Dimensions of Affine Deligne-Lusztig Varieties

in Affine Flag Varieties

Ulrich Görtz^{1}, Xuhua He^{2}

Received: October 7, 2010

Communicated by Peter Schneider

Abstract. Affine Deligne-Lusztig varieties are analogs of Deligne-Lusztig varieties in the context of an affine root system. We prove a conjecture stated in the paper [5] by Haines, Kottwitz, Reuman, and the first named author, about the question which affine Deligne-Lusztig varieties (for a split group and a basic σ-conjugacy class) in the Iwahori case are non-empty. If the underlying algebraic group is a classical group and the chosen basic σ-conjugacy class is the class of b = 1, we also prove the dimension formula predicted in op. cit. in almost all cases.

2010 Mathematics Subject Classification: 20F55, 20G25 Keywords and Phrases: Affine Deligne-Lusztig varieties

1 Introduction

1.1

Affine Deligne-Lusztig varieties, which are analogs of usual Deligne-Lusztig varieties [3] in the context of an affine root system, have been studied by several people, mainly because they encode interesting information about the reduction of Shimura varieties and specifically about the relation between the “Newton stratification” and the “Kottwitz-Rapoport stratification”. Their definition is purely group-theoretical. To recall it, we fix a split connected reductive group over the finite field 𝔽_{q} with q elements. Let 𝕜 be an algebraic closure of 𝔽_{q}, let L = 𝕜((ϵ)) be the field of formal Laurent series over 𝕜, and let σ be the automorphism of L defined by σ(∑ a_{n}ϵ^{n}) = ∑ a_{n}^{q}ϵ^{n}. We also denote the induced automorphism on the loop group G(L) by σ. Let T ⊂ G be a split maximal torus, and denote by W the corresponding Weyl group. Furthermore, let I ⊂ G(𝕜[[ϵ]]) be an Iwahori subgroup containing T (𝕜[[ϵ]]), and let W̃ be the extended affine Weyl group attached to these data. See Section 2.1 for details. For x ∈ W̃ and b ∈ G(L) the locally closed subscheme

1Partially supported by a Heisenberg grant and by the Sonderforschungsbereich TR 45 “Periods, Moduli spaces and Arithmetic of Algebraic Varieties” of the Deutsche Forschungsgemeinschaft.

2Partially supported by HKRGC grants 601409.

Documenta Mathematica 15 (2010) 1009–1028

X_{x}(b) = {g ∈ G(L)/I; g^{−1}bσ(g) ∈ IxI}

of the affine flag variety G(L)/I is called the affine Deligne-Lusztig variety attached to b and x. The X_{x}(b) are known to be finite-dimensional varieties (locally of finite type over 𝕜), but are possibly empty, and it is not in general easy to check whether X_{x}(b) = ∅ for a given pair x, b. In [5, Conjecture 9.5.1 (a)] , Haines, Kottwitz, Reuman and the first named author have stated the following conjecture, which extends a conjecture formulated earlier by Reuman. For simplicity, let us assume that G is quasi-simple of adjoint type. We denote by W̃′ the lowest two-sided cell in the sense of Kazhdan and Lusztig. In the terminology of [5], this is the union of the shrunken Weyl chambers. See Section 2.3. The notion of basic σ-conjugacy class can be characterized by saying that it contains an element of N(T)(L) which gives rise to a length 0 element of W̃ = N(T)(L)/T (𝕜[[ϵ]]). Equivalently, a σ-conjugacy class is basic if and only if its Newton vector is central. See [5, Lemma 7.2.1]. The σ-conjugacy class of b = 1 is always basic.

Conjecture 1.1.1. Suppose that the σ-conjugacy class of b is basic, and that x ∈ W̃′. If b and x are in the same connected component of G(L) and

η(x) ∈ W \ ⋃_{T ⊊S} W_{T} ,

then X_{x}(b) ≠ ∅ and

dim X_{x}(b) = \frac{1}{2} (ℓ(x) + ℓ(η(x)) − def_{G}(b)) .

Here S is the set of simple reflections, and for any T ⊂ S, W_{T} denotes the subgroup of W generated by T. Furthermore, η is the map W̃ → W given as follows: If x = vt^{μ}w with v, w ∈ W and such that the alcove t^{μ}w lies in the dominant chamber, then η(x) = wv. See Section 3.1. Finally, def_{G}(b) is the defect of b, see [11]. A strengthened version of the converse of the non-emptiness statement was proved in [5, Proposition 9.5.4]. Here we prove

Theorem 1.1.2. Suppose that the σ-conjugacy class of b is basic, and that x ∈ W̃′. Write x = vt^{μ}w as above. Assume that b and x are in the same connected component of G(L) and that

η(x) ∈ W \ ⋃_{T ⊊S} W_{T} .

1. Then X_{x}(b) ≠ ∅.

2. If v equals w_{0}, the longest element of W, or μ is regular, then

dim X_{x}(b) ⩽ \frac{1}{2} (ℓ(x) + ℓ(η(x)) − def_{G}(b)) .

3. If G is a classical group and b = 1, or of type A_{n} and b arbitrary basic,

then

dim X_{x}(b) ⩾ \frac{1}{2} (ℓ(x) + ℓ(η(x)) − def_{G}(b)) .

See Section 3.1 for more detailed statements and an outline of the proof. Roughly speaking, our methods are combinatorial (whereas in [5] the theory of ϵ-adic groups was used). Important ingredients are a refinement of the reduction method of Deligne and Lusztig (Section 2), and the results of the second named author about conjugacy classes in affine Weyl groups, see [9]. In [1], E. Beazley obtained similar results for groups of type A_{n}, C_{2} or G_{2} using a similar method, but using only results on conjugacy classes in finite Weyl groups. In Section 4, we briefly consider the case that x ∈ W̃ \ W̃′, but all in all this case remains unclear. Note however that the relation to stratifications of the wonderful compactification of G might provide further insight in this case, see [10]. Finally, a careful study of the reduction method also shows that affine Deligne-Lusztig varieties in the affine flag varieties are not equidimensional in general; we give a specific example in Section 5. Note that for affine Deligne-Lusztig varieties in the affine Grassmannian, equidimensionality is known. For b in the torus T (L), this was proved in [4, Proposition 2.17.1], and for basic b in [6, Theorem 1.2]; for the intermediate cases, see [7, Corollary 6.8 (a)].

2 Preliminaries

2.1 Notation

Let G be a split connected reductive group over 𝔽_{q}. We assume that G is quasi-simple of adjoint type. As explained in [4, 5.9], all problems about the dimension of affine Deligne-Lusztig varieties easily reduce to this case. Let L = 𝕜((ϵ)) be the field of formal Laurent series over 𝕜, and let σ be the automorphism on L defined by σ(∑ a_{n}ϵ^{n}) = ∑ a_{n}^{q}ϵ^{n}. We also denote the induced automorphism on the loop group G(L) by σ. Let T be a maximal torus of G, let B ⊃ T be a Borel subgroup of G, and let B^{−} be the opposite Borel subgroup so that T = B ∩ B^{−}. Let Φ be the set of roots and Y be the coweight lattice. We denote by Y_{+} the set of dominant coweights. Let (α_{i})_{i∈S} be the set of simple roots determined by (B, T). We denote by W the Weyl group N(T)/T. For i ∈ S, we denote by s_{i} the simple reflection corresponding to i.

For w ∈ W, we denote by supp(w) the set of simple reflections occurring in a reduced expression of w. So the condition w ∈ W \ ⋃_{T ⊊S} W_{T} is equivalent to supp(w) = S. For w ∈ W, we choose a representative in N(T) and also write it as w. For any J ⊂ S, let Φ_{J}^{+} (resp. Φ_{J}^{−}) be the positive (resp. negative) roots spanned by (α_{j})_{j∈J}. Let I be the inverse image of B^{−} under the projection map G(𝕜[[ϵ]]) → G sending ϵ to 0. Let W̃ = N(T (L))/(T (L) ∩ I) be the extended affine Weyl group of G(L). Then it is known that W̃ = W ⋉ Y = {wϵ^{χ}; w ∈ W, χ ∈ Y }. Let ℓ : W̃ → ℕ ∪ {0} be the length function. For x = wϵ^{χ} ∈ W̃, we also write x for the representative wϵ^{χ} in N(T (L)). Let X be the coroot lattice, and let W_{a} = W ⋉ X ⊂ W̃ be the affine Weyl group. Set S̃ = S ∪ {0} and s_{0} = ϵ^{θ^{∨}}s_{θ}, where θ is the largest positive root of G. Then (W_{a}, S̃) is a Coxeter system. Let κ : W̃ → W̃ /W_{a} be the natural projection. For any J ⊂ S̃, let W_{J} be the subgroup of W_{a} generated by J and W̃^{J} (resp. J W̃) be the set of minimal length coset representative of W̃ /W_{J} (resp. W_{J}\W̃). For example, ^{S}W̃ is the set of all elements for which the corresponding alcove is contained in the dominant chamber. In the case where J ⊂ S, we write W^{J} for W̃^{J} ∩ W and ^{J}W for ^{J} W̃ ∩ W. Let λ be a dominant coweight. Set I(λ) = {i ∈ S; ⟨λ, α_{i}⟩ = 0}, the “set of walls” that λ lies on. For J ⊂ S, let ρ_{J}^{∨} ∈ Y_{+} with

⟨ρ_{J}^{∨}, α_{i}⟩ = \begin{cases}1, & if j ∈ J \\ 0, & if j ∉ J \end{cases} .

We simply write ρ^{∨} for ρ_{S}^{∨}.

For any root α ∈ Φ, set δ_{α} = \begin{cases}1, & if α ∈ Φ^{−} \\ 0, & if α ∈ Φ^{+} \end{cases} .

2.2

Following [9, 1.4], we use the following notation: For x, x′ ∈ W̃ and i ∈ S̃, we write x ⟶^{s_{i}} x′ if x′ = s_{i}xs_{i} and ℓ(x′) ⩽ ℓ(x). We write x ˜→x′ if there is a sequence x = x_{0}, x_{1}, ⋯, x_{n} = x′ of elements in W̃ such that for all k,

x_{k} = τx_{k−1}τ^{−1} for some τ ∈ W̃ with ℓ(τ) = 0 or x_{k−1} ⟶^{s_{i}} x_{k} for some i ∈ S̃.

We write x ˜≈x′ if x ˜→x′ and x′ ˜→x.

2.3

Any element in W̃ can be written in a unique way as vt^{μ}w for μ ∈ Y_{+}, v ∈ W and w ∈ ^{I(μ)}W. Note that in this case t^{μ}w ∈ ^{S}W̃, and ℓ(vτ^{μ}w) = ℓ(v)+ℓ(t^{μ})− ℓ(w). Set

W̃′ = {vt^{μ}w; μ ∈ Y_{+}, w ∈ ^{I(μ)}W, ⟨μ, α_{i}⟩ + δ_{vα_{i}} − δ_{w^{−1}α_{i}} ≠ 0 ∀i ∈ S}.

It is proved by Lusztig [12], Shi [13] and Bédard [2] that W̃′ ∩ C is a two-sided cell for each W_{a}-coset C in W̃. It is called the lowest two-sided cell. It is also called the union of the shrunken Weyl chambers in [4] and [5].

2.4

We introduce a convenient notation for varieties of tuples of elements in Flag = G(L)/I (i.e., the affine flag variety of G over 𝕜). Instead of giving a rigorous definition, it is more useful to explain the notation by examples. We denote by O_{w} ⊂ Flag × Flag the locally closed subvariety of pairs (g, g′) such that the relative position of g and g′ is w. Then we set

\{ \xymatrix{g \ar[r]^{w} & g′′ \ar[r]^{w′} & g′} \} :=

{(g, g′, g′′) ∈ (Flag)^{3}; (g, g′′) ∈ O_{w}, (g′′, g′) ∈ O_{w′}}.

Similarly,

\{ \xymatrix{g \ar[r]^{w} \ar[rr]_{w′′} & g′′ \ar[r]^{w′} & g′} \} :=

{(g, g′, g′′) ∈ (Flag)^{3}; (g, g′′) ∈ O_{w}, (g′′, g′) ∈ O_{w′}, (g, g′) ∈ O_{w′′}}.

Finally, we need conditions on relative positions where elements g and bσ(g) occur both—the simplest case being the affine Deligne-Lusztig varieties themselves:

X_{x}(b) = \{ \xymatrix{g \ar[r]^{x} & bσ(g)} \}.

In all these cases, we do not distinguish between the sets given by the conditions on the relative position, and the corresponding locally closed sub-ind-schemes of the product of affine flag varieties. The following properties are easy to prove.

(1) Let x, y ∈ W̃. If l(xy) = l(x) + l(y), then the map (g, g′, g′′) ↦ (g, g′) gives an isomorphism

\{ \xymatrix{g \ar[r]^{x} \ar[rr]_{xy} & g′′ \ar[r]^{y} & g′} \} → \{ \xymatrix{g \ar[r]^{xy} & g′} \}.

(2) Let w ∈ W̃ and s ∈ S̃. If ws < w, then

\{ \xymatrix{g \ar[r]^{w} & g′′ \ar[r]^{s} & g′} \} = \{ \xymatrix{g \ar[r]^{w} \ar[rr]_{ws} & g′′ \ar[r]^{s} & g′} \} ⊔ \{ \xymatrix{g \ar[r]^{w} \ar[rr]_{w} & g′′ \ar[r]^{s} & g′} \},

where the first set on the right hand side of the equation is open, and the second one is closed. The projections (g, g′, g′′) ↦ (g, g′) give rise to Zariski-locally trivial fiber bundles

\{ \xymatrix{g \ar[r]^{w} \ar[rr]_{ws} & g′′ \ar[r]^{s} & g′} \} → \{ \xymatrix{g \ar[r]^{ws} & g′} \};

\{ \xymatrix{g \ar[r]^{w} \ar[rr]_{w} & g′′ \ar[r]^{s} & g′} \} → \{ \xymatrix{g \ar[r]^{w} & g′} \}.

with fibers isomorphic to 𝔸^{1} in the first case, and isomorphic to 𝔸^{1} \ {0} in the second case.

2.5 The reduction method of Deligne and Lusztig

Lemma 2.5.1 (He [8], Lemma 1). Let w, w′ ∈ W̃. Then the set {uu′; u ⩽ w, u′ ⩽ w′} has a unique maximal element, which we denote by w ∗ w′. We have ℓ(w ∗ w′) = ℓ(w) + ℓ(w^{−1}(w ∗ w′)) = ℓ((w ∗ w′)(w′)^{−1}) + ℓ(w′), and supp(w ∗ w′) = supp(w) ∪ supp(w′).

Note that the operation ∗ is associative.

Proposition 2.5.2. Let w, w′ ∈ W̃, and let w′′ ∈ {ww′, w ∗ w′}. All fibers of the projection

π: \{ \xymatrix{g \ar[r]^{w} \ar[rr]_{w′′} & g′′ \ar[r]^{w′} & g′} \} ⟶ \{ \xymatrix{g \ar[r]^{w′′} & g′} \}

which maps (g, g′, g′′) to (g, g′) have dimension

dim π^{−1}((g, g′)) ⩾ \{ \begin{matrix}ℓ(w) + ℓ(w′) − ℓ(w ∗ w′) & if w′′ = w ∗ w′, \\ \frac{1}{2}(ℓ(w) + ℓ(w′) − ℓ(ww′)) & if w′′ = ww′. \end{matrix}

Proof. We proceed by induction on ℓ(w′). If w′ = 1, then the statement is obvious. Now assume that l(w′) > 0. Then w′ = sw′_{1} for some s ∈ S̃ and w′_{1} ∈ W̃ with ℓ(w′) = ℓ(w′_{1}) + 1. Then

\{ \xymatrix{g \ar[r]^{w} \ar[rrr]_{w′′} & g′′ \ar[r]^{s} & g′′′ \ar[r]^{w′_{1}} & g′} \} ≅ \{ \xymatrix{g \ar[r]^{w} \ar[rr]_{w′′} & g′′ \ar[r]^{w′} & g′} \}

If ws > w, then

\{ \xymatrix{g \ar[r]^{w} \ar[rrr]_{w′′} & g′′ \ar[r]^{s} & g′′′ \ar[r]^{w′_{1}} & g′} \} ≅ \{ \xymatrix{g \ar[r]^{ws} \ar[rr]_{w′′} & g′′′ \ar[r]^{w′_{1}} & g′} .\}

We also have that ℓ(ws) + ℓ(w′_{1}) = ℓ(w) + 1 + ℓ(w′) − 1 = ℓ(w) + ℓ(w′), wsw′_{1} = ww′ and (ws) ∗ w′_{1} = w ∗ s ∗ w′_{1} = w ∗ w′. Now the statement follows from inductive hypothesis on w′_{1}. If ws < w, then

\{ \xymatrix{g \ar[r]^{w} \ar[rrr]_{w′′} & g′′ \ar[r]^{s} & g′′′ \ar[r]^{w′_{1}} & g′} \} = X_{1} ⊔ X_{2},

where

X_{1} = \{ \xymatrix{g \ar[r]^{w} \ar[rrr]_{w′′} \ar[rr]^{ws} & g′′ \ar[r]^{s} & g′′′ \ar[r]^{w′_{1}} & g′} \}

X_{2} = \{ \xymatrix{g \ar[r]^{w} \ar[rrr]_{w′′} \ar[rr]^{w} & g′′ \ar[r]^{s} & g′′′ \ar[r]^{w′_{1}} & g′} \}.

The projection from X_{1} to \{ \xymatrix{g \ar[r]^{w′′} & g′} \} factors through

X_{1} → \{ \xymatrix{g \ar[rr]_{w′′} \ar[r]^{ws} & g′′′ \ar[r]^{w′_{1}} & g′} \} → \{ \xymatrix{g \ar[r]^{w′′} & g′} \},

where the first map is a bundle map whose fibers are all of dimension 1. If w′′ = ww′, then by induction hypothesis on w′_{1}, the fibers of the second map all have dimension ⩾ \frac{1}{2}(ℓ(ws) + ℓ(w′_{1}) − l(ww′)). Notice that ℓ(ws) + ℓ(w′_{1}) = ℓ(w) − 1 + ℓ(w′) − 1 = ℓ(w) + ℓ(w′) − 2. So the fibers of the map X_{1} →

\{ \xymatrix{g \ar[r]^{w′′} & g′} \} all have dimension ⩾ \frac{1}{2}(ℓ(w) + ℓ(w′) − l(ww′)), which gives us the desired lower bound on the fiber dimension.

The projection from X_{2} to \{ \xymatrix{g \ar[r]^{w′′} & g′} \} factors through

X_{2} → \{ \xymatrix{g \ar[rr]_{w′′} \ar[r]^{w} & g′′′ \ar[r]^{w′_{1}} & g′} \} → \{ \xymatrix{g \ar[r]^{w′′} & g′} \},

where the first map is a bundle map whose fibers are all of dimension 1. If w′′ = w∗w′ = w∗s∗w′_{1} = w∗w′_{1}, then by induction hypothesis on w′_{1}, the fibers of the second map all have dimension ⩾ (ℓ(w) + ℓ(w′_{1}) − ℓ(w ∗ w′)). Notice that

ℓ(w) + ℓ(w′_{1}) = ℓ(w) + ℓ(w′) − 1. So the fibers of the map X_{2} → \{ \xymatrix{g \ar[r]^{w′′} & g′} \} all have dimension ⩾ (ℓ(w) + ℓ(w′) − ℓ(w ∗ (w′)).

As a corollary, we can prove the analog, in the affine context, of the “reduction method” of Deligne and Lusztig (see [3, proof of Theorem 1.6]). This result can of course be proved directly, along the lines of the proof of the proposition above, and was also worked out before by Haines at the suggestion of Lusztig.

Corollary 2.5.3. Let x ∈ W̃, and let s ∈ S̃ be a simple affine reflection.

1. If ℓ(sxs) = ℓ(x), then there exists a universal homeomorphism X_{x}(b) → X_{sxs}(b).

2. If ℓ(sxs) = ℓ(x)−2, then X_{x}(b) can be written as a disjoint union X_{x}(b) = X_{1} ⊔ X_{2} where X_{1} is closed and X_{2} is open, and such that there exist morphisms X_{1} → X_{sxs}(b) and X_{2} → X_{sx}(b) which are compositions of a Zariski-locally trivial fiber bundle with one-dimensional fibers and a universal homeomorphism.

Proof. By possibly exchanging, in case (1), x and sxs, we may assume that sx < x. By the proposition, the projection

X′ := \{ \xymatrix{g \ar[d]_{s} \ar[r]^{x} & bσ(g) \ar[d]^{s} \\ g_{1} \ar[ur]^{sx} & bσ(g_{1})} \} ⟶ \{ \xymatrix{g \ar[r]^{x} & bσ(g)} \} = X_{x}(b).

is an isomorphism, so we may replace X_{x}(b) by X′. We write X′ as the disjoint union

X′ = X_{1} ⊔ X_{2} := \{ \xymatrix{g \ar[d]_{s} \ar[r]^{x} & bσ(g) \ar[d]^{s} \\ g_{1} \ar[ur]^{sx} \ar[r]_{sxs} & bσ(g_{1})} \} ⊔ \{ \xymatrix{g \ar[d]_{s} \ar[r]^{x} & bσ(g) \ar[d]^{s} \\ g_{1} \ar[ur]^{sx} \ar[r]_{sx} & bσ(g_{1})} \}

Since we have ℓ(sx) < ℓ(x), the natural morphism

X_{1} → X′_{1} = \{ \xymatrix{g_{1} \ar[r]^{sx} \ar[rr]_{sxs} & g_{2} \ar[r]^{s} & bσ(g_{1})} \}

is a universal homeomorphism (note that the composition of X_{1} → X′_{1} with the projection to g_{2} is the map g ↦ bσ(g)). Now we distinguish between the two cases. In case (1), X_{2} = ∅, and applying the proposition once more, we find that in this case the projection

X′_{1} = \{ \xymatrix{g_{1} \ar[r]^{sx} \ar[rr]_{sxs} & g_{2} \ar[r]^{s} & bσ(g_{1})} \} ⟶ \{ \xymatrix{g_{1} \ar[r]^{sxs} & bσ(g_{1})} \} = X_{sxs}(b)

is an isomorphism. Next we come to case (2). The projection

X′_{1} = \{ \xymatrix{g_{1} \ar[r]^{sx} \ar[rr]_{sxs} & g_{2} \ar[r]^{s} & bσ(g_{1})} \} ⟶ \{ \xymatrix{g_{1} \ar[r]^{sxs} & bσ(g_{1})} \} = X_{sxs}(b)

has fibers of dimension \frac{1}{2}(ℓ(sx) + ℓ(s) − ℓ(sxs)) = 1, which proves the claim about X_{1}. Furthermore X_{2} can be replaced with

X′_{2} = \{ \xymatrix{g_{1} \ar[r]^{sx} \ar[rr]_{sx} & g_{2} \ar[r]^{s} & bσ(g_{1})} \}

up to a universal homeomorphism, and X′_{2} projects to X_{sx}(b) with 1dimensional fibers. The corollary is proved.

With slightly more care, one can show that in case (2) of the lemma, the fibers of the projection X_{1} → X_{sxs}(b) are all isomorphic to 𝔸^{1}, whereas the fibers of X_{2} → X_{sx}(b) are 𝔸^{1} \ {0}. This reflects the properties discussed at the end of subsection 2.4.

Lemma 2.5.4. Let x, τ ∈ W̃ with ℓ(τ) = 0. Then for any b ∈ G(L), X_{x}(b) is isomorphic to X_{τxτ^{−1}}(b).

Proof. Notice that X_{x}(b) = {gI; g^{−1}bσ(g) ∈ IxI}. Thus the isomorphism G(L)/I → G(L)/I, gI ↦ gIτ^{−1} = gτ^{−1}I gives an isomorphism from X_{x}(b) to {gI; (gτ)^{−1}bσ(gτ) ∈ IxI} = {gI; g^{−1}bσ(g) ∈ τIxIσ(τ)^{−1} = Iτxτ^{−1}I} = X_{τxτ^{−1}}(b).

Applying this lemma and the conjugation steps in the reduction method of Deligne and Lusztig, we obtain:

Corollary 2.5.5. Let x, x′ ∈ W̃, b ∈ G(L). If x ˜→x′, and X_{x′}(b) ≠ ∅, then X_{x}(b) ≠ ∅ and dim(X_{x}(b)) − dim(X_{x′}(b)) ⩾ \frac{1}{2}(ℓ(x) − ℓ(x′)).

2.6

In the sequel, we often use the following property of the Bruhat order: if α ∈ Φ^{+} with corresponding reflection s_{α}, and w ∈ W, then

ws_{α} > w if and only if wα > 0.

For further reference, we state the following version of the usual criterion for length additivity.

Lemma 2.6.1. Let w, y ∈ W such that wα < 0 for every α ∈ Φ^{+} with y^{−1}α < 0. Then ℓ(wy) = ℓ(w) − ℓ(y).

3 Proof of Reuman’s conjecture

3.1 Outline of the proof

We first state the result and give an outline of our strategy. Throughout this chapter, we fix b ∈ G(L), and we assume that whenever we consider X_{x}(b), then x and b are in the same connected component of G(L). We consider the following maps from the extended affine Weyl group W̃ to the finite Weyl group W:

η_{1} : W̃ = X_{∗}(T) ⋊ W → W, the projection  (1)
η_{2}, where η_{2}(x) is the unique element v such that v^{−1}x ∈ ^{S}W̃  (2)
η(x) = η_{2}(x)^{−1}η_{1}(x)η_{2}(x).  (3)

So if x = vt^{μ}w with μ dominant, v ∈ W, w ∈ ^{I(μ)}W, then η_{1}(x) = vw, η_{2}(x) = v, and η(x) = wv. Furthermore, for x ∈ W̃ (as always, in the same “connected component” as the fixed b ∈ G(L)) we define the virtual dimension:

d(x) = \frac{1}{2}(ℓ(x) + ℓ(η(x)) − def(b)).

As discussed above, it is conjectured in [5] that dim X_{x}(b) = d(x) for b basic, x ∈ W̃′ with X_{x}(b) ≠ ∅. We denote by w_{0} the longest element in W.

Theorem 3.1.1. Let x ∈ W̃ and assume that η_{2}(x) = w_{0} or that the translation part of x is given by a regular coweight. Then dim(X_{x}(b)) ⩽ d(x).

Theorem 3.1.2. Assume that b is basic. Let x ∈ W̃′ such that supp(η(x)) = S. Then X_{x}(b) ≠ ∅.

This proves the non-emptiness statement in Conjecture 9.5.1 (a) of [5]. Together with op. cit., Proposition 9.5.4, which states that the converse of the theorem holds as well, this completely settles the emptiness versus non-emptiness question for basic b and x in the shrunken Weyl chambers W̃′. The next theorem proves that the dimension of X_{x}(b) is at least as large as predicted by the conjecture if x ∈ W̃′, and G is a classical group and b = 1 or G is of type A_{n}:

Theorem 3.1.3. (1) Let G be a classical group, x ∈ W_{a} such that supp(η(x)) = S. If moreover, x ∈ W̃′ or η(x) is a Coxeter element of W, then dim X_{x}(1) ⩾ d(x).

(2) Let G = P GL_{n} and τ ∈ W̃ with ℓ(τ) = 0. Let x ∈ W_{a}τ such that supp(η(x)) = S. If moreover, x ∈ W̃′ or η(x) is a Coxeter element of W, then dim X_{x}(τ) ⩾ d(x).

The idea of the proofs of these theorems is to relate the given element x to other elements for which non-emptiness, a lower bound on the dimension, or an upper bound on the dimension, respectively, are known. These relations will mainly be shown using the reduction method of Deligne and Lusztig. To this end, we introduce the following notation:

Definition 3.1.4. Let x, y ∈ W̃ such that x, y are in the same W_{a}-coset. We write x ⇒ y if for every b,

dim X_{x}(b) − d(x) ⩾ dim X_{y}(b) − d(y).

Here by convention, we set the dimension of the empty set to be −∞. If the right hand side is −∞ then the inequality holds regardless of the left hand side. In the definition (and in the theorem below) we do not assume that b is basic. This is consistent with the expectation that whenever x ∈ W̃′ and X_{x}(b) ≠ ∅, the difference dim X_{x}(b) − d(x) is a constant depending only on b, but not on x. Note that this relation is transitive: If x ⇒ y, y ⇒ z, then x ⇒ z. By definition, if x ⇒ y and X_{y}(b) ≠ ∅, then X_{x}(b) ≠ ∅. In this case, the lower bound dim X_{y}(b) ⩾ d(y) implies the analogous bound for x, while the validity of the upper bound dim X_{x}(b) ⩽ d(x) implies the corresponding statement for y. We prove the following statements about the relation ⇒:

Theorem 3.1.5. 1. Let μ be a dominant coweight, v ∈ W and w ∈ ^{I(μ)}W. Assume that v = w_{0} or that μ is regular. Then

w_{0}t^{μ} ⇒ vt^{μ}w.

2. Let a ∈ W with supp(a) = S, and let μ ≠ 0 be a dominant coweight. Then there exists a Coxeter element c ∈ W such that

at^{μ} ⇒ t^{μ}c.

3. Assume that x ∈ W̃′, and that supp(η(x)) = S. Then there exist a dominant coweight λ and a ∈ W with supp(a) = S such that

x ⇒ at^{λ}.

4. Assume that G is a classical group and x ∈ W_{a} with η(x) a Coxeter

element of W, then

x ⇒ η(x).

Now the non-emptiness statement in Theorem 3.1.2 follows from Theorem 3.1.5 (2) & (3) and the following lemma (Lemma 9.3.3 in [5]), because Coxeter elements obviously are cuspidal.

Lemma 3.1.6. Let μ ∈ Y, and let w ∈ W be a cuspidal element (i. e. the conjugacy class of w does not meet any standard parabolic subgroup), let x = t^{μ}w, and let b be basic with κ_{G}(b) = κ_{G}(x). Then x is σ-conjugate to b, and in particular X_{x}(b) ≠ ∅.

The upper bound on the dimension stated in Theorem 3.1.1 follows from Theorem 3.1.5 (1) and the following lemma:

Lemma 3.1.7. Let x = w_{0}t^{μ}, where μ is a dominant coweight. Then dim X_{x}(b) ⩽ d(x).

Proof. By the dimension formula for affine Deligne-Lusztig varieties in the affine Grassmannian (see [4], [14]), we have

dim X_{μ}(b) = ⟨ρ, μ − ν_{b}⟩ − \frac{1}{2} def(b),

where X_{μ}(b) denotes the affine Deligne-Lusztig variety in the affine Grassmannian, and ν_{b} denotes the (dominant) Newton vector of b. Since b is basic, its Newton vector is central and hence does not actually contribute anything. On the other hand, denoting by π: Flag → Grass the projection, we have

π^{−1}(X_{μ}(b)) = ⋃_{x∈W t^{μ}W} X_{x}(b).

Therefore, for all w_{1}, w_{2} ∈ W

dim X_{w_{1}t^{μ}w_{2}}(b) ⩽ dim X_{μ}(b) + dim(G/B) = ⟨ρ, μ⟩ − \frac{1}{2} def(b) + ℓ(w_{0})

= \frac{1}{2}(ℓ(w_{0}t^{μ}) + ℓ(w_{0}) − def(b)) = d(w_{0}t^{μ}).

To prove the lower bound under the additional assumptions in Theorem 3.1.3 (1), we reduce to an element of the finite Weyl group. In fact, the following lemma (for τ = id) shows that it suffices to prove that x ⇒ c for some element c ∈ W. If η(x) is a Coxeter element, then this follows immediately from Theorem 3.1.5 (4). On the other hand, suppose x ∈ W̃′ and supp(η(x)) = S. We apply Theorem 3.1.5 (3). If the coweight λ is = 0, then we are done. Otherwise, we can use Theorem 3.1.5 (2) to see that there exists a dominant coweight λ and a Coxeter element c ∈ W such that x ⇒ t^{λ}c. Writing t^{λ}c = v_{1}t^{ν}v_{2} with t^{ν}v_{2} ∈ ^{S}W̃, i. e. η_{2}(t^{λ}c) = v_{1}, we have c = v_{1}v_{2} and η(t^{λ}c) = v_{2}v_{1}. Since c = v_{1}v_{2} is simply the decomposition into an element of W_{I(μ)} and an element of ^{I(μ)}W, we have ℓ(v_{1}) + ℓ(v_{2}) = ℓ(c) and since c is a Coxeter element, v_{2}v_{1} is also a Coxeter element of W. Therefore x ⇒ t^{λ}c ⇒ v_{2}v_{1} (using Theorem 3.1.5 (4)).

Lemma 3.1.8. Let τ ∈ W̃ with ℓ(τ) = 0. Let J ⊂ S with τ(J) = J. Then for any w ∈ W_{J}, dim X_{wτ} (τ) = ℓ(w).

Proof. By [9, Lemma 9.7], dim X_{wτ} (τ) = dim X_{τ} (τ) + ℓ(w). By [9, Prop 10.3], dim X_{τ} (τ) = 0. So dim X_{wτ} (τ) = ℓ(w).

Under the assumption in Theorem 3.1.3 (2), we have that τ = 1 or 0 < r < n and that τ is the length 0-element that corresponds to the r-th fundamental coweight of G. The case that τ = 1 is included in Theorem 3.1.3 (1). So we only need to consider the latter case. Similarly to the proof above, we have that x ⇒ t^{λ}c for some Coxeter element c of W. Let m = gcd(n, r). Then by [9, Prop 6.7 (2)], t^{λ}c ˜→(12 ⋯ m)τ. Hence by Corollary 2.5.5 and the Lemma above,

dim X_{t^{λ}c}(τ) ⩾ dim X_{(12···m)τ} (τ) + \frac{1}{2}(ℓ(t^{λ}c) − ℓ((12 ⋯ m)τ))

= \frac{1}{2}(ℓ(t^{λ}c) + m − 1).

Since c is a Coxeter element, η(t^{λ}c) is also a Coxeter element of W. We also have that def(τ) = n − m. Thus d(t^{λ}c) = \frac{1}{2}(ℓ(t^{λ}c) + n − 1 − def(τ)) = \frac{1}{2}(ℓ(t^{λ}c) + m − 1) ⩽ dim X_{t^{λ}c}(τ). So we obtain d(x) ⩽ dim X_{x}(τ). Therefore it remains to prove Theorem 3.1.5. This is the goal of the following sections.

3.2 Reduction of virtual dimension

Lemma 3.2.1. If x, x′ ∈ W̃ such that x ˜→x′ and ℓ(η(x)) = ℓ(η(x′)), then x ⇒ x′.

Proof. Since ℓ(η(x)) = ℓ(η(x′)), we have d(x) − d(x′) = \frac{1}{2}(ℓ(x) − ℓ(x′)). The Lemma now follows immediately from Corollary 2.5.5.

Lemma 3.2.2. Let x ∈ W̃. Let s ∈ S be a simple reflection such that ℓ(sxs) = ℓ(x) − 2. Then

d(x) ⩾ d(sx) + 1,

and equality holds if and only if ℓ(η(sx)) = ℓ(η(x)) − 1.

Proof. We write x as vt^{μ}w with v, w ∈ W and μ a dominant coweight such that t^{μ}w ∈ ^{S}W̃. Then η(x) = wv. Since sx < x, we must have that sv < v. If ws < w, then t^{μ}ws ∈ ^{S}W̃ and xs > x, which is a contradiction. Therefore ws > w. Let α denote the simple root corresponding to s, and write β = v^{−1}(−α), which is a positive root because sv < v. We then have wv(β) = w(−α) < 0 (since ws > w), and obtain

η(sx) = wsv = wvs_{β} < wv,

as desired.

Similarly,

Lemma 3.2.3. Let x = vt^{μ}w with v, w ∈ W and μ a dominant coweight such that t^{μ}w ∈ ^{S}W̃. Let s ∈ S be a simple reflection such that ℓ(sxs) = ℓ(x) − 2.

1. We have d(x) = d(xs) + 1 if and only if ℓ(η(xs)) = ℓ(η(x)) − 1.

2. If t^{μ}ws ∈ ^{S}W̃ or w = 1, then

d(x) ⩾ d(xs) + 1.

These results about the virtual dimension imply

Lemma 3.2.4. Let x ∈ W̃, s ∈ S such that ℓ(sxs) < ℓ(x). Then

1. If ℓ(η(sx)) = ℓ(η(x)) − 1, then x ⇒ sx.

2. If ℓ(η(xs)) = ℓ(η(x)) − 1, then x ⇒ xs.

Proof. For (1), we simply use the Deligne-Lusztig reduction (where we consider X_{2} in Corollary 2.5.3 (2)), and Lemma 3.2.2. For part (2), we first use the Deligne-Lusztig reduction from x to sx as in the first case. Then we use Corollary 2.5.3 (1) to reduce to xs = s(sx)s which has the same length as sx. Altogether we see that if X_{xs}(b) ≠ ∅, then X_{x}(b) ≠ ∅, and, using Lemma 3.2.3 (1),

dim X_{x}(b) − dim X_{xs}(b) ⩾ 1 = d(x) − d(xs).

3.3 Proof of Theorem 3.1.5 (1)

We write x = vt^{μ}w. First consider the case v = η_{2}(x) = w_{0}. Then we have that w_{0}t^{μ} ⇒ x = w_{0}t^{μ}w because we can successively apply Lemma 3.2.4 (2). Now we consider the case that μ is regular. Since μ is regular, t^{μ}wvw_{0} ∈ ^{S}W̃, so we can apply the “η_{2} = w_{0}”-case to the element w_{0}t^{μ}wvw_{0} and obtain that w_{0}t^{μ} ⇒ w_{0}t^{μ}wvw_{0}. Because μ is regular, Lemma 3.2.1 shows that

w_{0}t^{μ}wvw_{0} = w_{0}v^{−1}(vt^{μ}w)vw_{0} ⇒ vt^{μ}w = x.

3.4 Proof of Theorem 3.1.5 (2)

We prove the following stronger result: Let J ⊂ S and x = vt^{μ}w with v, w ∈ W, supp(v) = J, w is a Coxeter element in W_{S−J}, μ ≠ 0 and t^{μ}w ∈ ^{S}W̃. Then there exists a Coxeter element c of W such that vt^{μ}w ⇒ t^{μ}c. We proceed by induction on |J|. Suppose that the statement is true for all J′ ⊊ J, but not true for J. We may also assume that the claim of the proposition is true for all v′ with support supp(v′) = J and ℓ(v′) < ℓ(v). Let v = s_{i_{1}} ⋯ s_{i_{k}} be a reduced expression. If t^{μ}ws_{i_{1}} ∈ ^{S}W̃, then ℓ(t^{μ}ws_{i_{1}}) = ℓ(t^{μ})−ℓ(ws_{i_{1}}) = ℓ(t^{μ})−ℓ(w)−1 = ℓ(t^{μ}w)−1 and ℓ(s_{i_{1}}vt^{μ}ws_{i_{1}}) = ℓ(s_{i_{1}}v)+ℓ(t^{μ}ws_{i_{1}}) = ℓ(s_{i_{1}}v)+ℓ(t^{μ}w)−1 = ℓ(vt^{μ}w)−2. By Lemma 3.2.4 (1), vt^{μ}w ⇒ s_{i_{1}}vt^{μ}w. If supp(s_{i_{1}}v) = J, then by induction, there exists a Coxeter element c of W such that s_{i_{1}}vt^{μ}w ⇒ t^{μ}c. Hence vt^{μ}w ⇒ t^{μ}c. That is a contradiction. Now suppose that t^{μ}ws_{i_{1}} ∈ ^{S}W̃, but supp(s_{i_{1}}v) ⊊ J. In that case, we have ℓ(s_{i_{1}}vt^{μ}ws_{i_{1}}) = ℓ(vt^{μ}w) − 2. So vt^{μ}w ⇒ s_{i_{1}}vt^{μ}ws_{i_{1}} by Lemma 3.2.1. That is also a contradiction by induction hypothesis. Now we can assume that t^{μ}ws_{i_{1}} ∉ ^{S}W̃. Then we have that t^{μ}ws_{i_{1}} = s_{i′_{1}}t^{μ}w for some i′_{1} ∈ S. So ws_{i_{1}}w^{−1} = t^{−μ}s_{i′_{1}}t^{μ} is a reflection in W. So t^{μ} commutes with s_{i′_{1}} and ws_{i_{1}}w^{−1} = s_{i′_{1}} is a simple reflection. By our assumptions on w, it follows that i′_{1} = i_{1} and t^{μ}w commutes with s_{i_{1}}. In this case, if ℓ(s_{i_{1}}vs_{i_{1}}) < ℓ(v), then supp(s_{i_{1}}v) = J and using Lemma 3.2.4 (1) and the induction hypothesis as in the first case, we again have that vt^{μ}w ⇒ t^{μ}c for some Coxeter element c of W, which is a contradiction. Therefore we must have that s_{i_{1}} commutes with t^{μ}w and ℓ(s_{i_{1}}vs_{i_{1}}) = ℓ(v). In this case, vt^{μ}w ≈ s_{i_{1}}vs_{i_{1}}t^{μ}w. So dim X_{vt^{μ}w}(b) = dim X_{s_{i_{1}} vs_{i_{1}} t^{μ}w}(b) by Corollary 2.5.3 (1). We also have d(vt^{μ}w) = d(s_{i_{1}}vs_{i_{1}}t^{μ}w), because η(s_{i_{1}}vs_{i_{1}}t^{μ}w) = ws_{i_{1}}vs_{i_{1}} has length ℓ(w) + ℓ(s_{i_{1}}vs_{i_{1}}) (use that w is a Coxeter element in W_{S−J}). Applying the same argument to s_{i_{1}}vs_{i_{1}}t^{μ}w instead of vt^{μ}w, we have that s_{i_{2}} commutes with t^{μ}w. Repeating the same procedure, one can show that s_{i_{j}} commutes with t^{μ}w for all 1 ⩽ j ⩽ k. In particular, s_{k} commutes with w for all k ∈ J. Since G is quasi-simple, this is only possible if J = ∅ or J = S. If J = ∅, then v = 1 and w is a Coxeter element of W and the statement automatically holds. If J = S, then s_{i} commutes with t^{μ} for all i ∈ S. Thus μ = 0, which contradicts our assumption.

3.5 Proof of Theorem 3.1.5 (3)

Let x = vt^{μ}w ∈ W̃′ with μ ∈ Y_{+}, v ∈ W, w ∈ ^{I(μ)}W. We first give the definition of the elements γ and a that we use. Let J = {i ∈ S; s_{i}w < w}. Since w ∈ ^{I(μ)}W, J ∩I(μ) = ∅. Hence μ−ρ_{J}^{∨} ∈ Y_{+}. By definition, ⟨ρ_{J}^{∨}, α_{i}⟩−δ_{w^{−1}α_{i}} = 0 for any i ∈ S. Since x ∈ W̃′, we obtain ⟨μ − ρ_{J}^{∨}, α_{i}⟩ + δ_{vα_{i}} ≠ 0 for any i ∈ S. Let J′ = I(μ − ρ_{J}^{∨}). Then vα_{i} < 0 for any i ∈ J′. Thus v = v′w_{J′}^{0} for some v′ ∈ W^{J′}. Here w_{J′}^{0} is the largest element in W_{J′}. Now wv = (wv′)w_{J′}^{0} = w′z for some w′ ∈ W^{J′} and z ∈ W_{J′}. Define γ ∈ Y_{+} and y ∈ W^{I(γ)} by μ − ρ_{J}^{∨} + (w′)^{−1}ρ_{J}^{∨} = yγ. Furthermore, we define a = (y^{−1}z) ∗ (w′y). It has support supp(a) = S since S = supp(wv) ⊆ supp(w′y) ∪ supp(y^{−1}z). We show that

(a) ℓ(w′y) = ℓ(w′) − ℓ(y). Let α ∈ Φ^{+} with y^{−1}α < 0. Then ⟨γ, y^{−1}α⟩ ⩽ 0. If ⟨γ, y^{−1}α⟩ = 0, then y^{−1}α ∈ Φ_{I(γ)}^{−} and α = y(y^{−1}α) ∈ Φ^{−}. That is a contradiction. Hence ⟨μ−ρ_{J}^{∨} + (w′)^{−1}ρ_{J}^{∨}, α⟩ = ⟨yγ, α⟩ = ⟨γ, y^{−1}α⟩ < 0. Since μ − ρ_{J}^{∨} ∈ Y_{+}, ⟨μ − ρ_{J}^{∨}, α⟩ ⩾ 0. Thus ⟨ρ_{J}^{∨}, w′α⟩ = ⟨(w′)^{−1}ρ_{J}^{∨}, α⟩ < 0 and w′α < 0. Since w′α < 0 for any α ∈ Φ^{+} with y^{−1}α < 0, Lemma 2.6.1 shows that ℓ(w′y) = ℓ(w′) − ℓ(y). (a) is proved. Now set x_{1} = vz^{−1}t^{μ−ρ_{J}^{∨}}y and x_{2} = y^{−1}zt^{ρ_{J}^{∨}}w. Then x = x_{1}x_{2} and we claim that

(b) ℓ(x) = ℓ(x_{1}) + ℓ(x_{2}). By the proof of (a), y ∈ ^{J′}W. In fact, if for any j ∈ J′, y^{−1}α_{j} < 0, then by the proof of (a), w′α_{j} < 0, which contradicts that w′ ∈ W^{J′}. Hence t^{μ−ρ_{J}^{∨}}y ∈ SW̃ and ℓ(x_{1}) = ℓ(vz^{−1}) + ℓ(t^{μ−ρ_{J}^{∨}}) − ℓ(y) = ℓ(t^{μ−ρ_{J}^{∨}}) + ℓ(v) − ℓ(z) − ℓ(y). Also ℓ(x_{2}) = ℓ(y^{−1}z) + ℓ(t^{ρ_{J}^{∨}}) − ℓ(w) = ℓ(t^{ρ_{J}^{∨}}) + ℓ(y) + ℓ(z) − ℓ(w). Thus ℓ(x_{1}) + ℓ(x_{2}) = ℓ(t^{μ−ρ_{J}^{∨}}) + ℓ(t^{ρ_{J}^{∨}}) + ℓ(v) − ℓ(w) = ℓ(t^{μ}) + ℓ(v) − ℓ(w) = ℓ(x).

(b) is proved. Now

X_{x}(b) = \{ \xymatrix{g \ar[r]^{x} & bσ(g)} \} = \{ \xymatrix{g \ar[r]^{x_{1}} & g_{1} \ar[r]^{x_{2}} & bσ(g)} \}.

Set

X_{1} =\{ \xymatrix{g_{1} \ar[r]^{x_{2}} & g_{2} \ar[r]^{x_{1}} & bσ(g_{1})} \}
≅\{ \xymatrix{g_{1} \ar[r]^{y^{−1}z} & g_{3} \ar[r]^{t^{ρ^{∨}_{J}} w} & g_{2} \ar[r]^{x_{1}} & bσ(g_{1})} \}.

The map (g, g_{1}) ↦ (g_{1}, bσ(g)) is a universal homeomorphism from X_{x}(b) to X_{1}. Let

X_{2} = \{ \xymatrix{g_{1} \ar[r]^{y^{−1}z} & g_{3} \ar[r]^{t^{ρ^{∨}_{J}} w} \ar[rr]_{w′yt^{γ}} & g_{2} \ar[r]^{x_{1}} & bσ(g_{1})} \} ⊂ X_{1}.

Then we have that dim(X_{x}(b)) ⩾ dim(X_{2}). Now let

X_{3} = \{ \xymatrix{g_{1} \ar[r]^{y^{−1}z} & g_{3} \ar[r]^{w′yt^{γ}} & bσ(g_{1})} \},

and let f : X_{2} → X_{3} be the projection map. Notice that w′yt^{γ} = t^{ρ_{J}^{∨}}wx_{1}. Thus by 2.5.2 (1), the map is surjective and each fiber is of dimension

\frac{ℓ(t^{ρ^{∨}_{J}} w)+ℓ(x_{1})−ℓ(w′yt^{γ})}{2} = \frac{ℓ(x)−ℓ(y^{−1}z)−ℓ(w′y)−ℓ(t^{γ})}{2}. Hence

(c) dim(X_{x}(b)) ⩾ dim(X_{3}) + \frac{ℓ(x)−ℓ(y^{−1}z)−ℓ(w′y)−ℓ(t^{γ})}{2}. Notice that

X_{3} = \{ \xymatrix{g_{1} \ar[r]^{y^{−1}z} & g_{3} \ar[r]^{w′y} & g_{4} \ar[r]^{t^{γ}} & bσ(g_{1})} \}.

Documenta Mathematica 15 (2010) 1009–1028

Recall that a = (y^{−1}z) ∗ (w′y). We set

X_{4} =\{ \xymatrix{g_{1} \ar[r]^{y^{−1}z} \ar[rr]_{a} & g_{3} \ar[r]^{w′y} & g_{4} \ar[r]^{t^{γ}} & bσ(g_{1})} \},

X_{5} =\{ \xymatrix{g_{1} \ar[r]^{a} & g_{4} \ar[r]^{t^{γ}} & bσ(g_{1})} \}.

By 2.5.2 (2), dim(X_{3}) ⩾ dim(X_{4}) ⩾ dim(X_{5}) + ℓ(y^{−1}z) + ℓ(w′y) − ℓ(a). As we proved above, ℓ(y^{−1}z) + ℓ(w′y) = ℓ(w′) + ℓ(z) = ℓ(wv). Therefore ℓ(y^{−1}z) + ℓ(w′y) + \frac{ℓ(x)−ℓ(y^{−1}z)−ℓ(w′y)−ℓ(t^{γ})}{2} = \frac{ℓ(x)+ℓ(wv)−ℓ(t^{γ})}{2}. So

(d) dim(X_{x}(b)) ⩾ dim(X_{5}) + \frac{ℓ(x)+ℓ(wv)−ℓ(t^{γ})}{2} − ℓ(a). Notice that ℓ(at^{γ}) = ℓ(a) + ℓ(t^{γ}). Thus the map (g_{1}, g_{4}) ↦ g_{1} gives an isomorphism X_{5} ≅ X_{at^{γ}} (b). If X_{at^{γ}} (b) ≠ ∅, then we obtain that X_{x}(b) ≠ ∅, and

dim X_{x}(b) ⩾ dim X_{at^{γ}} (b) + d(x) − d(at^{γ}),

i.e., x ⇒ at^{γ}.

Example 3.5.1. Let us see what the elements a, γ are in the following two special cases:

1. If v = 1, then the assumption that x ∈ W̃′ implies that J′ = ∅. Therefore w′ = w, z = y = 1, so that γ = μ and a = w. The result in this case is that t^{μ}w ⇒ wt^{μ}.

2. If μ is “very regular”, then I(μ) = I(μ − ρ_{J}^{∨} + (w′)^{−1}ρ_{J}^{∨}) = ∅, so that a = w′ = wv. In this case we obtain vt^{μ}w ⇒ wvt^{γ}.

3.6 Proof of Theorem 3.1.5 (4)

Since η(x) is a Coxeter element of W, by [9, Prop 6.7 (1)], we have that x ˜→η(x). By Lemma 3.2.1, x ⇒ η(x).

4 Remarks on the critical strips

4.1 A sharpened criterion for non-emptiness

We have seen that the non-emptiness of X_{x}(b) for b basic and x ∈ W̃′ can be decided by looking at η(x) = η_{2}(x)^{−1}η_{1}(x)η_{2}(x). In fact, if supp(η(x)) ≠ S and the translation part of x is non-trivial in the sense that it is different from the Newton vector of b, then X_{x}(b) = ∅ ([5, Proposition 9.5.4]). If x ∈ W̃ \ W̃′, then the converse is not true anymore, but one could ask whether it would help to check whether supp(η′) = S for additional elements η′ ∈ W (depending on x), and specifically one could try to replace η_{2}(x) by a different element of W.

The following proposition gives a result in this direction, using the notion of P - alcove introduced in [5]. In the proposition, η_{2}(x) is replaced by an element of the form s_{α}η_{2}(x), where α depends on x. Recall that whenever x is a P -alcove sufficiently far away from the origin, then X_{x}(b) = ∅. Of course, the proposition is of particular interest, if ^{x}I ∩ U_{α} = I ∩ U_{α} (in particular x ∉ W̃′, and more specifically, x lies in the “critical strip” attached to α).

Proposition 4.1.1. Let x = vt^{μ}w ∈ W̃, w ∈ ^{I(μ)}W, let α be a finite root such that ^{x}I ∩ U_{α} ⊆ I ∩ U_{α}, and assume that −v^{−1}α is a simple root. If there exists j ∈ S such that

(s_{α}v)^{−1}η_{1}(x)(s_{α}v) ∈ W_{S\{j}},

then x is a P -alcove for P = ^{s_{α}v}P_{0}, P_{0} = M_{0}N_{0} the standard parabolic subgroup whose Levi component M_{0} is generated by S \ {j}.

Proof. As usual, we write P = MN for the Levi decomposition of P, where M is the Levi subgroup containing the fixed maximal torus. By definition of P, x ∈ W̃_{M}, so we have to show that ^{x}I ∩ U_{β} ⊆ I for every root β occurring in N. By assumption, −v^{−1}α is a simple root α_{i}. Denote by U the unipotent radical of the fixed Borel subgroup of G. Since the alcove v^{−1}x lies in the dominant chamber, we have^{v^{−1}x}I ∩U ⊆ I. Furthermore, by our normalization of I with respect to the dominant chamber, we have v(I ∩ U) ⊂ I, so altogether we obtain

xI ∩^{v}U ⊂ I.  (4)

The set R_{N} of roots occurring in N is

R_{N} = {vs_{i}γ = s_{α}vγ; γ a root in N_{0}}.

We distinguish two cases: If i ≠ j, i.e., s_{i} ∈ W_{M}, then s_{i} stabilizes the set of roots in N_{0}, and therefore R_{N} is the set of roots in ^{v}N_{0}. In this case our claim follows from (4). Now let us consider the case i = j, so that α_{i} ∈ R_{N_{0}}, and α = −vα_{i} = vs_{i}α_{i} ∈ R_{N}. For all β ∈ R_{N} \{α}, we have v^{−1}β > 0, so ^{x}I ∩U_{β} ⊂ I by (4), and finally we have ^{x}I ∩ U_{α} ⊂ I by assumption.

Even though the proposition yields examples of pairs (x, b) for which X_{x}(b) = ∅ although supp(η(x)) = S, it does not give rise to a sufficient criterion for nonemptiness in the critical strips, as can be shown by examples for G = SL_{4}.

5 An example of an affine Deligne-Lusztig variety which is not equidimensional

5.1

Looking again at the reduction method of Deligne and Lusztig, Corollary 2.5.3, we see that the situation in the affine case, in the shrunken Weyl chambers, is very much different from the classical situation: Whereas in the classical situation we always have dim X_{1} = dim X − 1, dim X_{2} = dim X (denoting by X the pertinent Deligne-Lusztig variety), in the affine shrunken case for the expected dimensions we have d(sxs) + 1 = d(x), d(sx) + 1 ⩽ d(x) — so the closed part X_{1}, if non-empty, always should have the same dimension as the affine Deligne-Lusztig variety itself, while the open part has at most this dimension. See the example below for a specific case where this inequality is strict and where one can produce examples of affine Deligne-Lusztig varieties which are not equidimensional.

5.2

To give an example of a non-equidimensional affine Deligne-Lusztig variety, we again use Corollary 2.5.3 (2). Let x = vt^{μ}w ∈ W_{a} be an element with μ dominant and very regular, and let s ∈ S such that

1. ℓ(sv) < ℓ(v), ℓ(ws) > ℓ(w),

2. ℓ(wsv) < ℓ(wv) − 1,

3. supp(wv) = supp(wsv) = S.

By (1), we have ℓ(sxs) = ℓ(x) − 2. We also have η(x) = η(sxs) = wv, η(sx) = wsv. Therefore as in Corollary 2.5.3 (2), we write X_{x}(1) = X_{1} ∪ X_{2}, where X_{1} is of relative dimension 1 over X_{sxs}(1), and X_{2} is of relative dimension 1 over X_{sx}(1). By the main result and assumption (3), we have

dim X_{2} = dim X_{sx}(1) + 1 = d(sx) + 1 < d(x) = dim X_{x}(1),

where the < in the second line holds because of (2) and Lemma 3.2.2. (We also see that dim X_{1} = dim X.) Since X_{2} is open in X_{x}(1) and has strictly smaller dimension, X_{x}(1) cannot be equidimensional. It remains to find elements v, w, and s that satisfy (1)–(3). But this is easy, for instance for type A_{3}, we can take

v = s_{1}s_{2},  w = s_{1}s_{2}s_{3}s_{2},  s = s_{1},

so that

wv = s_{1}s_{2}s_{3}s_{2}s_{1}s_{2}, wsv = s_{1}s_{2}s_{3}.

References

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Ulrich Görtz Institut für Experimentelle Mathematik Universität Duisburg-Essen Ellernstr. 29 45326 Essen Germany ulrich.goertz@uni-due.de Xuhua He Department of Mathematics The Hong Kong University of Science and Technology Clear Water Bay Kowloon Hong Kong maxhhe@ust.hk