ be an algebraically closed field in characteristic $p$. We let $K_{0}=W(k)[\frac{1}{p}]$. We fix an algebraic closure ${\overline{{K}}}_{0}$ of ${K}_{0}$, and we let ${C}_{p}$ denote the $p$-adic completion of ${\overline{{K}}}_{0}$. We use $K$ to denote subfields of ${C}_{p}$ of finite degree over $K_{0}$. We let $\Gamma_{K}$ denote the continuous automorphisms of ${C}_{p}$ that fix $K$. If ${\overline{K}}_{0}$ is the algebraic closure of $K_{0}$ in ${C}_{p}$ then $\Gamma_{K}$ is canonically isomorphic to ${{\mathrm{Gal}}}({\overline{K}}_{0}/K)$, since ${\overline{K}}_{0}$ is dense in ${C}_{p}$. We denote by $\Gamma_{K}^{{\mathrm{op}}}$ the opposite group of $\Gamma_{K}$ which we identify with the group of automorphisms of ${\mathrm{Spec}}({C}_{p})$ over ${\mathrm{Spec}}(K_{0})$. We let $W_{{K}_{0}}$ denote the subset of continuous automorphisms of ${\mathrm{Aut}}({C}_{p})$ that stabilize ${K}_{0}$ and act as an integral power of $\varphi$ on ${K}_{0}$. We topologize $W_{{K}_{0}}$ so that $\Gamma_{{K}_{0}}$ is an open subgroup. Suppose $E\subseteq{C}_{p}$ is a field of finite degree over ${\mathbb{Q}_{p}}$, and let $\mathbb{Q}_{p^{s}}$ be the maximal unramified extension of ${\mathbb{Q}_{p}}$ contained in $E$. The extension $E/\mathbb{Q}_{p^{s}}$ is totally ramified and $E\otimes_{\mathbb{Q}_{p^{s}}}{K}_{0}$ is canonically isomorphic to the compositum ${E_{0}}:=E\cdot{K}_{0}$ inside of ${C}_{p}$. We define an automorphism $\hat{\varphi}\in{\mathrm{Aut}}({E_{0}})$ as the automorphism that maps to ${\mathrm{Id}}\otimes\varphi$ under this identification. We let $W^{K_{0}}_{E}$ denote the continuous automorphisms of ${C}_{p}$ that stabilize ${E_{0}}$ and act on ${E_{0}}$ as $\hat{\varphi}^{s\cdot n}$ for some $n\in\mathbb{Z}$. Notice that $W^{K_{0}}_{E}$ fixes $E$. The case of interest is when $k={\overline{\mathbb{F}}}_{p}$ but some of arguments require us to pass to larger fields. When $k={\overline{\mathbb{F}}}_{p}$ then $K_{0}=\breve{\mathbb{Q}}_{p}$, ${C}_{p}=\mathbb{C}_{p}$, $E_{0}=\breve{E}$ and $W^{K_{0}}_{E}=W_{E}$.

Through out the text, $G$ will denote a connected reductive group over ${\mathbb{Q}_{p}}$. In certain subsections we will add the additional assumptions that $G$ is quasi-split or even stronger that it is unramified over ${\mathbb{Q}_{p}}$. We will point out when one of these two assumptions are taken. Whenever $G$ is quasi-split we will denote by $A$ a maximally split sub-torus of $G$ defined over ${\mathbb{Q}_{p}}$, $T$ will denote the centralizer of $A$ which is also a torus and $B$ will denote a ${\mathbb{Q}_{p}}$-rational Borel containing $T$. We will denote by $\mathscr{G}$ a parahoric model of $G$ over ${\mathbb{Z}_{p}}$. Sometimes we will assume $\mathscr{G}$ is hyperspecial in which case we will abuse notation and declare $G=\mathscr{G}$.

We will often work in the situation in which we are given an element $b\in G(K_{0})$ and/or a cocharacter $\mu:\mathbb{G}_{m}\to G_{K}$. In these circumstances $[b]$ always denotes the $\varphi$-conjugacy class of $b$ in $G({K}_{0})$ and $[\mu]$ denotes the unique geometric conjugacy class of cocharacters $[\mu]\in{\mathrm{Hom}}(\mathbb{G}_{m},G_{{{\overline{\mathbb{Q}}}_{p}}})$ that is conjugate to $\mu$ through the action of $G_{{C}_{p}}$. Moreover, we let $E/{\mathbb{Q}_{p}}$ denote the field extension contained in ${C}_{p}$ over which $[\mu]$ is defined. We let $E_{0}$ denote the compositum of $E$ and $K_{0}$ in ${C}_{p}$.