1 Introduction

Let G be a reductive group over the local field F, whose completion of the maximal unramified extension we denote by \(\breve{F}\). In the context of reduction of Shimura varieties, one would choose F to be a finite extension of p-adic rationals, whereas in the context of moduli spaces of shtukas, F would be the field of formal Laurent series over a finite field. In any case, the Galois group \({{\,\textrm{Gal}\,}}(\breve{F}/F)\) is (topologically) generated by the Frobenius \(\sigma \). We moreover pick a \(\sigma \)-stable Iwahori subgroup \(I\subseteq G(\breve{F})\). Then the affine Deligne–Lusztig variety associated to two elements \(x,b\in G(\breve{F})\) is defined as

$$\begin{aligned} X_x(b) = \{g\in G(\breve{F})/I\mid g^{-1}b\sigma (g)\in IxI\}. \end{aligned}$$

Evidently, the isomorphism type of \(X_x(b)\) only depends on the \(\sigma \)-conjugacy class

$$\begin{aligned} {[}b] = \{g^{-1}b\sigma (g)\mid g\in G(\breve{F})\}\subseteq G(\breve{F}) \end{aligned}$$

and the Iwahori double coset IxI. The latter Iwahori double cosets are typically indexed using the extended affine Weyl group \(\widetilde{W}\), so that each coset is given by IxI for a uniquely determined element \(x\in \widetilde{W}\). The set B(G) of \(\sigma \)-conjugacy classes has an important parametrization due to Kottwitz [9, 10], characterizing each \([b]\in B(G)\) by its Newton point \(\nu (b)\) and Kottwitz point \(\kappa (b)\).

Following [3, Section 2], we assume without loss of generality that the group G is quasi-split over F. We choose a maximal torus T whose unique parahoric subgroup \(T_0(\breve{F})\) is contained in I and a \(\sigma \)-stable Borel subgroup \(T\subset B\) such that, in the corresponding appartment of the Bruhat–Tits building of \(G_{\breve{F}}\), the alcove fixed by I is opposite to the dominant cone defined by B. With this notation, the Kottwitz point \(\kappa (b)\) for \(b\in G(\breve{F})\) lies in \((X_*(T)/\mathbb Z\Phi ^\vee )_{\Gamma }\), where \(\Phi ^\vee \) is the set of coroots and \(\Gamma \) the absolute Galois group of F. The dominant Newton point \(\nu (b)\) is an element of \(X_*(T)_{\Gamma _0}\otimes \mathbb Q\), where \(\Gamma _0\subset \Gamma \) is the absolute Galois group of \(\breve{F}\). We identify the extended affine Weyl group \(\widetilde{W}\) as the semidirect product of the finite Weyl group \(W = N_G(T)/T\) with \(X_*(T)_{\Gamma _0}\).

Geometric properties of the affine Deligne–Lusztig variety \(X_x(b)\) are closely related to those of the corresponding Newton stratum \([b]\cap IxI\subset G(\breve{F})\). Obviously, one is empty if and only if the other is empty. Further properties can be related following [12, Section 3.1].

2 Reduction to Levi subgroups

It is an important question to study which of these affine Deligne–Lusztig varieties are non-empty, i.e. to determine the set

$$\begin{aligned} B(G)_x = \{[b]\in B(G)\mid [b]\cap IxI\ne \emptyset \}. \end{aligned}$$

An important breakthrough result of Görtz–He–Nie [3] is a characterization of all elements \(x\in \widetilde{W}\) where \(B(G)_x\) contains the basic \(\sigma \)-conjugacy class. For this purpose, they introduce the notion of a \((J,w,\sigma )\)-alcove, generalizing the previous notion of a P-alcove known for split groups. Write \(\Delta \subseteq \Phi \) for the set of simple roots.

Definition 1

Let \(x\in \widetilde{W}, w\in W\) and \(J\subseteq \Delta \) such that \(J=\sigma (J)\). Then we say that x is a \((J,w,\sigma )\)-alcove element if the following conditions are both satisfied:

  1. (a)

    The element \(\tilde{x} = w^{-1} x\sigma (w)\) lies in the extended affine Weyl group \(\widetilde{W}_M\) of the standard Levi subgroup \(M=M_J\supseteq T\) defined by J.

  2. (b)

    For all positive roots \(\alpha \in \Phi ^+\) that are not in the root system \(\Phi _J\) generated by J, the corresponding root subgroup \(U_\alpha \subseteq G(\breve{F})\) satisfies

Observe that x is a \((J,w,\sigma )\)-alcove element if it is a \((J,w',\sigma )\)-alcove element for any \(w'\in wW_J\), where \(W_J\) is the subgroup of W generated by the simple reflections coming from J (or equivalently the finite Weyl group of \(M_J\)). Following Viehmann [13, Section 4], we say that x is a normalized \((J,w,\sigma )\)-alcove element if w has minimal length in its coset \(wW_J\). This extra assumption is independent of x, i.e. any \((J,w,\sigma )\)-alcove element will also be a normalized \((J, w', \sigma )\)-alcove element for a more refined choice of \(w'\).

Assume that x is a \((J,w,\sigma )\)-alcove element and write \(\tilde{x} = w^{-1} x\sigma (w)\in \widetilde{W}_M\). If \([b]\in B(G)_x\), it is a result of Görtz–He–Nie [3, Theorem 3.3.1] that each element in Newton stratum \(IxI\cap [b]\) is of the form \(i^{-1} w m \sigma (w^{-1} i)\) for some \(i\in I\) and

$$\begin{aligned} m\in \bigcup _{[b']} \Bigl ((I\cap M(\breve{F}))\tilde{x}(I\cap M(\breve{F}))\cap [b']\Bigr )\subseteq M(\breve{F}), \end{aligned}$$

with the union taken over all \([b']\in B(M)\) contained in [b]. Our main result states that, whenever x is a normalized \((J,w,\sigma )\)-alcove element, this union is spurious.

Theorem 2

Let x be a normalized \((J,w,\sigma )\)-alcove element and \(\tilde{x} {=} w^{-1}x\sigma (w){\in } \widetilde{W}_M\). Then we get a bijective map

$$\begin{aligned} B(M)_{\tilde{x}}\rightarrow B(G)_x, \end{aligned}$$

sending a \(\sigma \)-conjugacy class \([b]_M\in B(M)_{\tilde{x}}\) to the unique \(\sigma \)-conjugacy class \([b]_G\in B(G)\) with \([b]_M\subseteq [b]_G\). In this case, the dominant Newton points \(\nu _M(b)\) and \(\nu _G(b)\) agree as elements of \(X_*(T)_{\Gamma _0}\otimes \mathbb Q\).

Unfortunately, the relationship discussed here has been a source of confusion in the past. While surjectivity of the map \(B(M)_{\tilde{x}}\rightarrow B(G)_x\) follows from [3, Theorem 3.3.1], it should be noted that the natural map \(B(M)\rightarrow B(G)\) is neither injective nor surjective. In particular, for \([b]_G\in B(G)_x\), the intersection \([b]_G\cap M(\breve{F})\) may consist of more than one \(\sigma \)-conjugacy class of M. We would like to make the following remarks regarding previous literature:

  • The reader will notice that the published version of [3, Proposition 3.5.1] does not hold true in the claimed generality. It is actually only proved for basic \(\sigma \)-conjugacy classes, as explained in the erratumFootnote 1.

  • In [13, Example 5.4], Viehmann claims to provide a counterexample to the injectivity statement in Theorem 2. However, the situation considered there does not give rise to a normalized \((J,w,\sigma )\)-alcove. Her example does explain how crucial this assumption is for our present work.

We remark that for split G, the bijection of Theorem 2 is known from [1, Corollary 2.1.3].

We can use the correspondence in Theorem 2 to study affine Deligne–Lusztig varieties. In the setting of Theorem 2, [3, Theorem 3.3.1] gives us a closed immersion of the affine Deligne–Lusztig variety \(X_{\tilde{x}}(b)\) (for M) into \(X_x(b)\):

$$\begin{aligned} X_{\tilde{x}}(b)\rightarrow X_x(b),\qquad g(I\cap M)\mapsto gw^{-1} I. \end{aligned}$$

This map is usually not surjective. However, we can make it surjective as follows: For \(b\in G(\breve{F})\), denote its \(\sigma \)-centralizer by

$$\begin{aligned} \mathrm J_b(F) := \{g\in G(\breve{F})\mid g^{-1} b\sigma (g) = b\}. \end{aligned}$$

Observe that \(\mathrm J_b(F)\) acts on \(X_x(b)\) by left multiplication.

Corollary 3

The affine Deligne–Lusztig variety \(X_x(b)\) is a disjoint union of closed subsets

$$\begin{aligned} X_x(b) = \bigsqcup _{j\in \mathrm J_b(F) / (\mathrm J_b(F)\cap M(\breve{F}))} (j M(\breve{F})w^{-1} I/I\cap X_x(b)). \end{aligned}$$

For each \(j\in \mathrm J_b(F)\), we get an isomorphism

$$\begin{aligned} X_{\tilde{x}}(b)\rightarrow j M(\breve{F})w^{-1} I/I\cap X_x(b),\qquad g(I\cap M)\mapsto jgw^{-1} I. \end{aligned}$$

Proof

Similar to the proof of [1, Theorem 2.1.4], using Theorem 2 instead of [1, Corollary 2.1.3]. \(\square \)

The geometric correspondence between the affine Deligne–Lusztig varieties \(X_{\tilde{x}}(b)\) and \(X_x(b)\) is mirrored by a corresponding representation-theoretic result of He–Nie [7, Theorem C], comparing class polynomials of x and \(\tilde{x}\). These are certain structure constants describing the cocenter of the Iwahori–Hecke algebra of \(\widetilde{W}\) resp. \(\widetilde{W}_M\). If one knows all class polynomials for a given element \(x\in \widetilde{W}\), one can use these to determine many geometric properties the affine Deligne–Lusztig varieties \(X_x(b)\) for \([b]\in B(G)\), cf. [5, Theorem 6.1]. These properties include dimension as well as the number of top dimensional irreducible components up to the action of the \(\sigma \)-centralizer of \(b\in G(L)\). Moreover, in a certain sense, the number of rational points of the Newton stratum \(IxI\cap [b]\) can be expressed using these class polynomials [8, Proposition 3.7]. In this sense, it already follows from [7] that these numerical invariants agree for \(X_{\tilde{x}}(b)\) and \(X_x(b)\).

Proof of Theorem 2

We follow the reduction method of Deligne–Lusztig, adapted to the affine case by Görtz–He [2]. I.e. we do an induction on \(\ell (x)\).

If x is of minimal length in its \(\sigma \)-conjugacy class, then \(B(G)_x\) contains only one element, being the \(\sigma \)-conjugacy class defined by x. Moreover, He–Nie [7, Proposition 4.5] prove that in this case \(\tilde{x}\) is of minimal length in its \(\sigma \)-conjugacy class in \(\widetilde{W}_M\). Hence we only have to show that \(\nu _M(\tilde{x})\) agrees with \(\nu _G(x)\).

Following the definition of Newton points, we consider \(\sigma \)-twisted powers

$$\begin{aligned} x^{\sigma ,n} =x \sigma (x)\cdots \sigma ^{n-1}(x)\in \widetilde{W}. \end{aligned}$$

Observe that each \(x^{\sigma ,n}\) is a \((J,w,\sigma ^n)\)-alcove element. Let n be sufficiently large such that \(x^{\sigma ,n}\) is a pure translation element, i.e. equal to the image of some \(\mu \in X_*(T)_{\Gamma _0}\) in \(\widetilde{W}\), and such that \(\sigma ^n\) is the identity map on \(\widetilde{W}\). Then the Newton point \(\nu _G(x)\) is the unique dominant element in the W-orbit of \(\mu /n\). Similarly, the Newton point \(\nu _M(\tilde{x})\) is the unique dominant (with respect to \(B\cap M\)) element in the \(W_J\)-orbit of \(w^{-1}\mu /n\).

The fact that \(x^{\sigma ,n}\) is a (J, w, 1)-alcove element implies that \(\langle w^{-1}\mu ,\alpha \rangle \ge 0\) for all \(\alpha \in \Phi ^+{\setminus }\Phi _J\). Hence \(\nu _M(\tilde{x})\) is already dominant with respect to B, and thus \(\nu _G(x) = \nu _M(\tilde{x})\). This finishes the proof in case x has minimal length in its \(\sigma \)-conjugacy class.

If x is not of minimal length in its \(\sigma \)-conjugacy class, we can use [6, Theorem A] to obtain a sequence

$$\begin{aligned} x = x_1\xrightarrow {s_1}\cdots \xrightarrow {s_n} x_{n+1}, \end{aligned}$$

for simple affine reflections \(s_i\in \widetilde{W}\) and elements \(x_{i+1} = s_i x_i \sigma (s_i)\) such that \(\ell (x_1)=\cdots =\ell (x_{n}) >\ell (x_{n+1})\). From [7, Lemma 7.1], we find elements \(w_1, \dotsc , w_n\in W\) such that each \(x_i\) is a normalized \((J,w_i,\sigma )\)-alcove element. Denote the corresponding elements by \(\tilde{x}_i = w_i^{-1} x_i\sigma (w_i)\in \widetilde{W}_M\), so that the proof of [7, Corollary 4.4] shows

$$\begin{aligned} \ell _{\widetilde{W}_M}(\tilde{x}_1) = \cdots =\ell _{\widetilde{W}_M}(\tilde{x}_{n})>\ell _{\widetilde{W}_M}(\tilde{x}_{n+1}). \end{aligned}$$

Said proof moreover reveals that each \(\tilde{x}_{i+1}\) is conjugate to \(\tilde{x}_i\) either by a simple affine reflection in \(\widetilde{W}_M\) or a length zero element in \(\widetilde{W}_M\).

The Deligne–Lusztig reduction method of Görtz–He [2] yields

$$\begin{aligned} B(G)_x =&\cdots = B(G)_{x_{n}} = B(G)_{x_{n+1}}\cup B(G)_{s_n x_{n}},\\ B(M)_{\tilde{x}}=&\cdots = B(M)_{\tilde{x}_{n}}. \end{aligned}$$

We moreover know from the aforementioned article of He–Nie that \(\tilde{x}_{n+1} = \tilde{s} \tilde{x}_{n}\sigma (\tilde{s})\) for some simple affine reflection \(\tilde{s} = w_n s_n w_n^{-1}\in \widetilde{W}_M\) of M. Hence

$$\begin{aligned} B(M)_{\tilde{x}_{n}} = B(M)_{\tilde{x}_{n+1}}\cup B(M)_{\tilde{s}\tilde{x}_{n}}. \end{aligned}$$

By induction, we get bijective and Newton-point preserving maps

$$\begin{aligned} B(M)_{\tilde{x}_{n+1}}\rightarrow B(G)_{x_{n+1}},\qquad B(M)_{\tilde{s} \tilde{x}_{n}}\rightarrow B(G)_{s_nx_{n}}. \end{aligned}$$

We conclude that the map \(B(M)_{\tilde{x}}\rightarrow B(G)_x\) is well-defined, surjective and Newton-point preserving. If \([b_1]_M, [b_2]_M\in B(M)_{\tilde{x}}\) have the same image \([b]_G\in B(G)_x\) under this map, then \(\nu _M(b_1) = \nu _G(b) = \nu _M(b_2)\) and \(\kappa _M(b_1) = \kappa _M(\tilde{x}) = \kappa _M(b_2)\), hence \([b_1]_M = [b_2]_M\). This finishes the induction and the proof. \(\square \)

Let us note the following consequence of Theorem 2.

Corollary 4

If x is a \((J,w,\sigma )\)-alcove element and \([b_1], [b_2]\in B(G)_x\), then

$$\begin{aligned}&\qquad \qquad \qquad \qquad \qquad \nu _G(b_1)\equiv \nu _G(b_2)\pmod {\Phi _J^\vee }.\qquad \qquad \qquad \qquad \qquad \qquad \qquad \end{aligned}$$

\(\square \)

3 Lim’s conjecture on the nonemptiness of the basic locus

As an application of Corollary 4, we prove a conjecture of Dong-Gyu Lim [11, Conjecture 1], yielding an alternative criterion to the one from [3] for the non-emptiness of the basic Newton stratum in IxI.

Definition 5

Let \(x\in \widetilde{W}\) be written as \(x = \omega s_1\cdots s_{\ell (x)}\) for a length zero element \(\omega \) and simple affine reflections \(s_1,\dotsc ,s_{\ell (x)}\in \widetilde{W}\). We define the \(\sigma \)-support of x to be the smallest subset \(J\subseteq \widetilde{W}\) containing \(s_1,\dotsc ,s_{\ell (x)}\) and being closed under the action of the composite automorphism \(\sigma \circ \omega \). Denote it by \({{\,\textrm{supp}\,}}_\sigma (x)\). We say that x is spherically \(\sigma \)-supported if the subgroup of \(\widetilde{W}\) generated by \({{\,\textrm{supp}\,}}_\sigma (x)\) is finite.

It follows from [4, Proposition 5.6] that x has spherical \(\sigma \)-support if and only if \(B(G)_x = \{[b]\}\) for a basic \(\sigma \)-conjugacy class [b].

Proposition 6

Assume that the Dynkin diagram of \(\Phi \) is \(\sigma \)-connected, i.e. that the Frobenius \(\sigma \) acts transitively on the set of irreducible components of the root system \(\Phi \).

Let \(x\in \widetilde{W}\), and denote by \([b]\in B(G)\) the unique basic \(\sigma \)-conjugacy class with \(\kappa (b) = \kappa (x)\). Then \(X_x(b)=\emptyset \) if and only if the following two conditions are both satisfied:

  1. (a)

    The element x does not have spherical \(\sigma \)-support, i.e. \(B(G)_x\) contains a non-basic \(\sigma \)-conjugacy class.

  2. (b)

    There exists \(J\subsetneq \Delta \) and \(w\in W\) such that x is a \((J,w,\sigma )\)-alcove element.

Proof

If x has spherical \(\sigma \)-support, we get \(IxI\subseteq [b]\), so that indeed \(X_x(b)\ne \emptyset \). In the case that x is not a \((J,w,\sigma )\)-alcove element for any \(J\subsetneq \Delta \), we easily obtain \(X_x(b)\ne \emptyset \) by [3, Theorem A].

Assume now conversely that (a) and (b) both hold true, so we have to show \(X_x(b)=\emptyset \). Let (J, w) be as in (b) such that moreover x is a normalized \((J,w,\sigma )\)-alcove element. Let \([b_x]\in B(G)_x\) denote the generic \(\sigma \)-conjugacy class.

Assume that \([b]\in B(G)_x\). From (b) together with Corollary 4, we see

$$\begin{aligned}\nu (b)\equiv \nu (b_x)\pmod {\Phi _J^\vee }.\end{aligned}$$

In particular

$$\begin{aligned} \langle \nu _G(b_x)-\nu _G(b),2\rho -2\rho _J\rangle =0. \end{aligned}$$

Since \(\nu _G(b_x)\) is dominant and b is basic, we conclude \(\langle \nu _G(b_x),\alpha \rangle =0\) for all \(\alpha \in \Phi ^+{\setminus }\Phi _J\). Thus \(\Phi = \Phi _J\cup \Phi _{J'}\) where \(J'\subseteq \Delta \) is the stabilizer of \(\nu (b_x)\).

Each irreducible component of \(\Phi \) contains a unique longest root, and by \(\sigma \)-irreducibility, these longest roots form a single \(\sigma \)-orbit. Since \(\Phi _J\) and \(\Phi _{J'}\) are two \(\sigma \)-stable subsets of \(\Phi \) covering the entire root system, one of these two sets must contain all longest roots. This is only possible if \(J=\Delta \) or \(J'=\Delta \).

We assumed \(J\ne \Delta \) in (b), so we conclude that \(\nu _G(b_x)\) must be central. Thus \([b_x]\) is basic itself. This contradicts (a). \(\square \)