Finiteness properties of affine Deligne-Lusztig varieties

TL;DR

This paper establishes fundamental finiteness properties of affine Deligne-Lusztig varieties, showing they are locally of finite type and have global finiteness under minimal assumptions, advancing understanding of their geometric structure.

Contribution

It proves that affine Deligne-Lusztig varieties are locally of finite type and globally finite under minimal assumptions, generalizing previous special case results.

Findings

Affine Deligne-Lusztig varieties are locally of finite type.

They exhibit global finiteness related to group actions.

Results hold under minimal assumptions on the group.

Abstract

Affine Deligne-Lusztig varieties are closely related to the special fibre of Newton strata in the reduction of Shimura varieties or of moduli spaces of $G$-shtukas. In almost all cases, they are not quasi-compact. In this note we prove basic finiteness properties of affine Deligne-Lusztig varieties under minimal assumptions on the associated group. We show that affine Deligne-Lusztig varieties are locally of finite type, and prove a global finiteness result related to the natural group action. Similar results have previously been known for special situations.

Equations

$${\mathcal{F}\ell }_{\mathcal{G}}(k)=G(L)/P,$$

$$X_Z(b)(S)=\{g\in {\mathcal{F}\ell }_{\mathcal{G}}(S)\mid g_x^{-1}b\sigma (g_x)\in Z({\kappa }_x)\text{ for every geometric point }x\text{ of }S\}.$$

$${\tilde{X}}_Z(b)(S)=\{g\in LG(S)\mid g_x^{-1}b\sigma (g_x)\in Z({\kappa }_x)\text{ for every geometric point }x\text{ of }S\}.$$

$$J_b(R)=\{g\in G(R{\otimes }_{F}L)\mid gb=b\sigma (g)\}.$$

$$\tilde{W}:=N_0(L)/P_{T_0}\cong X_{\ast }(T_0{)}_{\mathrm{Gal}(\overline{L}/L)}⋊W,$$

$${\tilde{W}}^P\backslash \tilde{W}/{\tilde{W}}^P\stackrel{1:1}{\to }P\backslash G(L)/P.$$

$${\mathcal{B}}^e(G,L)=\mathcal{B}(G,L)\times V_0(G,L)$$

$$i:G(L)\to \mathrm{Isom}({\mathcal{B}}^e(G,L)),$$

$$f_{\ast }:{\mathcal{B}}^e(M_b,L)\to {\mathcal{B}}^e(G,L).$$

$${\mathcal{B}}^e(J_b,L)$$

$$S_w^{\circ }$$

$$S_w$$

$${\tilde{X}}_Z(b):=\{g\in G(L)\mid g^{-1}b\sigma (g)\in Z\}.$$

$$\iota :G(L)\to {\mathcal{B}}^e(G,L),g\mapsto g.p_0.$$

$$\mathrm{Min}(b\sigma ):=\{x\in {\mathcal{B}}^e(G,L)\mid d(x,b\sigma (x))\text{ attains its minimal possible value.}\},$$

$$d(x,b\sigma (x))=d(g^{-1}.x,g^{-1}.b\sigma (x))=d(p_0,g^{-1}b\sigma (g).p_0)<c_Z.$$

$$d((j^{-1}\cdot g).p_0,p_0)=d(x,j.p_0)<C_Z,$$

$$d(g,h)\le n⟺h^{-1}g\in \overline{P{\rho }^{\vee }({ϵ}^{2n})P}=\bigcup _{w\le 2n{\rho }^{\vee }}PwP$$

$${\mathcal{M}}_n(k):=\{x\in \mathcal{M}(k)\mid d(x,x_0)\le n\}.$$

$${\mathcal{M}}_n(y_0)(k):=\{y\in {\mathcal{M}}_n(k)\mid d(y_0,y)\le N_0\}.$$

$$U_n:={\mathcal{M}}_n(x_0)\setminus \bigcup _{\genfrac{}{}{0ex}{}{y\in \mathcal{M}(k_s)}{d(x_0,y)>N_0}}{\mathcal{M}}_n(y).$$

Full text

The authors were partially supported by ERC Consolidator Grant 770936: NewtonStrat.

1. Introduction

Let $F$ be a local field, $O_{F}$ its ring of integers, and $k_{F}=\mathbb{F}_{q}$ its residue field, a finite field of characteristic $p$. We denote by $L$ the completion of the maximal unramified extension of $F$, and by $O_{L}$ its ring of integers. Then the residue field $k$ of $L$ is an algebraic closure of $\mathbb{F}_{q}$. We denote by ${\epsilon}$ a uniformizer of $F$, which is then also a uniformizer of $L$. Let $\sigma$ be the Frobenius of $k$ over $k_{F}$ and also of $L$ over $F$. We denote by $I$ the inertia group of $F$.

We consider a smooth affine group scheme $\mathscr{G}$ over $O_{F}$ with reductive generic fibre. Let $P=\mathscr{G}(O_{L})$ and let $G=\mathscr{G}_{F}$.

We denote by ${\mathscr{F}\ell}_{\mathscr{G}}$ the base change to $k$ of the affine flag variety (over $k_{F}$) associated with $\mathscr{G}$ as in [PR08, § 1.c] and [BS17, Def. 9.4]. In particular, ${\mathscr{F}\ell}_{\mathscr{G}}$ is a sheaf on the fpqc-site of $k$-schemes ($\operatorname{char}F=p$) resp. of perfect $k$-schemes ($\operatorname{char}F=0$) with

[TABLE]

which is representable by an inductive limit of finite type schemes ($\operatorname{char}F=p$) resp. of perfectly of finite type schemes ($\operatorname{char}F=0$); see [PR08, Thm. 1.4],[BS17, Cor. 9.6]. Hence we can define an underlying topological space of ${\mathscr{F}\ell}_{\mathscr{G}}$, which is Jacobson. This means that by mapping a subset of ${\mathscr{F}\ell}_{\mathscr{G}}$ to its intersection with the subset of closed points ${\mathscr{F}\ell}_{\mathscr{G}}(k)\subset{\mathscr{F}\ell}_{\mathscr{G}}$ we obtain a bijection between the open subsets of ${\mathscr{F}\ell}_{\mathscr{G}}$ and the open subsets of ${\mathscr{F}\ell}_{\mathscr{G}}(k)$ (same for closed and for locally closed subsets). Moreover, being a base change from $k_{F}$, we have an action of $\sigma$ on ${\mathscr{F}\ell}_{\mathscr{G}}$.

To define affine Deligne-Lusztig varieties we fix an element $b\in G(L)$ and a locally closed subscheme $Z$ of the loop group $LG$ which is stable under $P$-$\sigma$-conjugation. Then we consider the functor on reduced $k$-schemes resp. reduced perfect $k$-schemes with

[TABLE]

Remark 1.1*.*

The functor $X_{Z}(b)$ defines a locally closed reduced sub-indscheme of ${\mathscr{F}\ell}_{\mathscr{G}}$: Consider the functor $\tilde{X}_{Z}(b)$ on reduced $k$-schemes resp. perfect $k$-schemes with

[TABLE]

Then $\tilde{X}_{Z}(b)$ is the inverse image of $Z$ under the morphism $LG\rightarrow LG$ with $g\mapsto g^{-1}b\sigma(g)$. Since $Z$ is locally closed, also $\tilde{X}_{Z}(b)$ defines a locally closed reduced sub-ind-scheme of $LG$. Furthermore, $X_{Z}(b)$ is the image of $\tilde{X}_{Z}(b)$ under the quotient map $LG\rightarrow{\mathscr{F}\ell}_{\mathscr{G}}$, which is an $L^{+}\mathscr{G}$-torsor. Hence it is again a locally closed sub-ind-scheme.

Let $J_{b}$ be the reductive group over $F$ whose $R$-valued points for any $F$-algebra $R$ are given by

[TABLE]

Then for every $Z$ there is a natural action of $J_{b}(F)$ on $X_{Z}(b)$ given by left multiplication. Our main result is

Theorem 1.2**.**

Assume in addition that $Z$ is bounded (see Section 3 for the definition of boundedness).

• (1)

$X_{Z}(b)$* is a scheme which is locally of finite type in the case that $\operatorname{char}F=p$ and locally perfectly of finite type in the case $\operatorname{char}F=0$.*

• (2)

The action of $J_{b}(F)$ on the set of irreducible components of $X_{Z}(b)$ has finitely many orbits.

This theorem is related to the fact that they are the underlying reduced subscheme of moduli spaces of local $G$-shtukas and to the general expectation for the arithmetic case that (at least in the minuscule case) affine Deligne-Lusztig varieties are the reduction modulo $p$ of integral models of local Shimura varieties. Their cohomology is conjectured to decompose according to the local Langlands and Jacquet-Langlands correspondences. In order to be able to apply the usual methods, one needs the cohomology groups to be finitely generated $J_{b}(F)$-representations, and thus the “infinite level” cohomology groups to be admissible. This follows from the above theorem by a formal argument once the integral model is constructed (see for example [Mie20, Thm. 4.4], [RV14, Prop. 6.1]).

Many particular cases of the theorem have been considered before. For the particular case of affine Deligne-Lusztig varieties arising as the underlying reduced subscheme of a Rapoport-Zink moduli space of $p$-divisible groups with additional structure of PEL type, questions as in Theorem 1.2 have been considered by several people. A recent general theorem along these lines is shown by Mieda [Mie20]. Also, the (rare) cases where an affine Deligne-Lusztig variety is even of finite type have been classified, compare [Gör10, Prop. 4.13].

In the case where $\mathscr{G}$ is reductive over $O_{F}$ and $Z$ is a single $P$-double coset, a complete description of the set of $J_{b}(F)$-orbits of irreducible components of $X_{Z}(b)$ is known. The present work was motivated by our own results in this direction in [HV18]. Recently, complete descriptions were given by Zhou and Zhu [ZZ] and by Nie [Nie].

The main tool to prove Theorem 1.2 is to relate the claimed finiteness statements to finiteness properties of certain subsets of the extended Bruhat-Tits building of $G$, using previous work of Cornut and Nicole [CN16].

Acknowledgement. We are grateful to G. Prasad for pointing out some of his work on Bruhat-Tits theory to us. We thank the referee for his/her helpful comments.

2. Reduction to the parahoric case

As a first step, we reduce to the case that $\mathscr{G}$ is a parahoric group scheme. While most assertions in the following still hold true in the general setup, the assertion that $\mathscr{G}$ is parahoric will simplify the proofs and the notation.

By the fixed point theorem [Tit79, 2.3.1] the group $P\rtimes\langle\sigma\rangle$ has a fixed point $x$ in the extended Bruhat-Tits building of $G_{L}$. We refer to the subsequent section for the relation between the extended Bruhat-Tits building and the “classical” Bruhat-Tits building. By definition the stabiliser $P_{x}$ of $x$ is $\sigma$-stable and contains $P$. We denote by $\mathscr{G}_{x}$ the corresponding group scheme over $O_{F}$ in the sense of [Prab, 1.9] and [Praa, 2].

Lemma 2.1**.**

The fpqc quotient $L^{+}\mathscr{G}_{x}/L^{+}\mathscr{G}$ is representable by a finitely presented (resp. perfectly finitely presented) scheme.

Proof.

We denote $P_{{x},n}\coloneqq\ker(\mathscr{G}_{x}(O_{L})\to\mathscr{G}_{x}(O_{L}/{\epsilon}^{n}))$. Since the $P_{{x},n}$ form a neighbourhood basis of the unit element in $G(L)$ we have $P_{{x},n}\subset P$ for some $n$. Thus the positive loop group $L^{+}P$ contains the kernel of the reduction map into the truncated positive loop group $L^{+}P_{x}\to L^{+}_{n}P_{x}$. Indeed, we have just shown that this is true on geometric points and the kernel is an infinite dimensional affine space by Greenberg’s structure theorem [Gre63, p. 263], thus in particular reduced. Hence we get $L^{+}\mathscr{G}_{x}/L^{+}\mathscr{G}\cong L^{+}_{n}\mathscr{G}_{x}/L^{+}_{n}\mathscr{G}$. Since the latter is a quotient of linear algebraic groups over $k_{F}$, the claim follows. ∎

Since $LG\to{\mathscr{F}\ell}_{\mathscr{G}}$ is an $L^{+}\mathscr{G}$-torsor, we get that ${\mathscr{F}\ell}_{\mathscr{G}}$ is étale locally isomorphic to ${\mathscr{F}\ell}_{\mathscr{G}_{x}}\times L^{+}\mathscr{G}_{x}/L^{+}\mathscr{G}$. In particular, the canonical projection ${\mathscr{F}\ell}_{\mathscr{G}}\to{\mathscr{F}\ell}_{\mathscr{G}_{x}}$ is relatively representable and of finite type. Thus Theorem 1.2 holds true for $\mathscr{G}$ if and only if it is true for $\mathscr{G}_{x}$, as it is enough to prove the theorem after enlarging $Z$ so that becomes stable under $P_{x}$-$\sigma$-conjugation. Let $\mathscr{G}_{x}^{\circ}\subset\mathscr{G}_{x}$ be the parahoric group scheme associated to $x$. Repeating the argument above, we see that it suffices to prove Theorem 1.2 for $\mathscr{G}_{x}^{\circ}$ instead of $\mathscr{G}$.

Therefore we can (and will) assume from now on that $\mathscr{G}$ is a parahoric group scheme.

3. Some properties of Bruhat-Tits buildings

We consider the following group theoretical setup. Let $S_{0}\subset G$ be a maximal $L$-split torus defined over $F$, let $T_{0}$ be its centraliser and let $N_{0}$ be the normaliser of $T_{0}$ in $G$. Then $T_{0}$ is a torus because $G$ is quasi-split over $L$. Thus $W=N_{0}(L)/T_{0}(L)$ is the relative Weyl group of $G$ over $L$. We denote by $P_{T_{0}}$ the unique parahoric subgroup of $T_{0}$. The extended affine Weyl group is defined as

[TABLE]

where $X_{\ast}(T_{0})_{\operatorname{Gal}(\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu/L)}$ denotes the group of Galois convariants of $X_{\ast}(T_{0})$ over $L$. We may choose $S_{0}$ such that $P$ stabilises a facet in the apartment of $S_{0}$ and denote $\tilde{W}^{P}=(N_{0}(L)\cap P)/P_{T_{0}}\subset\tilde{W}$. By [PR08, Appendix, Prop. 9] the embedding $N_{0}\hookrightarrow G$ induces a bijection

[TABLE]

We call a subset $\tilde{X}\subset G(L)$ bounded if it is contained in a finite union of $P$-double cosets. The bounded subsets form a bornology on $G(L)$, which does not depend on the choice of $P$.

Let $\mathcal{B}^{e}(G,L)$ be the extended Bruhat-Tits building of $G$ over $L$, that is

[TABLE]

where $\mathcal{B}(G,L)$ is the “classical” Bruhat-Tits building of $G$ and $V_{0}(G,L)\coloneqq X_{\ast}(G^{\rm ab})_{\mathbb{R}}^{\operatorname{Gal}(\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu/L)}\cong X_{\ast}(Z(G))_{\mathbb{R}}^{\operatorname{Gal}(\mkern 1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu/L)}$ with $Z(G)$ denoting the center of $G$. The extended apartment $\mathcal{A}^{e}(S,G)\subset\mathcal{B}^{e}(G,L)$ of a maximal $L$-split torus $S$ is defined as $\mathcal{A}(S,G;L)\times V_{0}(G,L)$ where $\mathcal{A}(S,G;L)$ denotes the apartment of $S$. We recall from Landvogt [Lan00, § 1.3] that $\mathcal{B}^{e}(G,L)$ is a polysimplicial complex with a metric $d$ and a $G(L)\rtimes\langle\sigma\rangle$-action by isometries. Moreover, one can canonically identify $\mathcal{B}^{e}(G,F)$ with the set of $\sigma$-invariants $\mathcal{B}^{e}(G,L)^{\langle\sigma\rangle}$.

We consider the canonical map

[TABLE]

where $\operatorname{Isom}(\mathcal{B}^{e}(G,L))$ denotes the space of self-isometries of $\mathcal{B}^{e}(G,L)$. A set $M\subset\operatorname{Isom}(\mathcal{B}^{e}(G,L))$ is called bounded if for some (or equivalently every) non-empty bounded set $A\subset\mathcal{B}^{e}(G,L)$ the set $\{f(x)\mid f\in M,x\in A\}\subset\mathcal{B}^{e}(G,L)$ is bounded. We have the following statement about the compatibility of bornological structure.

Proposition 3.2** ([BT84, Prop. 4.2.19]).**

A subset $\tilde{X}\subset G(L)$ is bounded if and only if its image under $i$ is.

We consider the following maps between extended Bruhat-Tits buildings. Let $f\colon G\to G^{\prime}$ be a morphism of reductive $F$-groups. A $G(L)$-equivariant map $g\colon\mathcal{B}^{e}(G,L)\to\mathcal{B}^{e}(G^{\prime},L)$ is called toral if for every maximal $L$-split torus $S\subset G_{L}$ there exists a maximal $L$-split torus $S^{\prime}\subset G^{\prime}_{L}$ such that $f(S)\subset S^{\prime}$ and $g$ restricts to an $X_{\ast}(S)_{\mathbb{R}}$-translation equivariant map between the apartments of $S$ and $S^{\prime}$. In [Lan00], Landvogt proves that there always exists a $G(L)\rtimes\langle\sigma\rangle$-invariant toral map, which becomes an isometry after normalising the metric on $\mathcal{B}^{e}(G^{\prime},L)$. However, this map depends on an auxiliary choice. We give a precise formulation of the result in the form and context that we need later on. For this consider the fixed element $b\in G(L)$ and denote by $\nu_{b}\in X_{\ast}(G)_{\mathbb{Q}}$ the Newton point of $b$ (see [Kot85, § 4] for its precise definition). We fix an integer $s\gg 0$ such that $s\cdot\nu_{b}\in X_{\ast}(G)$. Denote by $M_{b}\subset G$ the Levi subgroup centralising $\nu_{b}$ (and thus $s\cdot\nu_{b}$). Then $J_{b}$ is the inner form of $M_{b}$ obtained by twisting the action of the Frobenius by $b$. We can thus use the following result to relate the buildings of $G$ and $J_{b}$. A similar result is also shown in [CN16].

Proposition 3.3** ([Lan00, Prop. 2.1.5],[Rou77, Lemme 5.3.2]).**

Let $f\colon M_{b}\hookrightarrow G$. Then there exists a toral $M_{b}(L)\rtimes\langle\sigma\rangle$-equivariant injective map

[TABLE]

Moreover, $f_{\ast}$ is injective and unique up to translation by an element of $V_{0}(G,L)^{\langle\sigma\rangle}$. In particular, its image is the same for every choice of $f_{\ast}$ and equal to $\mathcal{B}^{e}(G,L)^{(s\cdot\nu_{b})(O_{L}^{\times})}$. After a suitable normalisation of the metric on $\mathcal{B}^{e}(G,L)$, this map becomes an isometry.

Remark 3.4*.*

Since $J_{b,L}\cong M_{b,L}$, we obtain an identification of $\mathcal{B}^{e}(J_{b},L)$ with $\mathcal{B}^{e}(G,L)^{(s\cdot\nu_{b})(O_{L}^{\times})}$. However, since $J_{b}$ is an inner twist of a Levi subgroup of $G$, this identification will not respect the action of the Frobenius in general. In order to distinguish it from the action on $\mathcal{B}^{e}(G,L)$, we denote the Frobenius action on $\mathcal{B}^{e}(J_{b},L)$ (and $J_{b}(L)$) by $\sigma_{b}$. More explicitely, we have \sigma_{b}={b\sigma}\raise-2.15277pt\hbox{|}_{\mathcal{B}(J_{b},L)}\times{\sigma}\raise-2.15277pt\hbox{|}_{V_{0}(J_{b},L)}. Indeed, by [Lan96, Lemma 3.3.1], the Frobenius action on the “classical” Bruhat-Tits building $\mathcal{B}(J_{b},L)$ is uniquely determined by the equation $\sigma_{b}(j.x)=\sigma_{b}(j).\sigma_{b}(x)=(b\sigma(j)b^{-1}).\sigma_{b}(x)$ and thus has to be equal to $b\sigma$. It follows from the explicit description in [Lan96, (3.3.2)], that the Frobenius action on $V_{0}(J_{b},L)$ remains the same.

Now assume that we have an embedding of reductive groups $f\colon G\hookrightarrow G^{\prime}$. The following statement is the main result of [Lan00].

Proposition 3.5** ([Lan00, Thm. 2.2.1]).**

There exists a $G(L)\rtimes\langle\sigma\rangle$-invariant toral map $f_{\ast}\colon\mathcal{B}^{e}(G,L)\to\mathcal{B}^{e}(G^{\prime},L)$. Furthermore the metric on $\mathcal{B}^{e}(G,L)$ can be normalised in a way such that $f_{\ast}$ becomes isometrical.

To simplify the notation, we identify $G$ with its image in $G^{\prime}$. Now $b$, considered as element of $G^{\prime}$, induces a group $J^{\prime}_{b}$ which is an inner form of the centraliser of $\nu_{b}$ in $G^{\prime}$. Since $f_{\ast}$ preserves the fixed points of $\nu_{b}(O_{L}^{\times})$, we obtain a commutative diagram by Proposition 3.3 and Remark 3.4,

[TABLE]

Lemma 3.7**.**

The restriction {f_{\ast}}\raise-2.15277pt\hbox{|}_{\mathcal{B}^{e}(J_{b},L)} is $\sigma_{b}$-equivariant.

A related statement is [CN16], 3.5. For the convenience of the reader, we provide the details of the proof here.

Proof.

We denote by $\sigma^{\prime}_{b}$ the canonical Frobenius action on $\mathcal{B}^{e}(J^{\prime}_{b},L)$. Note that the action of $b\sigma$ and the actions of $\sigma_{b},\sigma_{b}^{\prime}$ differ by the translations $t_{b},t_{b}^{\prime}$ induced by the action of $b$ on $V_{0}(J_{b},L)$ and $V_{0}(J_{b}^{\prime},L)$ respectively. Since $f_{\ast}$ is $b\sigma$-equivariant, it suffices to show that $f_{\ast}\circ t_{b}=t_{b}^{\prime}\circ f_{\ast}$.

To prove this, consider the composition of $f_{\ast}$ with the canonical projection $\mathcal{B}^{e}(J_{b}^{\prime},L)\twoheadrightarrow V_{0}(J_{b}^{\prime},L)$. We claim that this map factors through $V_{0}(J_{b},L)$. This can be checked on extended apartments. Let $S\subset J_{b},S^{\prime}\subset J_{b}^{\prime}$ be maximal split tori over $L$ with $f(S)\subset S^{\prime}$ and $f_{\ast}(\mathcal{A}^{e}(S,J_{b};L))\subset\mathcal{A}^{e}(S^{\prime},J^{\prime}_{b};L)$. For the intersections with the derived groups of $G,G^{\prime}$ we have $S^{\rm der}\subset S^{\prime\rm der}$. Hence the composition $\mathcal{A}^{e}(S,J_{b};L)\hookrightarrow\mathcal{A}^{e}(S^{\prime},J^{\prime}_{b};L)\twoheadrightarrow V_{0}(J_{b}^{\prime};L)$ is $S^{\rm der}(L)$-invariant and thus factors through $\mathcal{A}^{e}(S,J_{b};L)/\mathcal{A}^{e}(S^{\rm der},J_{b}^{\rm der};L)=V_{0}(J_{b},L)$.

Thus we obtain a commutative diagram

{\mathcal{B}^{e}(J_{b},L)}$${\mathcal{B}^{e}(J_{b}^{\prime},L)}$${V_{0}(J_{b},L)}$${V_{0}(J_{b}^{\prime},L)}$$\scriptstyle{f_{\ast}}$$\scriptstyle{p}$$\scriptstyle{p^{\prime}}$$\scriptstyle{f_{\ast}^{\rm ab}}

Since $p,p^{\prime}$ and $f_{\ast}$ commute with the action of $b$, so does $f_{\ast}^{\rm ab}$. Thus $f_{\ast}^{\rm ab}\circ t_{b}=t_{b}^{\prime}\circ f_{\ast}^{\rm ab}$, proving $f_{\ast}\circ t_{b}=t_{b}^{\prime}\circ f_{\ast}$. ∎

4. Boundedness properties on the affine flag variety

We denote by $w_{G}\colon G(L)\to\pi_{1}(G)_{I}$ the Kottwitz homomorphism. For any subset $X\subset G(L)$ and $\omega\in\pi_{1}(G)_{I}$, we define $X^{\omega}\coloneqq X\cap w_{G}^{-1}(\{\omega\})$. We remark that by [PR08, Thm. 5.1] and [Zhu17, Prop. 1.21] the connected components of ${\mathscr{F}\ell}_{\mathscr{G}}$ are precisely the subsets of the form ${\mathscr{F}\ell}_{\mathscr{G}}^{\omega}$.

For further considerations, it will be useful to fix a presentation of ${\mathscr{F}\ell}_{\mathscr{G}}^{\rm red}$ as a limit of schemes. For any $w\in\tilde{W}^{P}\backslash\tilde{W}/\tilde{W}^{P}$ we denote by

[TABLE]

the Schubert cell and the Schubert variety associated with $w$, respectively. Here, $\leq$ denotes the Bruhat order on $\widetilde{W}$ induced by any fixed choice of an Iwahori subgroup of $P$. By [PR08, § 8] and [BS17, Thm. 9.3] each Schubert variety (resp. cell) is a closed (resp. locally-closed) quasi-compact subscheme of ${\mathscr{F}\ell}_{\mathscr{G}}$, which is of finite type in the case $\operatorname{char}F=p$ and perfectly of finite type in the case $\operatorname{char}F=0$. Note that by (3.1), we have that ${\mathscr{F}\ell}_{\mathscr{G}}=\bigcup S_{w}^{\circ}$ is a decomposition into locally closed subsets, hence we can write ${\mathscr{F}\ell}_{\mathscr{G}}^{\rm red}=\varinjlim S_{w}$.

We equip ${\mathscr{F}\ell}_{\mathscr{G}}(k)$ with the bornology induced by the canonical projection $G(L)\twoheadrightarrow G(L)/P={\mathscr{F}\ell}_{\mathscr{G}}(k)$, that is a subset $X\subset{\mathscr{F}\ell}_{\mathscr{G}}(k)$ is bounded, if it is contained in a finite union of Schubert varieties. We obtain the following geometric characterisation of bounded subsets.

Lemma 4.1**.**

A subset $X\subset{\mathscr{F}\ell}_{\mathscr{G}}(k)$ is bounded if and only if it is relatively quasi-compact (i.e. contained in a quasi-compact subset). In this case $X$ is even quasi-compact itself.

Proof.

Since the $S_{w}$ are quasi-compact, any bounded subset of ${\mathscr{F}\ell}_{\mathscr{G}}$ is relatively quasi-compact. The $S_{w}$ are Noetherian, thus their subsets are quasi-compact themselves.

On the other hand, assume that $X$ is not bounded. We prove that $X$ is not quasi-compact by constructing an infinite discrete closed subset $Y\subset X$. By definition, the set $T\coloneqq\{w\in\widetilde{W}\mid X\cap S_{w}^{\circ}\not=\emptyset\}$ is infinite. For each $w\in T$, choose an element $x_{w}\in X\cap S_{w}^{\circ}$. Then $Y\coloneqq\{x_{w}\mid w\in T\}$ is infinite and discrete. Its intersection with every $S_{w}$ for $w\in\widetilde{W}$ is closed, hence $Y$ is closed. ∎

Lemma 4.2**.**

Let $X\subset{\mathscr{F}\ell}_{\mathscr{G}}$ be a locally closed reduced sub-ind-scheme. Then $X$ is a scheme if and only if every point of $X(k)$ has an open neighbourhood which is bounded as subset of ${\mathscr{F}\ell}_{\mathscr{G}}(k)$. In this case $X$ is locally of finite type if $\operatorname{char}F=p$, respectively locally of perfectly finite type if $\operatorname{char}F=0$.

Proof.

The “only if” direction follows from the previous lemma because every point of a scheme has a quasi-compact open neighbourhood.

To prove the “if” direction, we may assume that $X$ is bounded, since its representability is a Zariski-local property. Then the embedding $X(k)\hookrightarrow{\mathscr{F}\ell}_{\mathscr{G}}$ factors through some finite union of Schubert varieties by the previous lemma, in particular $X(k)$ is a locally closed subvariety of this union. Since the Schubert varieties are (perfectly) of finite type, so is $X$. ∎

Remark 4.3*.*

The analogous assertions of Lemmas 4.1 and 4.2 in $LG(k)$, the loop group of $G$, also hold true (with the exception of the last statement of Lemma 4.2). Indeed, since a set $X\subset G(L)$ is bounded if and only if $X\cdot P$ is bounded, it suffices to prove the assertion in the case that $X$ is right $P$-invariant. Then the claim follows from the above lemmas since $LG\to{\mathscr{F}\ell}_{\mathscr{G}}$ is an $L^{+}\mathscr{G}$-torsor and thus relatively representable and quasi-compact.

5. Affine Deligne Lusztig varieties

We now prove that the first part of Theorem 1.2 implies the second. By Lemma 4.2 together with the first part of the theorem, its second assertion is equivalent to the following proposition, which we prove below.

Proposition 5.1**.**

Let $Z$ a bounded subset of $G(L)$ and denote

[TABLE]

Then there exists a bounded subset $\tilde{X}_{0}\subset\tilde{X}_{Z}(b)$ such that $\tilde{X}_{Z}(b)=J_{b}(F)\cdot\tilde{X}_{0}$.

For the proof of the proposition we need some preparation.

Lemma 5.2**.**

The $\sigma$-conjugacy class of $b\in G(L)$ has a decent representative for which $\mathcal{B}^{e}(J_{b},F)\cap\mathcal{B}^{e}(G,F)\neq\emptyset$ (viewed as subspaces of $\mathcal{B}^{e}(G,L)$).

Here, an element $b\in G(L)$ is called decent if there is a natural number $s$ with $(b\sigma)^{s}=s\nu(\epsilon)\sigma^{s}$.

Proof.

In Remark 3.4 we identified the extended Bruhat-Tits building $\mathcal{B}^{e}(J_{b},L)$ with $\mathcal{B}^{e}(M_{b},L)$. We fix a maximal $L$-split torus $S\subset M_{b}$, denote by $T$ its centraliser and by $\tilde{W}_{M_{b}}$ the associated extended affine Weyl group of $M_{b}$. Since any reductive group over $F$ is residually quasi-split by [BT87, Thm. 4.1], there exists a $\sigma$-stable alcove $\mathfrak{a}$ in $\mathcal{A}(S,M_{b},L)$. The Kottwitz homomorphism maps the stabiliser $\Omega\subset\tilde{W}_{M_{b}}$ of $\mathfrak{a}$ isomorphically onto $\pi_{1}(G)_{I}$. Since any basic $\sigma$-conjugacy class is uniquely determined by its Kottwitz point, we may assume that $b$ (after replacing it by a $M_{b}(L)$-$\sigma$-conjugate if necessary) is a representative in $M_{b}(L)$ of an element of $\Omega$. By [Kim19, Lemma 2.2.10] we may assume this representative to be decent. It now follows from the explicit description of $\sigma_{b}$ in Remark 3.4 that we may take $p_{0}\coloneqq(p_{b},p_{v})$ where $p_{b}\in\mathcal{B}(M_{b},L)$ is the barycenter of $\mathfrak{a}$ and $p_{v}\in V_{0}(M_{b},L)$ is any point fixed by $\sigma$. Then $p_{0}\in\mathcal{B}^{e}(J_{b},F)\cap\mathcal{B}^{e}(G,F).$ ∎

Thus after replacing $b$ by a $\sigma$-conjugate if necessary, we fix $p_{0}\in\mathcal{B}^{e}(J_{b},F)\cap\mathcal{B}^{e}(G,F)$. In order to relate the bornologies on $G(L)$ and on $\mathcal{B}^{e}(G,L)$ directly, we consider the map

[TABLE]

By the choice of $p_{0}$, the map $\iota$ is $G(L)\rtimes\langle\sigma\rangle$-equivariant and the restriction to $J_{b}(L)$ is moreover $\sigma_{b}$-equivariant, cf. Remark 3.4. By Proposition 3.2, for any $C^{\prime}>0$ the set $Z_{C^{\prime}}\coloneqq\{g\in G(L)\mid d(p_{0},\iota(g))<C^{\prime}\}$ is a bounded set and for any bounded $Z\subset G(L)$ the constant $c_{Z}\coloneqq\sup\{d(p_{0},\iota(y))\mid y\in Z\}$ is finite.

The following lemma translates the results of [CN16] into our terms.

Lemma 5.3**.**

Let $G$ be a reductive group over $F$ and $b\in G(L)$.

• (a)

For any $c>0$ there exists a $C>0$ such that if $x\in\mathcal{B}^{e}(G,L)$ satisfies $d(x,b\sigma(x))<c$ then there exists $x_{0}\in\mathcal{B}^{e}(J_{b},F)$ with $d(x,x_{0})<C$.

• (b)

For any $c>0$ there exists a $C>0$ such that if $x\in\iota(G(L))$ satisfies $d(x,b\sigma(x))<c$ then there exists $x_{0}\in\iota(J_{b}(F))$ with $d(x,x_{0})<C$.

Proof.

Assertion (a) is proven in [CN16] by an elegant geometrical argument. By Theorem 3.3 of loc. cit. $f_{\ast}$ identifies $\mathcal{B}^{e}(J_{b},F)$ with the set

[TABLE]

Thus the statement (a) claims that if $d(x,b\sigma(x))$ is bounded, so is the distance to ${\rm Min}(b\sigma)$. This (together with an upper bound for $C$) is proven in [CN16, Prop. 8].

To show that (a) implies (b), we have to show that the distance of a point $x\in\mathcal{B}^{e}(G,L)$ to $\iota(G(L))$ is bounded above, or equivalently that there exists a bounded subset $M\subset\mathcal{B}^{e}(G,L)$ such that $G(L)\cdot M=\mathcal{B}^{e}(G,L)$ as well as the analogous assertion for $J_{b}(F)$. For this, we fix an isomorphism $X_{\ast}(Z)^{I}\cong\mathbb{Z}^{r}$, which yields an identification $V_{0}(G,L)=\mathbb{R}^{r}$. Then we may choose $M=\mathfrak{a}\times[0,1]^{r}$, where $\mathfrak{a}$ is any alcove of the usual Bruhat-Tits building $\mathcal{B}(G,L)$. ∎

Proof of Proposition 5.1.

Let $Z\subset G(L)$ be bounded. We fix $g\in\tilde{X}_{Z}(b)$ and denote $x\coloneqq\iota(g)$. Then

[TABLE]

By Lemma 5.3(b), there exist a $C_{Z}>0$ depending only on $Z$ and a $j\in J_{b}(F)$ such that

[TABLE]

i.e. $j^{-1}\cdot g\in Z_{C_{Z}}$. Hence $\tilde{X}_{Z}(b)=J_{b}(F)\cdot(\tilde{X}_{Z}(b)\cap Z_{C_{Z}})$. ∎

It remains to prove the first part of Theorem 1.2. By Lemma 4.2 it is equivalent to the following proposition.

Proposition 5.4**.**

Every $x_{0}\in X_{Z}(b)(k)$ has a bounded open neighbourhood.

Proof.

The proof follows by an analogous argument as the last part of [HV11, Thm. 6.3]. Since the situation simplifies a lot by considering only the reduced structure, and since in loc. cit. only split groups, hyperspecial $P$, and certain $Z$ are considered, we give the complete proof for the reader’s convenience.

Let $\omega=w_{G}(x_{0})$ and let again $X_{Z}(b)^{\omega}=X_{Z}(b)\cap w_{G}^{-1}(\omega)$. Since $X_{Z}(b)^{\omega}\subset X_{Z}(b)$ is open and closed, it suffices to prove the claim for $X_{Z}(b)^{\omega}$. We can define a $G(L)$- and $\sigma$-invariant semi-metric $d\colon G(L)^{\omega}\to\mathbb{N}\cup\{0\}$ by

[TABLE]

where $\rho^{\vee}$ denotes the half-sum of the positive coroots and $w\in\tilde{W}^{P}\backslash\tilde{W}/\tilde{W}^{P}$. Obviously this semi-metric descends to ${\mathscr{F}\ell}_{G}^{\omega}$. Then a subset is bounded with respect to the bornology defined before Lemma 4.1 if and only if it is bounded with respect to $d$.

We choose $b$ as in Lemma 5.2. Let $s\in\mathbb{N}$ be as in the decency equation, i.e. $(b\sigma)^{s}=(s\cdot\nu)({\epsilon})$. Enlarging $s$ and $Z$ if necessary, we assume that $\omega$ and $Z$ are both $\sigma^{s}$-invariant. Then $X_{Z}(b)^{\omega}$ is $\sigma^{s}$-stable and thus is defined over the extension $k_{s}$ of degree $s$ of $k_{F}$. The closed point $x_{0}$ defined over some finite extension of $k_{F}$. By enlarging $k_{s}$ further if necessary, we assume that $x_{0}$ is a $k_{s}$-rational point. We denote by $\mathcal{M}$ the model of $X_{Z}(b)^{\omega}$ over $k_{s}$ and for every $n\in\mathbb{N}$ we define the closed sub-ind-scheme

[TABLE]

Note that $\mathcal{M}_{n}$ is actually a (perfectly) finite type scheme by Lemma 4.2 and moreover defined over $k_{s}$ since $d$ is $\sigma$-invariant. Also note that $\mathcal{M}=\varinjlim\mathcal{M}_{n}.$

The decency of $b$ implies that $J_{b}(F)\subset G(F_{s})$ where $F_{s}$ is the unramified extension of $F$ of degree $s$. Thus the $J_{b}(F)^{0}$-action stabilises $\mathcal{M}(k_{s})$. Together with Proposition 5.1 (which we proved independently of the first assertion of Theorem 1.2) we obtain that there exists an $N_{0}\in\mathbb{N}$ such that for every $x\in\mathcal{M}(k)$ there exists a $y_{0}\in\mathcal{M}(k_{s})$ with $d(x,y_{0})\leq N_{0}$. For every $y_{0}\in\mathcal{M}(k_{s})$ define the closed subscheme $\mathcal{M}_{n}(y_{0})\subset\mathcal{M}_{n}$ by

[TABLE]

Now consider the open subset of $\mathcal{M}_{n}$

[TABLE]

The union on the right hand side is indeed finite (and hence closed): By the triangular inequality $\mathcal{M}_{n}(y)$ is empty unless $y\in\mathcal{M}_{N_{0}+n}(k_{s})$; the latter set is finite since $\mathcal{M}_{N_{0}+n}$ is (perfectly) of finite type. We claim that the chain $U_{1}\subseteq U_{2}\subseteq\dotsm$ stabilises at $U_{2N_{0}}$ at the latest. To prove this, let $x\in U_{n}(k)$ for some $n$. We choose a rational point $y_{0}\in\mathcal{M}(k_{s})$ with $d(x,y_{0})\leq N_{0}$. By definition of $U_{n}$, we must have $d(x_{0},y_{0})\leq N_{0}$. Thus $d(x,x_{0})\leq d(x,y_{0})+d(y_{0},x_{0})\leq 2N_{0}$, i.e. $x\in U_{2N_{0}}$.

Since $\mathcal{M}=\varinjlim\mathcal{M}_{n}$, the subset $U_{2N_{0}}=\varinjlim U_{n}$ is open in $\mathcal{M}$. It is moreover bounded and contains $x_{0}$. It is thus a bounded open neighbourhood of $x_{0}$. ∎