DIMENSIONS OF SOME AFFINE DELIGNE-LUSZTIG VARIETIES

ULRICH GÖRTZ, THOMAS J. HAINES, ROBERT E. KOTTWITZ, AND DANIEL C. REUMAN

1. Introduction

Let k be a finite field with q elements, and let k̄ be an algebraic closure of k. We consider the field L := k̄((ϵ)) and its subfield F := k((ϵ)). We write σ : x ↦ x^{q} for the Frobenius automorphism of k̄/k, and we also regard σ as an automorphism of L/F in the usual way, so that σ(∑ a_{n}ϵ^{n}) = ∑ σ(a_{n})ϵ^{n}. We write 𝔬 for the valuation ring k̄[[ϵ]] of L.

Let G be a split connected reductive group over k, and let A be a split maximal torus of G. Put 𝔞 := X_{∗}(A)_{ℝ}. Write W for the Weyl group of A in G. Fix a Borel subgroup B = AU containing A with unipotent radical U. For λ ∈ X_{∗}(A) we write ϵ^{λ} for the element of A(F) obtained as the image of ϵ ∈ 𝔾_{m}(F) under the homomorphism λ : 𝔾_{m} → A.

This paper concerns the dimensions of certain affine Deligne-Lusztig varieties, both in the affine Grassmannian and in the affine flag manifold. We begin with the affine Grassmannian.

Put K := G(𝔬). We denote by X the affine Grassmannian X = G(L)/K and by x_{0} its obvious base-point. The group G(L) acts by left translation on X. By the Cartan decomposition G(L) is the disjoint union of the subsets Kϵ^{μ}K, with μ running over the dominant elements in X_{∗}(A). For b ∈ G(L) and a dominant coweight μ ∈ X_{∗}(A) the affine Deligne-Lusztig variety X_{μ}(b) = X_{μ}^{G}(b) is the locally closed subset of X defined by

(1.0.1)  X_{μ}(b) := {x ∈ G(L)/K : x^{−1}bσ(x) ∈ Kϵ^{μ}K}.

For g ∈ G(L) it is clear that x ↦ gx yields an isomorphism X_{μ}(b) → X_{μ}(gbσ(g)^{−1}), so the isomorphism class of X_{μ}(b) depends only on the σ-conjugacy class of b.

Let 𝔻 be the diagonalizable group over F with character group ℚ. Just as in [Kot85], the element b determines a homomorphism ν_{b} : 𝔻 → G over L, and b is said to be basic if ν_{b} factors through the center of G. For g ∈ G(L) we have

(1.0.2)  ν_{gbσ(g)^{−1}} = Int(g) ∘ ν_{b},

where Int(g) denotes the inner automorphism x ↦ gxg^{−1} of G over L, and since it is harmless to σ-conjugate b, we may as well assume that ν_{b} factors through A, and that the corresponding element ν̄_{b} ∈ X_{∗}(A)_{ℚ} ⊂ 𝔞 is dominant. Following [RR96] we refer to ν̄_{b} as the Newton point of b. The centralizer M_{b} of ν_{b} in G is then a Levi subgroup of G over F.

1991 Mathematics Subject Classification. Primary 14L05; Secondary 11S25, 20G25, 14F30.

Haines was partially supported by NSF Grant DMS-0303605 and a Sloan Research Fellowship.

Kottwitz was partially supported by NSF Grant DMS-0245639.

Reuman was supported by NSF grants DEB-9981552 and DMS-0443803.

Just as in [Kot85], [RZ96, 1.12], [Kot97], there is an inner form J of M_{b} whose R-valued points (for any F -algebra R) are given by

J(R) = {g ∈ G(R ⊗_{F} L) : g^{−1}bσ(g) = b}.

The group J(F) acts by left multiplication on X_{μ}(b). Note that J(F) ⊂ M_{b}(L) because of (1.0.2).

Let Λ_{G} denote the quotient of X_{∗}(A) by the coroot lattice for G. We denote by p_{G} the canonical surjection X_{∗}(A) ↠ Λ_{G}. There is a canonical homomorphism η_{G} : G(L) ↠ Λ_{G}, which is trivial on K = G(𝔬) and hence induces a surjection, also denoted η_{G}, from X = G(L)/K to Λ_{G}. The fibers of η_{G} : X ↠ Λ_{G} are the connected components of the affine Grassmannian X. As in [Kot97, 7.7], the restriction of the homomorphism η_{M_{b}} : M_{b}(L) ↠ Λ_{M_{b}} to J(F) is surjective, which implies that the restriction of η_{G} : G(L) ↠ Λ_{G} to J(F) is also surjective. Using the action of J(F) on X_{μ}(b), we then see that the intersections X_{μ}(b) ∩ η_{G}^{−1}(λ) (λ ∈ Λ_{G}) of X_{μ}(b) with the various connected components of X are all isomorphic to each other.

As in [KR] (see also [Lei02], [Kot03]), there is a simple criterion for X_{μ}(b) to be non-empty (see Proposition 5.6.1). Here we mention only that when b is basic, X_{μ}(b) is non-empty if and only if η_{G}(b) = p_{G}(μ). When X_{μ}(b) is non-empty, there is a conjectural formula for its dimension, due to Rapoport [Rap02], which we now recall, reformulating it slightly (see [Kot05]) along the way.

This formula involves a non-negative integer def_{G}(b) attached to b. By definition def_{G}(b) is the F -rank of G minus the F -rank of J. Clearly def_{G}(b) depends only on the σ-conjugacy class of b. (As usual the F -rank of G is the common dimension of all maximal F -split tori in G.) We write ρ ∈ X^{∗}(A)_{ℚ} for the half-sum of the positive roots.

Conjecture 1.0.1 (Rapoport). Assume that X_{μ}(b) is non-empty. Then its dimension is given by

dim X_{μ}(b) = ⟨ρ, μ − ν̄_{b}⟩ − \frac{1}{2} def_{G}(b).

In [Reu04] Reuman proves the conjecture for G = SL_{2}, SL_{3}, Sp_{4} and b = 1. In [Mie] Mierendorff proves (in the context of ℚ_{p} rather than k((ϵ))) that Rapoport’s conjecture is true for GL_{n} and minuscule μ.

In this paper we prove Rapoport’s conjecture for all b ∈ A(L) (see Theorem 2.15.1, noting that def_{G}(b) = 0 when b ∈ A(L)), and in fact we show in this case that X_{μ}(b) is equidimensional (see Proposition 2.17.1), answering a question of Rapoport. Moreover, returning to general elements b ∈ G(L), we show in Theorem 5.8.1 that, if M is a Levi subgroup of G such that b ∈ M(L) and b is basic in M(L), and if Rapoport’s conjecture holds for (M, b), then Rapoport’s conjecture holds for (G, b).

Consequently, in order to prove Rapoport’s conjecture in general, it would be enough to prove it for superbasic elements b, by which we mean those for which no σ-conjugate is contained in a proper Levi subgroup of G. As we verify in 5.9, superbasic elements are very special, and for simple groups they exist only in type A_{n}. The upshot is that it would be enough to prove Rapoport’s conjecture for basic elements b ∈ GL_{n}(L) such that the valuation of det(b) is relatively prime to n.

Now we turn to affine Deligne-Lusztig varieties inside the affine flag manifold G(L)/I, where I is the Iwahori subgroup of G(L) obtained from an alcove a_{1} in the apartment associated to A. Given b ∈ G(L) and an element x ∈ W̃ = W ⋉ X_{∗}(A), we get the affine Deligne-Lusztig variety

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

For the groups G = SL_{2}, SL_{3}, Sp_{4} and b = 1 the dimension of X_{x}(b) was computed by Reuman [Reu04]. For suitably general x (those in the union of the “shrunken Weyl chambers”) Reuman gives a simple formula for dim X_{x}(b) and conjectures that it holds in general (for b = 1 and suitably general x).

In this article we prove a formula (see Theorem 6.3.1) for dim X_{x}(b) when b ∈ A(L). Unfortunately this formula does not suffice to establish Reuman’s conjecture, since it involves the unknown dimension of the intersection of I- and U(L)- orbits in the affine flag manifold. (The reason we had better luck with the affine Grassmannian is that the dimensions of intersections of G(𝔬)- and U(L)-orbits in the affine Grassmannian are known, thanks to Mirković-Vilonen [MV1].)

However there is an algorithm for computing the dimensions of such intersections, and in section 7 we describe the results of computer calculations made using this algorithm. Reuman’s conjecture (see subsection 7.2) turns out to hold in all cases checked by the computer. In the case of rank 2 groups, the results can be presented in the form of pictures. Figures 1,2,3 show the dimensions for b = 1 and A_{2}, C_{2}, G_{2} respectively, while figures 11,12 show dimensions for two elements b ∈ A(L), b ≠ 1, one for type A_{2}, and one for type C_{2}.

The results in these last two figures support Conjecture 7.5.1, an extension of Reuman’s conjecture to elements b ∈ A(L). We finish this introduction by mentioning that there is mounting evidence that an analog of Reuman’s conjecture holds for all b ∈ G(L) (for suitably general x ∈ W̃).

2. Affine Deligne-Lusztig varieties inside the affine Grassmannian

2.1. Further preliminaries. For ν in X_{∗}(A) or 𝔞 we write ν_{dom} for the unique dominant element in the W -orbit of ν. For cocharacters μ, ν ∈ X_{∗}(A) we say that ν ≤ μ if μ − ν is a non-negative integral linear combination of positive coroots.

Any b ∈ A(L) is σ-conjugate to an element of the form ϵ^{ν}. By [KR] the set X_{μ}(ϵ^{ν}) is non-empty if and only if ν_{dom} ≤ μ; we now assume that this is the case. We are going to calculate the dimension of X_{μ}(ϵ^{ν}), using the obvious fact that X_{μ}(ϵ^{ν}) is preserved by the action of A(F).

2.2. Topology on X. We view X as an ind-scheme in the usual way. Each Korbit on X is finite dimensional, and we denote by Z_{n} the union of all K-orbits having dimension less than or equal to n. Each Z_{n} is a projective variety, and the increasing family Z_{0} ⊂ Z_{1} ⊂ Z_{2} ⊂ Z_{3} ⊂ … exhausts X. We put the direct limit topology on X, so that a subset Y of X is closed (respectively, open) if and only if for all n the intersection Y ∩Z_{n} is closed (respectively, open) in the Zariski topology on Z_{n}. Each Z_{n} is closed in X. If Y is locally closed in X, then each intersection Y ∩ Z_{n} is locally closed in Z_{n}.

2.3. Dimensions of locally closed subsets of X. We write Z for the family of subsets Z of X for which there exists n such that Z is a closed subset of Z_{n}. Each Z ∈ Z is a projective variety, and the family Z is stable under the action of G(L). For any locally closed subset Y of X we put

(2.3.1)  dim(Y) := sup{dim(Y ∩ Z) : Z ∈ Z}.

Of course dim(Y) might be +∞, as happens for example when Y = X. Clearly dim(gY) = dim(Y) for all g ∈ G(L).

2.4. U(L)-orbits on X. For any λ ∈ X_{∗}(A) we put x_{λ} := ϵ^{λ}x_{0} ∈ X. We write X_{λ} for the U(L)-orbit of x_{λ}; by the Iwasawa decomposition we have

(2.4.1)  X = ∐_{λ∈X_{∗}(A)} X_{λ},

set-theoretically. The sets

(2.4.2)  X_{≤λ} := ∐_{λ′:λ′≤λ} X_{λ′}

are closed in X, and the U(L)-orbits X_{λ} are locally closed in X. Taking λ = 0 we get X_{0} = U(L)/U(𝔬).

2.5. Dimensions of A(F)-stable locally closed subsets of X. Let Y be an A(F)-stable locally closed subset of X. We claim that

(2.5.1)  dim(Y) = dim(Y ∩ X_{λ})

for any λ ∈ X_{∗}(A). Clearly X_{λ} = ϵ^{λ}X_{0}, and because Y is A(F)-stable and hence satisfies Y = ϵ^{λ}Y, we have Y ∩ X_{λ} = ϵ^{λ}(Y ∩ X_{0}), so that

(2.5.2)  dim(Y ∩ X_{λ}) = dim(Y ∩ X_{0})

for all λ. Thus we need only show that

(2.5.3)  dim(Y) = dim(Y ∩ X_{0}).

The inequality dim(Y ∩ X_{0}) ≤ dim(Y) is clear. For the reverse inequality we must show that

(2.5.4)  dim(Y ∩ Z) ≤ dim(Y ∩ X_{0})

for any Z ∈ Z. It is well-known (and easy) that there exists a finite subset S of X_{∗}(A) such that Z is contained in

⋃_{λ∈S} X_{λ}.

Therefore

dim(Y ∩ Z) = sup{dim(Y ∩ Z ∩ X_{λ}) : λ ∈ S}

≤ sup{dim(Y ∩ X_{λ}) : λ ∈ S}. (2.5.5)

Combining (2.5.5) and (2.5.2), we see that dim(Y ∩ Z) ≤ dim(Y ∩ X_{0}), as desired.

Since X_{μ}(ϵ^{ν}) is A(F)-stable, the remarks above show that its dimension is the same as that of

X_{μ}(ϵ^{ν}) ∩ X_{0} = {u ∈ U(L)/U(𝔬) : u^{−1}ϵ^{ν}σ(u) ∈ Kϵ^{μ}K}.

2.6. Root subgroups of U. For any positive root α we write U_{α} for the root subgroup of U corresponding to α. Enumerate the positive roots (in any order) as α_{1}, …, α_{r}. Then, as is well-known, the map (u_{1}, …, u_{r}) ↦ u_{1} … u_{r} is an isomorphism

∏_{i=1}^{r} U_{α_{i}} → U

of algebraic varieties over k. We now fix, for each positive root α, an isomorphism U_{α} ≃ 𝔾_{a} over k. Thus we may identify U(L) with L^{r} and U(𝔬) with 𝔬^{r}.

2.7. Subgroups U_{n} of U(𝔬). For any n ≥ 0 the ring homomorphism 𝔬 ↠ 𝔬/ϵ^{n}𝔬 induces a surjective group homomorphism U(𝔬) ↠ U(𝔬/ϵ^{n}𝔬), whose kernel we denote by U_{n}. Thus we have a descending chain U(𝔬) = U_{0} ⊃ U_{1} ⊃ U_{2} ⊃ … of normal subgroups of U(𝔬). Under our identification of U(𝔬) with 𝔬^{r} the subgroup U_{n} becomes identified with (ϵ^{n}𝔬)^{r}.

2.8. Subgroups U(m) of U(L). We now fix a dominant regular coweight λ_{0} ∈ X_{∗}(A); thus ⟨α, λ_{0}⟩ > 0 for every positive root α of A. Put a := ϵ^{λ_{0}}, and then for m ∈ ℤ define a subgroup U(m) of U(L) by

U(m) := a^{m}U(𝔬)a^{−m}.

There is a chain of inclusions

⋯ ⊃ U(−2) ⊃ U(−1) ⊃ U(0) ⊃ U(1) ⊃ U(2) ⊃ ⋯

and moreover

U(L) = ⋃_{m∈ℤ} U(m).

Under our identification of U(L) with L^{r}, the subgroup U(m) becomes identified with

∏_{i=1}^{r} ϵ^{m⟨α_{i},λ_{0}⟩}𝔬.

Clearly the filtrations U_{0} ⊃ U_{1} ⊃ U_{2} ⊃ … and U(0) ⊃ U(1) ⊃ U(2) ⊃ … define the same topology on U(𝔬).

2.9. Dimensions of locally closed subsets Y of X_{0}. Recall that

X_{0} = U(L)/U(𝔬).

Therefore X_{0} can be written as the increasing union

X_{0} = ⋃_{m≥0} U(−m)/U(0)

of its closed subspaces U(−m)/U(0).

Now consider a locally closed subset Y of X_{0}. Then Y is locally closed in X and by definition its dimension is sup{dim(Y ∩ Z) : Z ∈ Z}. It is clear that for any Z ∈ Z there exists m ≥ 0 such that Z ∩ X_{0} ⊂ U(−m)/U(0). It is equally clear that for any m ≥ 0 there exists Z ∈ Z such that U(−m)/U(0) ⊂ Z ∩ X_{0}. We conclude that

(2.9.1)  dim(Y) = sup{dim Y ∩ (U(−m)/U_{0}) : m ≥ 0}.

We of course are particularly interested in the dimension of the locally closed subset

(2.9.2)  Y_{μ,ν} := X_{μ}(ϵ^{ν}) ∩ X_{0} = {u ∈ U(L)/U(𝔬) : u^{−1}ϵ^{ν}σ(u) ∈ Kϵ^{μ}K},

defined for any coweights μ, ν with μ dominant. In other words

(2.9.3)  Y_{μ,ν} = f_{ν}^{−1}(Kϵ^{μ}Kϵ^{−ν} ∩ U(L))/U(𝔬),

where f_{ν} is the map U(L) → U(L) defined by f_{ν}(u) := u^{−1}ϵ^{ν}σ(u)ϵ^{−ν}.

2.10. Dimensions for admissible and ind-admissible subsets of U(L). In view of (2.9.3) we see that it would be useful to introduce a notion of dimension for suitable subsets V of U(L) and then to compute the dimension of f_{ν}^{−1}V in terms of that of V.

For m, n ≥ 0 the quotient U(−m)/U(n) is the set of k̄-points of an algebraic variety over k. We say that a subset V of U(−m) is admissible if there exists n ≥ 0 and a locally closed subset V′ of U(−m)/U(n) such that V is the full inverse image of V′ under U(−m) ↠ U(−m)/U(n). We say that a subset V of U(L) is admissible if there exists m ≥ 0 such that V is an admissible subset of U(−m). We say that a subset V of U(L) is ind-admissible if V ∩ U(−m) is admissible for all m ≥ 0. Obviously admissible subsets are also ind-admissible.

For any admissible subset V of U(L) we choose n ≥ 0 such that V is preserved by right multiplication by U(n) and then put

(2.10.1)  dim V := dim(V/U(n)) − dim(U(0)/U(n));

this is clearly independent of the choice of n. In this definition we could equally well have used the subgroups U_{n} instead of U(n). Clearly dim U(𝔬) = 0, and of course dim V can be negative.

For any ind-admissible subset V of U(L) we put

(2.10.2)  dim V := sup{dim V ∩ U(−m) : m ≥ 0}.

Of course dim(V) can be +∞, as happens in case V = U(L).

2.11. Warm-up exercise and key proposition. The following familiar lemma is what makes the next proposition work.

Lemma 2.11.1. Let a, b be non-negative integers and consider the group homomorphism f : 𝔬 → 𝔬 defined by f(x) := ϵ^{a}σ(x) − ϵ^{b}x. Then the image of f is the subgroup ϵ^{c}𝔬, where c := min{a, b}.

Proof. Clearly it is enough to treat the case in which c = 0. Then at least one of a, b is zero. If (a, b) ≠ (0, 0), then f preserves the filtration

𝔬 ⊃ ϵ𝔬 ⊃ ϵ^{2}𝔬 ⊃ …

of 𝔬 and induces an isomorphism on the associated graded group. Therefore f is bijective in this case.

If (a, b) = (0, 0), then f preserves the filtration above and induces on each successive quotient k̄ the map x ↦ σ(x) − x = x^{q} − x, which is surjective since k̄ is algebraically closed. Therefore f maps 𝔬 onto 𝔬. □

Now we come to the key proposition. It involves two dominant coweights ν and ν′. Define homomorphisms ϕ, ψ : U(L) → U(L) by ϕ(u) := ϵ^{ν′}uϵ^{−ν′} and ψ(u) = ϵ^{ν}σ(u)ϵ^{−ν}. The dominance of ν, ν′ implies that both ϕ, ψ preserve the normal subgroups U_{n} of U(𝔬) and hence induce homomorphisms ϕ_{n}, ψ_{n} : U(𝔬/ϵ^{n}𝔬) → U(𝔬/ϵ^{n}𝔬).

We use ϕ, ψ to define a right action, denoted ∗, of U(𝔬) on itself: the action of an element u ∈ U(𝔬) upon an element u′ ∈ U(𝔬) is given by u′ ∗ u := ϕ(u)^{−1}u′ψ(u). Similarly, we use ϕ_{n}, ψ_{n} to define a right action, again denoted ∗, of U(𝔬/ϵ^{n}𝔬) on itself: the action of an element u ∈ U(𝔬/ϵ^{n}𝔬) upon an element u′ ∈ U(𝔬/ϵ^{n}𝔬) is given by u′ ∗ u := ϕ_{n}(u)^{−1}u′ψ_{n}(u).

We regard U(𝔬/ϵ^{n}𝔬) as the set of k̄-points of an algebraic group over k in the usual way. In particular the underlying variety is simply an affine space of dimension n dim U. In the next proposition the dimensions are those of varieties over k̄. In the proposition, and throughout the paper, we write sums over the set of positive roots as ∑_{α>0}.

Proposition 2.11.2. Let ν and ν′ be dominant coweights, and let n be a nonnegative integer large enough that

n ≥ min{⟨α, ν⟩, ⟨α, ν′⟩}

for every positive root α. Then we have:

(1) The codimension of the U(𝔬/ϵ^{n}𝔬)-orbit of 1 ∈ U(𝔬/ϵ^{n}𝔬) is equal to

∑_{α>0} min{⟨α, ν⟩, ⟨α, ν′⟩},

which in turn is equal to the dimension of the stabilizer S_{n} of 1 ∈ U(𝔬/ϵ^{n}𝔬) in U(𝔬/ϵ^{n}𝔬). Note that S_{n} = {u ∈ U(𝔬/ϵ^{n}𝔬) : ϕ_{n}(u) = ψ_{n}(u)}.

(2) The U(𝔬)-orbit of 1 ∈ U(𝔬) coincides with the inverse image under U(𝔬) ↠ U(𝔬/ϵ^{n}𝔬) of the U(𝔬/ϵ^{n}𝔬)-orbit of 1 ∈ U(𝔬/ϵ^{n}𝔬). In particular the U(𝔬)- orbit of 1 ∈ U(𝔬) contains U_{n}.

Proof. To simplify notation we sometimes write H for the group U(𝔬/ϵ^{n}𝔬). We begin by proving the first statement. The codimension of the H-orbit of 1 ∈ H is

dim H − dim(H/S_{n}) = dim S_{n}.

Under our identification 𝔾_{a}^{r} ≃ U, the subgroup S_{n} of U(𝔬/ϵ^{n}𝔬) goes over to

∏_{i=1}^{r}{x ∈ 𝔬/ϵ^{n}𝔬 : ϵ^{⟨α_{i},ν′⟩}x = ϵ^{⟨α_{i},ν⟩}σ(x)},

and the dimension of the i-th factor in this product is equal to the codimension of the image of the homomorphism x ↦ ϵ^{⟨α_{i},ν⟩}σ(x) − ϵ^{⟨α_{i},ν′⟩}x from 𝔬/ϵ^{n}𝔬 to itself, and by Lemma 2.11.1 (and our hypothesis on n) this codimension is obviously min{⟨α_{i}, ν⟩, ⟨α_{i}, ν′⟩}.

Now we prove the second statement. For u ∈ U(𝔬) we write ū for the image of u under U(𝔬) ↠ H. It is clear that the ∗-actions on U(𝔬) and H are compatible, in the sense that

(2.11.1)  \overline{u ∗ u′} = ū ∗ ū′,

from which it follows that U(𝔬) ↠ H maps the U(𝔬)-orbit of 1 ∈ U(𝔬) into the H-orbit of 1 ∈ H. We must prove that if u ∈ U(𝔬) has the property that ū lies in the H-orbit of 1, then u lies in the U(𝔬)-orbit of 1. Replacing u by u ∗ u′ for suitable u′ ∈ U(𝔬), we may assume that ū = 1. In other words it remains only to show that any u ∈ U_{n} lies in the U(𝔬)-orbit of 1 ∈ U(𝔬). Some care is needed since the ∗-action of U(𝔬) on itself does not preserve the subgroup U_{n}.

Until now we have been working with any ordering α_{1}, …, α_{r} of the positive roots. Now let us order them so that ht(α_{1}) ≤ ht(α_{2}) ≤ ⋯ ≤ ht(α_{r}), where ht(α) is the number of simple roots needed in order to write the positive root α as a sum of simple roots. We then get a decreasing chain

U = U[1] ⊃ U[2] ⊃ U[3] ⊃ ⋯ ⊃ U[r + 1] = {1}

of normal subgroups U[j] of U, with U[j] defined as the subgroup consisting of elements whose projections onto the first j − 1 root subgroups U_{α_{1}}, …, U_{α_{j−1}} are all equal to 1.

Now return to our element u ∈ U_{n}, and let u_{1} denote the image of u in U[1](𝔬)/U[2](𝔬) = U_{α_{1}}(𝔬). By Lemma 2.11.1 (and our hypothesis on n) we may choose v ∈ U_{α_{1}}(𝔬) so that u_{1} ∗ v = 1. It follows that u ∗ v ∈ U[2](𝔬). We claim that u ∗ v ∈ U_{n}. For this we must check that \overline{u ∗ v} = 1, which is clear since

\overline{u ∗ v} = ū ∗ v̄ = 1 ∗ v̄ = ū_{1} ∗ v̄ = \overline{u_{1} ∗ v} = 1.

Replacing u by u ∗ v, we may now assume that u ∈ U[2](𝔬) ∩ U_{n}. Using U_{α_{2}} the same way we just used U_{α_{1}}, we may push u down into U[3](𝔬) ∩ U_{n}. Continuing in this way, we eventually end up with u ∈ U[r + 1](𝔬) = {1}, at which point we are done. □

2.12. Formula for the dimension of f_{ν,ν′}^{−1}V. Now let ν, ν′ ∈ X_{∗}(A) and define a map f_{ν,ν′} : U(L) → U(L) by

(2.12.1)  f_{ν,ν′}(u) := (ϵ^{ν′}uϵ^{−ν′})^{−1} · (ϵ^{ν}σ(u)ϵ^{−ν}) = ϵ^{ν′}u^{−1}ϵ^{ν−ν′}σ(u)ϵ^{−ν}.

Note that when ν′ = 0, we get back the map f_{ν} considered earlier. Moreover, when ν, ν′ are both dominant and u ∈ U(𝔬), we have f_{ν,ν′}(u) = 1 ∗ u, with ∗ denoting the action of U(𝔬) on itself introduced in 2.11.

Now let λ ∈ X_{∗}(A). We will also need the conjugation map c_{λ} : U(L) → U(L) defined by

(2.12.2)  c_{λ}(u) := ϵ^{λ}uϵ^{−λ}.

Proposition 2.12.1. Let λ, ν, ν′ ∈ X_{∗}(A). Then we have:

(1) f_{ν,ν′} c_{λ} = f_{λ+ν,λ+ν′} = c_{λ} f_{ν,ν′}.

(2) Let V be an admissible (respectively, ind-admissible) subset of U(L). Then c_{λ}^{−1}V is admissible (respectively, ind-admissible) and

dim c_{λ}^{−1}V = dim V + ∑_{α>0}⟨α, λ⟩.

(3) Let V be an admissible subset of U(L). Then f_{ν,ν′}^{−1}V is ind-admissible and

dim f_{ν,ν′}^{−1}V = dim V + ∑_{α>0} min{⟨α, ν⟩, ⟨α, ν′⟩}.

Proof. The first statement of the proposition is an easy calculation. Next we prove the second statement. Assume that V is an admissible subset of U(L), and pick n big enough that V is stable under right multiplication by U_{n}. Since c_{λ} induces an isomorphism from the variety c_{λ}^{−1}V/c_{λ}^{−1}U_{n} to the variety V/U_{n}, we see that

dim c_{λ}^{−1}V − dim c_{λ}^{−1}U_{n} = dim V − dim U_{n},

from which it follows that it is enough to prove the second statement of the proposition for V = U_{n}. Using the root subgroups U_{α}, we are reduced to the obvious fact that

dim ϵ^{a}𝔬/ϵ^{b}𝔬 = b − a.

Finally we prove the third statement of the proposition. Using the first two parts of the proposition, one sees easily that the third part is true for (ν, ν′) if and only if it is true for (ν + λ, ν′ + λ). Therefore we may assume without loss of generality that both ν, ν′ are dominant. From now on we abbreviate f_{ν,ν′} to f. Note that the dominance of ν, ν′ implies that f(U_{0}) ⊂ U_{0}; indeed, for u ∈ U_{0} we have f(u) = 1 ∗ u in the notation of 2.11, so that f(U_{0}) coincides with the U_{0}-orbit of 1 ∈ U_{0}, something we have analyzed in Proposition 2.11.2. We write f_{0} for the map f_{0} : U_{0} → U_{0} obtained by restriction from f.

As in Proposition 2.11.2 we now choose a non-negative integer n_{0} large enough that

n_{0} ≥ min{⟨α, ν⟩, ⟨α, ν′⟩}

for every positive root α. Using the first two parts of Proposition 2.12.1 again, we see that the third part of the proposition is true for V if and only if it is true for ϵ^{λ}V ϵ^{−λ}. Therefore we may assume without loss of generality that V ⊂ U_{n_{0}}.

Under this assumption we are going to prove that for all m ≥ 0 the intersection U(−m) ∩ f^{−1}V is admissible of dimension

dim V + ∑_{α>0} min{⟨α, ν⟩, ⟨α, ν′⟩},

which will be enough to prove the proposition.

Recall that U(−m) = ϵ^{−mλ_{0}}U_{0}ϵ^{mλ_{0}}. Once again using the first two parts of this proposition, we see that

U(−m) ∩ f^{−1}V = ϵ^{−mλ_{0}}(U_{0} ∩ f^{−1}(ϵ^{mλ_{0}}V ϵ^{−mλ_{0}}))ϵ^{mλ_{0}}

and that

dim U(−m) ∩ f^{−1}V − dim V = dim U_{0} ∩ f^{−1}(ϵ^{mλ_{0}}V ϵ^{−mλ_{0}}) − dim ϵ^{mλ_{0}}V ϵ^{−mλ_{0}}.

Since ϵ^{mλ_{0}}V ϵ^{−mλ_{0}} is still contained in U_{n_{0}}, we may without loss of generality take m = 0.

Thus we are now reduced to proving that if V ⊂ U_{n_{0}}, then f_{0}^{−1}V is admissible of dimension

dim V + ∑_{α>0} min{⟨α, ν⟩, ⟨α, ν′⟩}.

Choose n ≥ n_{0} such that V is invariant under right multiplication by U_{n}, and once again put H := U_{0}/U_{n}. Then V is obtained as the inverse image under

p : U_{0} ↠ H

of a locally closed subset V̄ of H contained in U_{n_{0}}/U_{n} ⊂ H.

Writing f̄ for the map H → H defined by f̄(h) := 1 ∗ h, we have pf_{0} = f̄p, from which it follows that

f_{0}^{−1}V = p^{−1}f̄^{−1}V̄ .

In particular f_{0}^{−1}V is admissible of dimension

dim U_{n} + dim f̄^{−1}V̄ .

Recall from before that S_{n} denotes the stabilizer group-scheme of 1 ∈ H for the ∗-action of H on itself. Now the morphism f̄ induces an isomorphism from the fpqc quotient S_{n}\H to the H-orbit of 1 ∈ H (see Lemma 2.17.2), and by the second part of Proposition 2.11.2, applied to n_{0}, we have

V̄ ⊂ U_{n_{0}}/U_{n} ⊂ f̄(H).

Therefore by Lemma 2.17.2

dim f̄^{−1}V̄ = dim V̄ + dim S_{n},

from which it follows that

dim f_{0}^{−1}V = dim U_{n} + dim V̄ + dim S_{n} = dim V + dim S_{n}.

Using the formula for dim S_{n} given in the first part of Proposition 2.11.2, we see that

dim f_{0}^{−1}V = dim V + ∑_{α>0} min{⟨α, ν⟩, ⟨α, ν′⟩},

as desired. □

2.13. Formula for ⟨ρ, ν − ν_{dom}⟩. Let ρ ∈ X^{∗}(A)_{ℚ} be the half-sum of the positive roots of A. Recall that ν_{dom} denotes the unique dominant element in the W -orbit of ν.

Lemma 2.13.1. For any ν ∈ 𝔞 there is an equality

(2.13.1)  ⟨ρ, ν − ν_{dom}⟩ = ∑_{α>0} min{⟨α, ν⟩, 0}.

Proof. The lemma follows immediately from the equations

(2.13.2)  ⟨ρ, ν⟩ = ∑_{α>0}⟨α, ν⟩/2

and

(2.13.3)  ⟨ρ, ν_{dom}⟩ = ∑_{α>0} |⟨α, ν⟩|/2.
□

2.14. Mirković-Vilonen dimension formula. Recall that X_{ν} denotes the U(L)- orbit of x_{ν} = ϵ^{ν}x_{0} in X. For dominant μ ∈ X_{∗}(A) we also consider the K-orbit X^{μ} := Kx_{μ} of x_{μ} in X. We are interested in the intersection X^{μ} ∩ X_{ν}. We assume that ν_{dom} ≤ μ, since otherwise the intersection is empty (see [BT72, 4.4.4]). As before ρ is the half-sum of the positive roots. Then we have

Proposition 2.14.1 (Mirković-Vilonen). For any dominant μ ∈ X_{∗}(A) and any ν ∈ X_{∗}(A) such that ν_{dom} ≤ μ there is an equality

(2.14.1)  dim X^{μ} ∩ X_{ν} = ⟨ρ, μ + ν⟩.

This is proved in [MV1, MV2]. Another proof in the present context of finite residue fields is given in section 5, see Remark 5.5.

We will need the following reformulation of their result.

Proposition 2.14.2. For any dominant μ ∈ X_{∗}(A) and any ν ∈ X_{∗}(A) such that ν_{dom} ≤ μ there is an equality

(2.14.2)  dim Kϵ^{μ}Kϵ^{−ν} ∩ U(L) = ⟨ρ, μ − ν⟩.

Proof. The variety X^{μ} ∩ X_{ν} is isomorphic to the quotient

(Kϵ^{μ}Kϵ^{−ν} ∩ U(L))/ϵ^{ν}U_{0}ϵ^{−ν},

and Kϵ^{μ}Kϵ^{−ν} ∩ U(L) is obviously an admissible subset of U(L). Thus, according to our discussion of dimension theory for admissible subsets of U(L), we have

(2.14.3)  dim X^{μ} ∩ X_{ν} = dim Kϵ^{μ}Kϵ^{−ν} ∩ U(L) − dim ϵ^{ν}U_{0}ϵ^{−ν}.

From the second part of Proposition 2.12.1 we obtain

(2.14.4)  dim ϵ^{ν}U_{0}ϵ^{−ν} = −2⟨ρ, ν⟩.

Combining (2.14.1), (2.14.3), (2.14.4), we get the desired result. □

2.15. Main result. Now we are ready to calculate the dimension of X_{μ}(ϵ^{ν}). Recall that ρ denotes the half-sum of the positive roots of A.

Theorem 2.15.1. Let μ, ν be coweights of A. Assume that μ is dominant and that ν_{dom} ≤ μ. Then

dim X_{μ}(ϵ^{ν}) = ⟨ρ, μ − ν_{dom}⟩.

Proof. In 2.5 we saw that

(2.15.1)  dim X_{μ}(ϵ^{ν}) = dim Y_{μ,ν},

where Y_{μ,ν} denotes the intersection X_{μ}(ϵ^{ν}) ∩ X_{0}. From (2.9.3) we have

(2.15.2)  Y_{μ,ν} = f_{ν}^{−1}(Kϵ^{μ}Kϵ^{−ν} ∩ U(L))/U(𝔬).

Applying Proposition 2.12.1 with ν′ = 0 (so that f_{ν,ν′} = f_{ν}), we see that

(2.15.3)  dim Y_{μ,ν} = dim Kϵ^{μ}Kϵ^{−ν} ∩ U(L) + ∑_{α>0} min{⟨α, ν⟩, 0}.

Using the Mirković-Vilonen dimension formula (Proposition 2.14.2) and Lemma 2.13.1, we obtain

(2.15.4)  dim Y_{μ,ν} = ⟨ρ, μ − ν⟩ + ⟨ρ, ν − ν_{dom}⟩ = ⟨ρ, μ − ν_{dom}⟩,

as desired. □

2.16. Remark. In the last proof we only used Proposition 2.12.1 in the special case ν′ = 0. The reason we took arbitrary ν′ in Proposition 2.12.1 was to allow a reduction to the case of dominant coweights in the proof of that proposition. When ν′ = 0 and ν is dominant, this reduction step is not needed, and since X_{μ}(ϵ^{ν}) depends only on the σ-conjugacy class of ϵ^{ν}, hence only on the W -orbit of ν, it would have been enough to consider only dominant ν.

Thus one might think it a waste of effort to have formulated and proved Proposition 2.12.1 in the generality that we did. However, later in the paper, when we switch from the affine Grassmannian to the affine flag manifold, we will need Proposition 2.12.1 for ν′ = 0 and arbitrary ν. For non-dominant ν no simplification occurs when ν′ = 0, and the most natural formulation of Proposition 2.12.1 seems to be the one we have given.

2.17. Equidimensionality. Next we will prove that X_{μ}(ϵ^{ν}) is equidimensional, in a sense we will make precise below. The equidimensionality follows easily from the corresponding fact about the intersections X^{μ} ∩ X_{ν} due to Mirković-Vilonen (see Lemma 2.17.4), and from a basic lemma concerning fpqc (or fppf) quotients of algebraic groups by stabilizer subgroup schemes.

We say an ind-admissible set Y ⊂ U(L) is irreducible if for all m ≥ 0, the intersection Y ∩ U(−m) is the full inverse image of an irreducible locally closed subset \overline{Y} ⊂ U(−m)/U(n), for some n ≥ 0. Note that if Y ⊂ U(−m) is the full inverse image of a locally closed subset \overline{Y}_{n} ⊂ U(−m)/U(n), then \overline{Y}_{n} is irreducible if and only if its inverse image \overline{Y}_{n′} ⊂ U(−m)/U(n′) is irreducible for some (equivalently, all) n′ ≥ n. This holds because the canonical projection U(−m)/U(n′) → U(−m)/U(n) can be realized in suitable coordinates as a projection 𝔸^{a} × 𝔸^{b} → 𝔸^{a} for some nonnegative integers a and b.

We say an ind-admissible set Y ⊂ U(L) is equidimensional if for each m ≥ 0, the intersection Y ∩U(−m) is the full inverse image of an equidimensional locally closed subvariety of U(−m)/U(n) for some (equivalently, all) sufficiently large n ≥ 0.

Finally, an A(F)-stable locally closed subset Y ⊂ X will be termed equidimensional provided that the full inverse image in U(L) of Y ∩ X_{0} ⊂ U(L)/U(𝔬) is an equidimensional ind-admissible set.

Proposition 2.17.1. The affine Deligne-Lusztig variety X_{μ}(ϵ^{ν}) is equidimensional.

To prove this proposition, we will need a few lemmas.

To state the first lemma, suppose that H is an algebraic group over k, that is, a reduced and irreducible k-group scheme of finite type. Suppose H acts (on the left) on a variety, or more generally on a finite-type k-scheme 𝔛. For a closed point x ∈ 𝔛(k), we let O denote the H-orbit of x, a locally-closed subset of 𝔛 which we give the reduced subscheme structure. Let H_{x} ⊂ H denote the subgroup scheme which stabilizes the point x, and let H/H_{x} denote the fppf-sheaf associated to the fppf-presheaf R ↦ H(R)/H_{x}(R). Finally, let 𝔭_{x} : H → O be the morphism h ↦ hx.

Lemma 2.17.2. (1) The morphism 𝔭_{x} : H → O is fppf, hence it is an epimorphism of fppf-sheaves and induces an isomorphism of fppf-sheaves

H/H_{x} ⟶^{∼} O.

(2) For every irreducible subset Y ⊂ O, the inverse image scheme 𝔭_{x}^{−1}(Y) is a finite union of irreducible subsets Z, each having dimension

dim Z = dim Y + dim H_{x}.

Proof. Part (1) is contained in [DG], II, §3, 5.2. For part (2), we have by loc. cit. III, §3, 5.5 that dim O = dim H − dim H_{x}. Since H → O is a dominant morphism of irreducible varieties, there exists a dense open subset U ⊂ O with the following property: for every irreducible set Y which meets U and every irreducible component W of 𝔭_{x}^{−1}(Y) which meets 𝔭_{x}^{−1}(U), we have

dim W − dim Y = dim H − dim O = dim H_{x},

see e.g. [Mum], I §8, Thm. 3. Since H acts transitively on O, the latter is covered by open subsets having the same property, and this implies (2). □

Lemma 2.17.3. Let ν, ν′ ∈ X_{∗}(A). If V ⊂ U(L) is an equidimensional indadmissible set, then so is f_{ν,ν′}^{−1}(V).

Proof. We may assume V is irreducible. We may also assume that V is admissible, since the intersection f_{ν,ν′}^{−1}(V) ∩ U(−m) does not change when V is replaced by V ∩ U(−m′) for suitably large m′.

As in the proof of Proposition 2.12.1, we may replace (ν, ν′) with (λ+ν, λ+ν′) for a suitably dominant λ and thereby reduce to the case where ν and ν′ are dominant and f = f_{ν,ν} satisfies f(U_{0}) ⊂ U_{0}. Also, we may replace V with ϵ^{λ}V ϵ^{−λ} and thus we may assume V ⊂ U_{n_{0}}, where n_{0} is chosen as in Proposition 2.12.1. We claim that for any m ≥ 0, the intersection U(−m) ∩ f^{−1}V is a union of finitely many irreducible admissible sets all having the same dimension. As in Proposition 2.12.1, it is enough to treat the case m = 0.

Recall that H := U_{0}/U_{n} and f_{0} induces the map f̄: H → H, which is the right action of H on 1 ∈ H: f̄(h) = 1 ∗ h. Now as before for any n ≥ n_{0} we have the commutative diagram

\xymatrix{U_{0} \ar[r]^{f_{0}} \ar[d]^{p} & U_{0} \ar[d]^{p} \\ U_{0}/U_{n} \ar[r]^{f̄} & U_{0}/U_{n}}

and \overline{V} ⊂ U_{n_{0}}/U_{n} ⊂ f̄(H), where all notation is as in Proposition 2.12.1, except that \overline{V} is now an irreducible subset of U_{0}/U_{n}.

Since f_{0}^{−1}(V) = p^{−1}f̄^{−1}(\overline{V}), we need only show that f̄^{−1}(\overline{V}) is the union of finitely many irreducible locally closed subsets of U_{0}/U_{n}, all having the same dimension. This follows from Lemma 2.17.2. □

The next lemma was established by Mirković and Vilonen as part of their proof of the geometric Satake isomorphism.

Lemma 2.17.4 (Mirković-Vilonen). Let μ be G-dominant and let ν ∈ X_{∗}(A) be an element such that ν_{dom} ≤ μ. Then the variety X^{μ} ∩ X_{ν} is equidimensional.

Proof. We shall deduce the lemma from the following result. Let IC_{μ} denote the intersection complex on the closure of X^{μ}, normalized in such a way that IC_{μ}[⟨2ρ, μ⟩](⟨ρ, μ⟩) is a self-dual perverse sheaf of weight 0. Then the complex RΓ_{c}(X_{ν}, IC_{μ}) is concentrated in cohomological degree ⟨2ρ, μ + ν⟩. This is a part of the geometric Satake isomorphism [MV1, MV2], and it was also proved in a direct fashion by Ngô-Polo in [NP]. Our goal is simply to take it as an input and show how it implies the equidimensionality statement.

Suppose X^{μ} ∩ X_{ν} is not equidimensional, and choose an irreducible component C of dimension d < ⟨ρ, μ + ν⟩. Let U denote the complement in X^{μ} ∩ X_{ν} of all the other irreducible components. Thus U is open and dense in C and it is also open in \overline{X^{μ}} ∩ X_{ν}. Let Z denote the complement of U in \overline{X^{μ}} ∩ X_{ν}.

Consider the exact sequence

H_{c}^{2d−1}(Z, IC_{μ}) → H_{c}^{2d}(U, IC_{μ}) → H_{c}^{2d}(\overline{X^{μ}} ∩ X_{ν}, IC_{μ}).

The first map is identically zero for weight reasons: the complex IC_{μ} has weights ≤ 0, and so by [D], the left-most group has weights ≤ 2d − 1. On the other hand the middle group is pure of weight 2d, since IC_{μ}|_{U} is the constant sheaf. This shows that the non-zero middle group injects into H_{c}^{2d}(X_{ν}, IC_{μ}), a contradiction. □

Finally, note that Proposition 2.17.1 follows immediately from Lemmas 2.17.3 and 2.17.4 and the following identities we already proved:

X_{μ}(ϵ^{ν}) ∩ X_{0} = f_{ν}^{−1}(Kϵ^{μ}Kϵ^{−ν} ∩ U(L))/U(𝔬)
X^{μ} ∩ X_{ν} = (Kϵ^{μ}Kϵ^{−ν} ∩ U(L))/ϵ^{ν}U(𝔬)ϵ^{−ν}.

3. Dimension theory for admissible subsets of L^{n}

Let V be a finite dimensional vector space over L. We discuss admissibility and dimensions for subsets of V. This will be needed in the next section.

3.1. Lattices. Recall that a lattice Λ in V is a finitely generated 𝔬-submodule of V that generates V as vector space. If V is n-dimensional, then every lattice in V is free of rank n as 𝔬-module. Given two lattices Λ_{1}, Λ_{2} in V, there exist j, l ∈ ℤ such that ϵ^{j}Λ_{2} ⊂ Λ_{1} ⊂ ϵ^{l}Λ_{2}, and moreover Λ_{1} ∩ Λ_{2} and Λ_{1} + Λ_{2} are again lattices.

3.2. Admissible subsets of V. We say that a subset Y ⊂ V is admissible if there exist lattices Λ_{2} ⊂ Λ_{1} and a Zariski locally closed subset Ȳ of the finite dimensional k̄-vector space Λ_{1}/Λ_{2} such that Y is the inverse image under Λ_{1} ↠ Λ_{1}/Λ_{2} of Ȳ; in this situation we say that Y comes from Ȳ. We say that a subset Y ⊂ V is ind-admissible if for every lattice Λ the intersection Y ∩ Λ is admissible. Note that a subset Y is ind-admissible if Y ∩ Λ_{α} is admissible for all Λ_{α} in a cofinal family of lattices.

3.3. Dimension theory for admissible subsets of V. Let Λ be a lattice and let Y be an admissible subset of V. Choose lattices Λ_{2} ⊂ Λ_{1} such that Λ_{2} ⊂ Λ and Y comes from Ȳ ⊂ Λ_{1}/Λ_{2}, and put

dim_{Λ} Y := dim Ȳ − dim Λ/Λ_{2},

a number independent of the choice of Λ_{1}, Λ_{2}. This notion of dimension depends of course on Λ, as the notation indicates. We have normalized things so that dim Λ = 0. However, for any two admissible subsets Y_{1}, Y_{2} the difference

dim_{Λ} Y_{1} − dim_{Λ} Y_{2}

is independent of Λ, and so we will permit ourselves to write such differences as simply

dim Y_{1} − dim Y_{2}.

For any ind-admissible subset Y ⊂ V we define dim_{Λ} Y to be the supremum of the numbers dim_{Λ} Y ∩ Λ_{1} as Λ_{1} ranges through all lattices.

Lemma 3.3.1. Let g be an L-linear automorphism of V. Then for any admissible subset Y ⊂ V the inverse image g^{−1}Y is admissible and

dim g^{−1}Y − dim Y = val det g.

Proof. Easy. □

4. Relative dimension of certain morphisms f : V → V

This section contains a generalization of Lemma 2.11.1 that will be needed in the next section.

4.1. Review of F -spaces. Recall that an F -space is a pair (V, Φ), with V a finite dimensional L-vector space and Φ a σ-linear bijection V → V. The category of F -spaces is an F -linear abelian category in which every object is a finite direct sum of simple objects. The isomorphism classes of simple objects are in natural bijection with ℚ, and a simple F -space corresponding to λ ∈ ℚ is said to have slope λ. For λ ∈ ℚ we write V_{λ} for the sum of all the simple sub-F -spaces of V having slope λ. We then have the slope decomposition

V = ⨁_{λ∈ℚ} V_{λ}.

Suppose that V is n-dimensional. We get an unordered family of n rational numbers by including each λ ∈ ℚ in the family with multiplicity equal to dim V_{λ}. These n rational numbers are called the slopes of V.

Let λ = s/r ∈ ℚ, with r ≥ 1 and (r, s) = 1. We get a simple F -space of slope λ by taking V = L^{r} and Φ_{λ}(a_{1}, …, a_{r}) = (ϵ^{s}σa_{r}, σa_{1}, …, σa_{r−1}). It follows that for any F -space V the number λ dim V_{λ} is an integer.

4.2. Definition of f : V → V and d(V, Φ). For any F -space (V, Φ) we define an F -linear map f : V → V by

f(v) := Φ(v) − v.

Moreover we define an integer d(V, Φ) by

d(V, Φ) := ∑_{λ<0} λ dim V_{λ}.

Lemma 4.2.1. The following statements hold for V, Φ, f, d(V, Φ) as above. (1) The map f is surjective.

(2) For any lattices Λ_{2} ⊂ Λ_{1} the group

(Λ_{1} ∩ ker f)/(Λ_{2} ∩ ker f)

is finite.

(3) There exists a lattice Λ_{0} such that (a) Λ′_{0} := fΛ_{0} is a lattice, and

(b) dim Λ_{0} − dim Λ′_{0} = d(V, Φ).

(4) For any lattice Λ there exist lattices Λ_{1}, Λ_{2} such that Λ_{1} ⊂ fΛ ⊂ Λ_{2}.

(5) For any lattice Λ′ the inverse image f^{−1}Λ′ contains a lattice and is indadmissible.

(6) If Λ, Λ′ are lattices such that fΛ ⊃ Λ′, then

dim Λ ∩ f^{−1}Λ′ − dim Λ′ = d(V, Φ).

(7) For any lattice Λ′ there is an equality

dim f^{−1}Λ′ − dim Λ′ = d(V, Φ).

Proof. We begin by proving the first three parts of the lemma for the simple F - spaces (L^{r}, Φ_{λ}) described above. Recall that λ = s/r with (r, s) = 1. We consider three cases.

First suppose that s > 0. Then f preserves the standard filtration ϵ^{n}𝔬^{r} on L^{r}, and induces the obviously bijective map

f̄(a_{1}, …, a_{r}) = (−a_{1}, σa_{1} − a_{2}, …, σa_{r−1} − a_{r})

on each successive quotient k̄^{r}. Therefore f is bijective and f(𝔬^{r}) = 𝔬^{r}. In this case (1) and (2) are clear, and (3) holds for Λ_{0} = 𝔬^{r}.

Next suppose that s = 0. Then r = 1 and we are once again looking at f : L → L defined by f(x) = σ(x) − x. From previous work we know that f(𝔬) = 𝔬 and hence (since f commutes with multiplication by ϵ^{a}) that f(ϵ^{a}𝔬) = ϵ^{a}𝔬 for all a ∈ ℤ. Therefore f is surjective. Its kernel is obviously F. So (1) and (2) are clear and (3) holds for Λ_{0} = 𝔬.

Finally suppose that s < 0. Put Λ_{0} = 𝔬^{r} and Λ′_{0} = ϵ^{s}𝔬 ⊕ 𝔬^{r−1}. It is then clear that

fΛ_{0} ⊂ Λ′_{0}.

Again using that f commutes with multiplication by powers of ϵ, we see that

f(ϵ^{a}Λ_{0}) ⊂ ϵ^{a}Λ′_{0}

for all a ∈ ℤ. Thus f induces a map from the associated graded group for the filtration ϵ^{a}Λ_{0} to the one for the filtration ϵ^{a}Λ′_{0}. All successive quotients for both filtrations can be identified with k̄^{r} in an obvious way, and when this is done the map induced by f on the successive quotients is always the obviously bijective map

f̄(a_{1}, …, a_{r}) = (σa_{r}, σa_{1} − a_{2}, …, σa_{r−1} − a_{r}).

We conclude that f is bijective and that fΛ_{0} = Λ′_{0}. In this case (1) and (2) are clear and (3) holds for the lattice Λ_{0} we have chosen, since dim Λ_{0}−dim Λ′_{0} = s = d(V, Φ).

Next we prove the first three parts of the lemma in general. Write V as a direct sum of simple F -spaces. Then f decomposes accordingly as a direct sum. Since each summand of f is surjective, so is f. As for (2) we are free to enlarge Λ_{1} and shrink Λ_{2}, so we may assume that both Λ_{1} and Λ_{2} are direct sums of lattices in the simple summands of V, which allows us to reduce to the simple case that has already been treated. To prove (3) we note that we can find a suitable Λ_{0} by taking a direct sum of suitable lattices in the simple summands.

Now we are going to deduce the remaining parts of lemma from the three parts we have already proved. We will no longer need to reduce to the simple case. We begin by choosing a lattice Λ_{0} as in (3). In particular Λ′_{0} := fΛ_{0} is a lattice. To see that (4) is true we pick a, b such that

ϵ^{a}Λ_{0} ⊂ Λ ⊂ ϵ^{b}Λ_{0}.

Then

ϵ^{a}Λ′_{0} ⊂ fΛ ⊂ ϵ^{b}Λ′_{0}.

Next we prove (5). Pick j such that ϵ^{j}Λ′_{0} ⊂ Λ′. Then f^{−1}Λ′ contains the lattice ϵ^{j}Λ_{0}. To show that f^{−1}Λ′ is ind-admissible it is enough to show that for l ≥ 0 the set Y_{l} := ϵ^{j−l}Λ_{0} ∩ f^{−1}Λ′ is admissible. This is clear since Y_{l} comes from the closed subset Ȳ_{l} of ϵ^{j−l}Λ_{0}/ϵ^{j}Λ_{0} obtained as the kernel of the homomorphism

ϵ^{j−l}Λ_{0}/ϵ^{j}Λ_{0} → ϵ^{−l}Λ′/Λ′

induced by f.

Now we prove the key statement (6). Let L be the set of pairs (Λ, Λ′) of lattices such that fΛ ⊃ Λ′. For any pair (Λ, Λ′) ∈ L we put

d(Λ, Λ′) := dim Λ ∩ f^{−1}Λ′ − dim Λ′.

We must show that d(Λ, Λ′) = d(V, Φ) for all (Λ, Λ′) ∈ L. Since we chose Λ_{0} so as to satisfy (3), we see that (Λ_{0}, Λ′_{0}) ∈ L and d(Λ_{0}, Λ′_{0}) = d(V, Φ). Thus it is enough to show that

d(Λ, Λ′) = d(Λ_{0}, Λ′_{0}).

In fact we claim that for any j ≤ 0 such that Λ ⊂ ϵ^{j}Λ_{0} there is a chain of equalities

d(Λ, Λ′) = d(ϵ^{j}Λ_{0}, Λ′) = d(ϵ^{j}Λ_{0}, Λ′ ∩ Λ′_{0}) = d(ϵ^{j}Λ_{0}, Λ′_{0}) = d(Λ_{0}, Λ′_{0}).

Each of these equalities is a consequence of one of the following two statements.

Statement 1. If (Λ, Λ′) ∈ L and Λ′′ ⊂ Λ′, then (Λ, Λ′′) ∈ L and d(Λ, Λ′) = d(Λ, Λ′′).

Statement 2. If (Λ, Λ′) ∈ L and Λ ⊂ Λ_{1}, then (Λ_{1}, Λ′) ∈ L and d(Λ, Λ′) = d(Λ_{1}, Λ′).

Statement 1 follows from the fact that f induces an isomorphism

\frac{Λ ∩ f^{−1}Λ′}{Λ ∩ f^{−1}Λ′′} → \frac{Λ′}{Λ′′} .

Statement 2 follows from the fact that

\frac{Λ_{1} ∩ ker f}{Λ ∩ ker f} ≃ \frac{Λ_{1} ∩ f^{−1}Λ′}{Λ ∩ f^{−1}Λ′} ,

which implies (by part (2) of this lemma) that Λ_{1} ∩ f^{−1}Λ′ is a finite union of cosets of Λ ∩ f^{−1}Λ′.

Part (7) of the lemma follows immediately from part (6). □

Now we come to the key proposition. Let V, V′ be finite dimensional L-vector spaces of the same dimension. Let ϕ : V → V′ be an L-linear bijection and let ψ : V → V′ be a σ-linear bijection. Put f := ψ − ϕ, an F -linear map V → V′. Since (V, ϕ^{−1}ψ) is an F -space, it makes sense to put d := d(V, ϕ^{−1}ψ).

In the proposition below we will be considering the kernel ker f̄ of a certain homomorphism f̄ of algebraic groups over k̄. We take this kernel in the naive sense, that is to say, we take the reduced subscheme structure on the closed set ker f̄. We then write (ker f̄)^{0} for the identity component of ker f̄.

Proposition 4.2.2. For any lattice Λ in V there exists a lattice Λ′ in V′ and a non-negative integer j such that

ϵ^{j}Λ′ ⊂ fΛ ⊂ Λ′.

For any such Λ′, j, and for any l ≥ j we consider the homomorphism

f̄: Λ/ϵ^{l}Λ → Λ′/ϵ^{l}Λ′

induced by f. Then

(1) im f̄ ⊃ ϵ^{j}Λ′/ϵ^{l}Λ′.
(2) dim ker f̄ = d + dim ϕ^{−1}Λ′ − dim Λ.
(3) (ker f̄)^{0} ⊂ ϵ^{l−j}Λ/ϵ^{l}Λ.

Proof. We may use ϕ to identify V′ with V. Thus it is enough to prove this proposition when V′ = V and ϕ = id_{V}. Thus (V, ψ) is an F -space and f now has the familiar form ψ − id.

The existence of Λ′, j is a consequence of part (4) of Lemma 4.2.1. Part (1) of the lemma follows from our assumption that ϵ^{j}Λ′ ⊂ fΛ. To prove (2) we note that

ker f̄ = \frac{Λ ∩ f^{−1}(ϵ^{l}Λ′)}{ϵ^{l}Λ},

a set having dimension d + dim Λ′ − dim Λ by part (6) of Lemma 4.2.1, which can be used since fΛ ⊃ ϵ^{l}Λ′.

To prove (3) it is enough to show that

(ker f̄) ∩ (ϵ^{l−j}Λ/ϵ^{l}Λ)

has the same dimension as ker f̄. For this we note that

(ker f̄) ∩ (ϵ^{l−j}Λ/ϵ^{l}Λ) = \frac{ϵ^{l−j}Λ ∩ f^{−1}(ϵ^{l}Λ′)}{ϵ^{l}Λ},

a set again having dimension d + dim Λ′ − dim Λ by part (6) of Lemma 4.2.1, which can be used since f(ϵ^{l−j}Λ) ⊃ ϵ^{l}Λ′. □

5. Reduction step for more general elements b

So far we have treated only elements b ∈ A(L). Now we will prove Rapoport’s conjectural formula for dim X_{μ}^{G}(b) assuming its validity for basic elements.

5.1. Notation related to P = MN. We continue with G ⊃ B = AU and W as in the introduction. In addition we consider a parabolic subgroup P = MN containing B. Here N is the unipotent radical of P, and M is the unique Levi factor of P that contains A. Put B_{M} := B ∩ M and U_{M} := U ∩ M, so that B_{M} = AU_{M} is a Borel subgroup of M containing A. We write W_{M} for the Weyl group of A in M and view W_{M} as a subgroup of W.

We write A_{M} for the identity component of the center of M. Then A_{M} is a subtorus of A, and X_{∗}(A_{M}) is a subgroup of X_{∗}(A). Put 𝔞_{M} := X_{∗}(A_{M})_{ℝ}, a linear subspace of 𝔞.

Let R_{N} denote the set of roots of A in Lie(N). We write ρ_{N} ∈ X^{∗}(A)_{ℚ} for the half-sum of the roots in R_{N}. Thus ρ = ρ_{M} + ρ_{N}, where ρ (respectively, ρ_{M}) is the half-sum of the roots of A in Lie(U) (respectively, Lie(U_{M})).

For a coweight μ of A we will need to distinguish between the two relevant notions of dominance. We say that μ is G-dominant (respectively, M-dominant) if ⟨α, μ⟩ ≥ 0 for every root α of A in Lie(U) (respectively, Lie(U_{M})).

Similarly we need to distinguish between the two relevant partial orders on X_{∗}(A). When μ − ν is a non-negative integral linear combination of simple coroots for G (respectively, M), we write ν ≤ μ (respectively, ν ≤_{M} μ).

Recall from the introduction the canonical surjections p_{G} : X_{∗}(A) ↠ Λ_{G} and η_{G} : X^{G} = G(L)/K → Λ_{G}. We apply these definitions to M as well as G, obtaining p_{M} : X_{∗}(A) ↠ Λ_{M} and

η_{M} : X^{M} = M(L)/K_{M} ↠ Λ_{M},

where K_{M} denotes M(𝔬). There is an obvious embedding X^{M} ↪ X^{G}. In particular we view x_{0} as the base-point for X^{M} as well as for X^{G}.

Note that the composition X_{∗}(A_{M}) ↪ X_{∗}(A) ↠ Λ_{M} is injective and identifies X_{∗}(A_{M}) with a subgroup of finite index in Λ_{M}, so that 𝔞_{M} can also be identified with Λ_{M} ⊗_{ℤ} ℝ; in this way we obtain a canonical homomorphism

(5.1.1)  Λ_{M} → 𝔞_{M}.

5.2. Dimensions for admissible and ind-admissible subsets of N(L). In 2.8 we defined subgroups

⋯ ⊃ U(−2) ⊃ U(−1) ⊃ U(0) ⊃ U(1) ⊃ U(2) ⊃ …

of U(L), and in 2.10 we used these subgroups to define (ind)-admissible subsets of U(L), as well as dimensions of such sets.

For m ∈ ℤ we now put

N(m) := N(L) ∩ U(m)

and define (ind)-admissibility and dimensions for subsets of N(L) just as we did for U(L), using the subgroups N(m) in place of U(m).

5.3. Relative dimension of f_{m}. As before, for ν ∈ 𝔞 we write ν_{dom} for the unique G-dominant element in the W -orbit of ν.

Proposition 5.3.1. Let m ∈ M(L) and let ν be the M-dominant element of 𝔞 defined as the Newton point of the σ-conjugacy class of m in M(L). Define a map f_{m} : N(L) → N(L) by f_{m}(n) = n^{−1}mσ(n)m^{−1}. Let Y be an admissible subset of N(L). Then f_{m}^{−1}Y is ind-admissible and

(5.3.1)  dim f_{m}^{−1}Y − dim Y = ∑_{α∈R_{N}} min{⟨α, ν⟩, 0}

= ⟨ρ, ν − ν_{dom}⟩. (5.3.2)

Moreover f_{m} is surjective.

Proof. It would be awkward to prove this proposition as it stands, since conjugation by m need not preserve N(𝔬). Therefore we are going to reformulate the proposition in a way that will make it easier to prove. □

For m_{1}, m_{2} ∈ M(L) we now define a map f_{m_{1},m_{2}} : N(L) → N(L) by

f_{m_{1},m_{2}}(n) := m_{1}n^{−1}m_{1}^{−1} · m_{2}σ(n)m_{2}^{−1},

which gives back f_{m} when (m_{1}, m_{2}) = (1, m). Put 𝔫 := Lie N. For any m ∈ M(L) we denote by Ad_{𝔫}(m) the adjoint action of m on 𝔫(L). Now Ad_{𝔫}(m_{1})^{−1} Ad_{𝔫}(m_{2})σ is a σ-linear bijection from 𝔫(L) to itself, so (𝔫(L), Ad_{𝔫}(m_{1})^{−1} Ad_{𝔫}(m_{2})σ) is an F -space, and we may put

d(m_{1}, m_{2}) := d(𝔫(L), Ad_{𝔫}(m_{1})^{−1} Ad_{𝔫}(m_{2})σ) + val det Ad_{𝔫}(m_{1}).

Note that

d(1, m) = d(𝔫(L), Ad_{𝔫}(m)σ)

= ∑_{α∈R_{N}} min{⟨α, ν⟩, 0}

since by its very definition the Newton point ν of m has the property that the slopes of the F -space (𝔫(L), Ad_{𝔫}(m)σ) are the numbers ⟨α, ν⟩ for α ∈ R_{N}. Moreover it follows from Lemma 2.13.1 that

∑_{α∈R_{N}} min{⟨α, ν⟩, 0} = ⟨ρ, ν − ν_{dom}⟩,

since ⟨α, ν⟩ ≥ 0 for all positive roots α in M. Therefore the next proposition does indeed generalize the previous one, and it will be enough to prove it.

Proposition 5.3.2. The map f_{m_{1},m_{2}} is surjective. Moreover, for any admissible subset Y of N(L) the inverse image f_{m_{1},m_{2}}^{−1}Y is ind-admissible and

dim f_{m_{1},m_{2}}^{−1}Y − dim Y = d(m_{1}, m_{2}).

Proof. For any m ∈ M(L) we have

f_{mm_{1},mm_{2}} = c_{m} ∘ f_{m_{1},m_{2}},

where c_{m} : N(L) → N(L) denotes the conjugation map n ↦ mnm^{−1}. It is easy to see that for any admissible subset Y of N(L) the inverse image c_{m}^{−1}Y is admissible, and that

(5.3.3)  dim c_{m}^{−1}Y − dim Y = val det Ad_{𝔫}(m).

Therefore the proposition is true for m_{1}, m_{2} if and only if it is true for mm_{1}, mm_{2}. So this proposition is not more general than the previous one in any serious way, but it is more convenient to prove, since we are free to improve the given pair (m_{1}, m_{2}) as follows.

Let M(L)_{+} denote the union of all K_{M}-double cosets K_{M}ϵ^{μ_{M}} K_{M} as μ_{M} ranges over all M-dominant coweights such that ⟨α, μ_{M}⟩ ≥ 0 for all α ∈ R_{N}.

What is the point of this definition? For any algebraic group I over k and any j ≥ 0 the kernel of the homomorphism I(𝔬) ↠ I(𝔬/ϵ^{j}𝔬) is a normal subgroup of I(𝔬) that we will denote by I_{j}. Thus

I(𝔬) = I_{0} ⊃ I_{1} ⊃ I_{2} ⊃ …

with I_{0}/I_{j} = I(𝔬/ϵ^{j}𝔬). Applying this definition to N (later we’ll apply it to some other groups), we get normal subgroups N_{j} of N(𝔬), and the point is that for any m ∈ M(L)_{+} and j ≥ 0

mN_{j}m^{−1} ⊂ N_{j}.

There exists a ∈ A_{M}(L) such that am_{1}, am_{2} ∈ M(L)_{+}. Replacing (m_{1}, m_{2}) by (am_{1}, am_{2}), we see that it is enough to prove the proposition under the additional assumption that m_{1}, m_{2} ∈ M(L)_{+}, an assumption we will make from now on.

Define homomorphisms ϕ, ψ : N(L) → N(L) by ϕ(n) := m_{1}nm_{1}^{−1} and ψ(n) := m_{2}σ(n)m_{2}^{−1}. Note that ϕ(N_{j}) ⊂ N_{j} and ψ(N_{j}) ⊂ N_{j} for all j ≥ 0. Thus we may define a right action of N(𝔬) on itself, called the ∗-action, with n ∈ N(𝔬) acting on n′ ∈ N(𝔬) by

n′ ∗ n := ϕ(n)^{−1}n′ψ(n).

The homomorphisms

ϕ_{j}, ψ_{j} : N_{0}/N_{j} → N_{0}/N_{j}

induced by ϕ, ψ respectively allow us to define a right ∗-action of N_{0}/N_{j} on itself by

n′ ∗ n := ϕ_{j}(n)^{−1}n′ψ_{j}(n).

Denoting by n ↦ n̄ the canonical surjection N_{0} ↠ N_{0}/N_{j}, we have the obvious compatibility

\overline{n′ ∗ n} = n̄′ ∗ n̄.

We need to linearize our problem by means of a suitable filtration

N = N[1] ⊃ N[2] ⊃ …

of N by normal subgroups of P such that each successive quotient

N⟨i⟩ := N[i]/N[i + 1]

is abelian. To this end we introduce the following coweight δ_{N} of the quotient A/Z of A by the center Z of G. In X_{∗}(A/Z) we have the ℤ-basis of fundamental coweights ϖ_{α}, one for each simple root α of A. Define the coweight δ_{N} to be the sum of the fundamental coweights ϖ_{α} with α ranging through the simple roots of A that occur in 𝔫 = Lie N. Then for i ≥ 1 we let N[i] be the product of all root subgroups U_{α} for which ⟨α, δ_{N}⟩ ≥ i.

Put 𝔫[i] := Lie N[i] and 𝔫⟨i⟩ := Lie N⟨i⟩. Then there is a P -equivariant isomorphism

(5.3.4)  N⟨i⟩ ≅ 𝔫⟨i⟩

of algebraic groups over k, and the action of P on both groups factors through P ↠ M. (To construct this isomorphism first work with split groups over ℤ.)

The homomorphisms ϕ, ψ preserve the subgroups N[i](L) and therefore induce homomorphisms

ϕ⟨i⟩, ψ⟨i⟩ : N⟨i⟩(L) → N⟨i⟩(L)

as well, so we get ∗-actions of N[i](𝔬) and N⟨i⟩(𝔬) on themselves. Moreover, under the identification (5.3.4) of N⟨i⟩(L) with 𝔫⟨i⟩(L), the homomorphism ϕ⟨i⟩ (respectively, ψ⟨i⟩) goes over to the L-linear bijection Ad_{𝔫⟨i⟩}(m_{1}) (respectively, the σ-linear bijection Ad_{𝔫⟨i⟩}(m_{2})σ), where Ad_{𝔫⟨i⟩}(·) denotes the adjoint action of M on 𝔫⟨i⟩ = 𝔫[i]/𝔫[i + 1].

Thus for each i we are in the linear situation considered in Proposition 4.2.2 (with V = V′ = 𝔫⟨i⟩(L)), and as in that proposition we define an F -linear map

f⟨i⟩ : 𝔫⟨i⟩(L) → 𝔫⟨i⟩(L)

by f⟨i⟩ := ψ⟨i⟩ − ϕ⟨i⟩.

Since m_{1}, m_{2} ∈ M(L)_{+}, both ϕ⟨i⟩ and ψ⟨i⟩ carry the lattice 𝔫⟨i⟩(𝔬) into itself; therefore f⟨i⟩ does too. By Proposition 4.2.2 there exists j ≥ 0 such that

(5.3.5)  f⟨i⟩(𝔫⟨i⟩(𝔬)) ⊃ ϵ^{j}(𝔫⟨i⟩(𝔬)).

Since 𝔫⟨i⟩ = 0 for large i, we can even choose j so that (5.3.5) holds for all i. We also need to look at this slightly differently. For a ∈ A_{M}(F) define

c_{a} : N(L) → N(L)

by c_{a}(n) := ana^{−1}. The homomorphisms c_{a} preserve the subgroups N[i](L) and hence induce homomorphisms

c_{a} : N⟨i⟩(L) → N⟨i⟩(L).

For a ∈ A_{M}(F)_{+} := A_{M}(F) ∩ M(L)_{+} we have c_{a}N_{0} ⊂ N_{0}, c_{a}N[i]_{0} ⊂ N[i]_{0}, c_{a}N⟨i⟩_{0} ⊂ N⟨i⟩_{0}, and it is easy to see that

c_{a}(n′ ∗ n) = c_{a}(n′) ∗ c_{a}(n)

for all n, n′ ∈ N_{0}.

Claim 1. There exists a ∈ A_{M}(F)_{+} such that for any i ≥ 1

(5.3.6)  c_{a}N[i]_{0} ⊂ 1 ∗ N[i]_{0}.

Now we prove Claim 1. We begin by proving the weaker statement that for each i ≥ 1 there exists a_{i} ∈ A_{M}(F)_{+} such that (5.3.6) holds for a_{i} and i, and this we prove by descending induction on i, starting with any i such that N[i] is trivial, so that the statement we need to prove becomes trivial. Assume the statement is true for i + 1 and prove it for i. Thus we are assuming that there exists a_{i+1} ∈ A_{M}(F)_{+} such that (5.3.6) holds for a_{i+1} and i + 1.

Applying Proposition 4.2.2 just as we did to find j, we see that there exists a′ ∈ A_{M}(F)_{+} such that

(5.3.7)  c_{a′}N⟨i⟩_{0} ⊂ 1 ∗ N⟨i⟩_{0}.

We will now check that (5.3.6) holds for a_{i} := a′a_{i+1} and i.

Indeed, consider an element x = c_{a_{i}}n with n ∈ N[i]_{0}. The image of c_{a′}n under N[i]_{0} ↠ N⟨i⟩_{0} lies in c_{a′}N⟨i⟩_{0}, and therefore (5.3.7) guarantees the existence of n′ ∈ N[i]_{0} such that (c_{a′}n) ∗ n′ ∈ N[i + 1]_{0}, whence

c_{a_{i+1}}((c_{a′}n) ∗ n′) ∈ c_{a_{i+1}}N[i + 1]_{0}.

Since (5.3.6) holds for a_{i+1} and i + 1, there exists n′′ ∈ N[i + 1]_{0} such that

(5.3.8)  (c_{a_{i+1}}((c_{a′}n) ∗ n′)) ∗ n′′ = 1.

The left side of equation (5.3.8) works out to

c_{a_{i}}n ∗ (c_{a_{i+1}}(n′)n′′),

showing that c_{a_{i}}n ∈ 1 ∗ N[i]_{0}, as desired. This proves the weaker statement.

To prove the claim itself we first note that for a, a′ ∈ A_{M}(F)_{+} it is clear that aa′ ∈ A_{M}(F)_{+} and that

c_{aa′}N[i]_{0} = c_{a}(c_{a′}N[i]_{0}) ⊂ c_{a}N[i]_{0}.

Therefore if (5.3.6) holds for a and i, it also holds for aa′ and i. We already know that for each of the finitely many values of i for which N[i] is non-trivial, we can find an element a_{i} for which (5.3.6) holds for a_{i} and i, and taking the product of all these elements a_{i}, we get an element a such that (5.3.6) holds for all i. This proves the claim.

Now we fix a ∈ A_{M}(F)_{+} as in Claim 1. Choose j ≥ 0 as before, so that (5.3.5) holds for all i. For reasons that will soon become apparent, we now consider any l ≥ j large enough that for all i ≥ 1

(5.3.9)  N[i]_{l−j} ⊂ c_{a}N[i]_{0}.

We denote by H, H[i], H⟨i⟩ the groups of 𝔬/ϵ^{l}𝔬-points of N, N[i], N⟨i⟩ respectively, regarded as algebraic groups over k̄. Note that H = H[1] and H⟨i⟩ = H[i]/H[i + 1]. On H[i], H⟨i⟩ we have descending filtrations H[i]_{β}, H⟨i⟩_{β} coming from the powers ϵ^{β} of ϵ; more precisely, for 0 ≤ β ≤ l we put

H[i]_{β} := ker[N[i](𝔬/ϵ^{l}𝔬) → N[i](𝔬/ϵ^{β}𝔬)]

and similarly for H⟨i⟩. The ∗-actions on N[i](𝔬) and N⟨i⟩(𝔬) induce compatible ∗-actions on H[i] and H⟨i⟩.

The homomorphism c_{a} preserves both N_{0} and N_{l} and hence induces a homomorphism c_{a} : H → H. Since Claim 1 holds for a, we see that for any i ≥ 1

(5.3.10)  c_{a}H[i] ⊂ 1 ∗ H[i].

Moreover (5.3.9) implies immediately that

(5.3.11)  H[i]_{l−j} ⊂ c_{a}H[i].

Also note that

(5.3.12)  H[i] ∩ c_{a}H = c_{a}H[i],

as one sees easily from the description of H as a product of copies of 𝔬/ϵ^{l}𝔬, one for each root in R_{N}.

Write S[i] (respectively, S⟨i⟩) for the stabilizer, for the ∗-action, of 1 ∈ H[i] in H[i] (respectively, of 1 ∈ H⟨i⟩ in H⟨i⟩). Note that S := S[1] is the stabilizer, for the ∗-action, of 1 ∈ H in H. Here we are taking naive stabilizers: put the reduced subscheme structure on the set-theoretic stabilizers.

Claim 2. dim S = d(m_{1}, m_{2}). We begin by proving that

(5.3.13)  dim S[i] = dim S[i + 1] + dim S⟨i⟩.

For this it is enough to prove that

S⟨i⟩^{0} ⊂ im[S[i] → H⟨i⟩] ⊂ S⟨i⟩.

The second inclusion being clear, it is enough to prove the first one. So let s ∈ S⟨i⟩^{0}. By Proposition 4.2.2 we have

S⟨i⟩^{0} = (ker \overline{f⟨i⟩})^{0} ⊂ H⟨i⟩_{l−j}.

Pick h ∈ H[i]_{l−j} such that h ↦ s. By (5.3.11) there exists h_{0} ∈ H[i] such that h = c_{a}h_{0}. Now 1 ∗ h ↦ 1 ∗ s = 1 ∈ H⟨i⟩, showing that 1 ∗ h ∈ H[i + 1]. Moreover 1∗h = c_{a}(1∗h_{0}), showing that 1∗h ∈ c_{a}H. From (5.3.12) and (5.3.10) we conclude that

1 ∗ h ∈ c_{a}H[i + 1] ⊂ 1 ∗ H[i + 1]

which means that there exists h_{1} ∈ H[i + 1] such that 1 ∗ hh_{1} = 1. Thus hh_{1} ∈ S[i], and clearly hh_{1} ↦ s, proving that s lies in the image of S[i], as desired. This proves the equality (5.3.13).

It follows from (5.3.13) that

dim S = ∑_{i≥1} dim S⟨i⟩.

Moreover, by part (2) of Proposition 4.2.2

dim S⟨i⟩ = d(𝔫⟨i⟩(L), ϕ⟨i⟩^{−1}ψ⟨i⟩) + dim ϕ⟨i⟩^{−1}(𝔫⟨i⟩(𝔬)) − dim 𝔫⟨i⟩(𝔬).

Therefore

dim S = d(𝔫(L), Ad_{𝔫}(m_{1})^{−1} Ad_{𝔫}(m_{2})σ) + dim Ad_{𝔫}(m_{1})^{−1}𝔫(𝔬) − dim 𝔫(𝔬)
= d(m_{1}, m_{2}),

completing the proof of Claim 2.

Now consider the map f_{0} : N(𝔬) → N(𝔬) defined by f_{0}(n) := 1 ∗ n. Equivalently, f_{0} is the restriction of f_{m_{1},m_{2}} to N(𝔬).

Claim 3. Suppose that Y is an admissible subset of c_{a}N_{0} (with a again as in Claim 1). Then f_{0}^{−1}Y is admissible and

dim f_{0}^{−1}Y − dim Y = d(m_{1}, m_{2}).

Now we prove Claim 3. The set Y comes from a locally closed subset Ȳ of H = N(𝔬/ϵ^{l}𝔬) for some suitably large l. By increasing l we may assume that Claim 2 holds for l. We have a commutative diagram

N(𝔬) ⟶^{f_{0}} N(𝔬)
p↓  ^{p}↓
H ⟶^{f̄} H

where p is the canonical surjection and f̄ is induced by f_{0}. By Claim 1 Ȳ is contained in the image of f̄. Every non-empty (reduced) fiber of f̄ is isomorphic to the stabilizer group S considered above. Therefore

dim f̄^{−1}Ȳ = dim Ȳ + dim S
= dim Ȳ + d(m_{1}, m_{2}).

Since f_{0}^{−1}Y = p^{−1}f̄^{−1}Ȳ, we see that f_{0}^{−1}Y is admissible and

dim f_{0}^{−1}Y − dim Y = d(m_{1}, m_{2}),

proving Claim 3.

Now we finish the proof. We abbreviate f_{m_{1},m_{2}} to f and d(m_{1}, m_{2}) to d. Let Y be any admissible subset of N(L).

For any a′ ∈ A_{M}(F) it is clear that f commutes with c_{a′}. Thus it follows from Claim 3 together with (5.3.3) that f^{−1}Y ∩ a^{′−1}N(𝔬)a′ is admissible of dimension dim Y + d for all a′ ∈ A_{M}(F) such that a′Y a^{′−1} ⊂ aN_{0}a^{−1} (with a as before, so that Claims 1 and 3 hold for it).

Pick a coweight λ_{0} ∈ X_{∗}(A_{M}) such that ⟨α, λ_{0}⟩ > 0 for all α ∈ R_{N}, and for t ∈ ℤ put a_{t} := λ_{0}(ϵ^{t}). There exists t_{0} ≥ 0 such that a_{t}Y a_{t}^{−1} ⊂ aN_{0}a^{−1} for all t ≥ t_{0}. Therefore f^{−1}Y ∩ a_{t}^{−1}N(𝔬)a_{t} is admissible of dimension dim Y + d for all t ≥ t_{0}, and since any of the subgroups N(i) (i ∈ ℤ) used in 5.2 to define dimensions is contained in a_{t}^{−1}N(𝔬)a_{t} for sufficiently large t, we see that f^{−1}Y is ind-admissible and that

dim f^{−1}Y − dim Y = d,

as desired.

The last point is the surjectivity of f. We already know by Claim 1 that f(N_{0}) contains aN_{0}a^{−1}. Therefore f(a_{t}^{−1}N_{0}a_{t}) contains a_{t}^{−1}aN_{0}a^{−1}a_{t} for all t ∈ ℤ. Since the union of the subgroups a_{t}^{−1}aN_{0}a^{−1}a_{t} (t ∈ ℤ) is all of N(L), we see that f is indeed surjective. □

5.4. Dimensions of intersections of N(L)- and K- orbits on X^{G}. As before we put x_{λ} = ϵ^{λ}x_{0}. Let μ be a G-dominant coweight.

For m ∈ M(L) we are interested in the intersection

(5.4.1)  N(L)mx_{0} ∩ Kx_{μ}

of the N(L)-orbit of mx_{0} and the K-orbit of x_{μ} in the affine Grassmannian X^{G}. Let μ_{M} be the unique M-dominant coweight such that m ∈ K_{M}ϵ^{μ_{M}} K_{M}. Then there exists k_{M} ∈ K_{M} such that mx_{0} = k_{M}x_{μ_{M}}, and left multiplication by k_{M} defines an isomorphism

(5.4.2)  N(L)x_{μ_{M}} ∩ Kx_{μ} ≃ N(L)mx_{0} ∩ Kx_{μ}.

For a given G-dominant coweight μ we write S_{M}(μ) for the set of M-dominant coweights μ_{M} for which the intersection N(L)x_{μ_{M}} ∩Kx_{μ} is non-empty. For ν ∈ Λ_{M} we then put

S_{M}(μ, ν) := {μ_{M} ∈ S_{M}(μ) : p_{M}(μ_{M}) = ν}.

The next lemma will help us understand the finite sets S_{M}(μ) and S_{M}(μ, ν). Define a finite subset Σ(μ) of X_{∗}(A) as follows: μ′ ∈ Σ(μ) if and only if μ′_{dom} ≤ μ, where, as before, μ′_{dom} denotes the unique G-dominant element in the W -orbit of μ′. Then denote by Σ(μ)_{M- dom} the set of M-dominant elements in Σ(μ). Finally, denote by Σ(μ)_{M- max} the set of all elements in Σ(μ)_{M- dom} that are maximal in Σ(μ)_{M- dom} with respect to the partial order ≤_{M} on X_{∗}(A).

Lemma 5.4.1. For any G-dominant coweight μ there are inclusions

Σ(μ)_{M- max} ⊂ S_{M}(μ) ⊂ Σ(μ)_{M- dom}.

Moreover, all three sets Σ(μ)_{M- max}, S_{M}(μ), Σ(μ)_{M- dom} have the same image under the map p_{M} : X_{∗}(A) ↠ Λ_{M}. In particular S_{M}(μ, ν) is non-empty if and only if ν lies in the image of Σ(μ)_{M- dom} under p_{M}.

Proof. First recall [Mat] (see also [Rap00, H]) that a coweight μ′ lies in Σ(μ) if and only if U(L)ϵ^{μ′} meets Kϵ^{μ}K. By definition μ′ ∈ S_{M}(μ) if and only if μ′ is M-dominant and N(L)ϵ^{μ′} meets Kϵ^{μ}K. Since N ⊂ U, it is clear that S_{M}(μ) ⊂ Σ(μ)_{M- dom}.

Now let μ′ ∈ Σ(μ)_{M- max}. Thus there exists u ∈ U(L) such that uϵ^{μ′} ∈ Kϵ^{μ}K. Write u = nu_{M} with n ∈ N(L), u_{M} ∈ U_{M}(L). Then N(L)u_{M}ϵ^{μ′} meets Kϵ^{μ}K, so there exists μ_{M} ∈ S_{M}(μ) such that u_{M}ϵ^{μ′} ∈ K_{M}ϵ^{μ_{M}} K_{M}. Now applying the fact recalled at the beginning of the proof to M rather than G, we see that μ′ ≤_{M} μ_{M},

and of course μ_{M} lies in Σ(μ)_{M- dom}, since we have already proved that S_{M}(μ) ⊂ Σ(μ)_{M- dom}. By maximality of μ′ we then conclude that μ′ = μ_{M}, proving that μ′ ∈ S_{M}(μ), as desired.

For the second statement of the lemma it is enough to show that Σ(μ)_{M- max} and Σ(μ)_{M- dom} have the same image under the map p_{M} : X_{∗}(A) ↠ Λ_{M}. So suppose that p_{M}(μ) = ν for some μ ∈ Σ(μ)_{M- dom}. Clearly there exists μ′ ∈ Σ(μ)_{M- max} such that μ ≤_{M} μ′. Since μ ≤_{M} μ′ implies p_{M}(μ′) = p_{M}(μ), we are done.

The last statement of the lemma follows from the second statement. □

For G-dominant μ and μ_{M} ∈ S_{M}(μ) we denote by d(μ, μ_{M}) the dimension of the intersection N(L)x_{μ_{M}} ∩ Kx_{μ}. From (5.4.2) we see that

(5.4.3)  dim N(L)mx_{0} ∩ Kx_{μ} = d(μ, μ_{M})

for all m ∈ K_{M}ϵ^{μ_{M}} K_{M}.

Our next task is to give an estimate for the numbers d(μ, μ_{M}). At this point we need to introduce some new notation. Recall that we have fixed a Borel B = AU in G, and this gives rise to a based root system. Let G^{∨} ⊃ B^{∨} ⊃ A^{∨} denote the complex reductive group together with a Borel and a maximal torus, whose root system is dual to that determined by G ⊃ B ⊃ A. The Levi subgroup M is determined by a subset Δ_{M} of the simple B-positive roots for G. We let M^{∨} denote the Levi subgroup of G^{∨} determined by the set of simple B^{∨}-positive roots {α^{∨} | α ∈ Δ_{M}}. Then M^{∨} is a dual group for M. Given a G-dominant coweight μ ∈ X_{∗}(A), we will simultaneously think of it also as a weight μ ∈ X^{∗}(A^{∨}). Let χ_{μ}^{G} denote the character of the unique irreducible G^{∨}-module with highest weight μ. For an M-dominant coweight μ_{M} ∈ X_{∗}(A), the symbol χ_{μ_{M}}^{M} has the analogous meaning.

Proposition 5.4.2. Let μ be a G-dominant coweight. Then for all μ_{M} ∈ S_{M}(μ) there is an inequality

(5.4.4)  d(μ, μ_{M}) ≤ ⟨ρ, μ + μ_{M}⟩ − 2⟨ρ_{M}, μ_{M}⟩,

and equality holds if and only if χ_{μ_{M}}^{M} occurs in the restriction of χ_{μ}^{G} to M^{∨}. Moreover, the number of irreducible components of the intersection N(L)x_{μ_{M}} ∩ Kx_{μ} having dimension ⟨ρ, μ+μ_{M}⟩−2⟨ρ_{M}, μ_{M}⟩ is the multiplicity a_{μ_{M} μ} with which χ_{μ_{M}}^{M} occurs in χ_{μ}^{G}.

Proof. To simplify notation, during this proof we temporarily use the symbol λ instead of μ_{M} to denote an M-dominant coweight.

Recall that F = k((ϵ)) where k denotes the finite field with q elements. We will use the symbols G, M, B, A etc. to denote the groups of F -points G(F), M(F), B(F), A(F) etc., and furthermore here we write K (resp. K_{M}) for the compact open subgroup G(k[[ϵ]]) (resp. M(k[[ϵ]]) of G(F) (resp. M(F)). The intersection N(F)x_{λ}∩Kx_{μ} is the set of k-points of the variety N(L)x_{λ}∩G(𝔬)x_{μ}, and an estimate for the dimension of the latter will follow from a suitable estimate for the growth of the number of points

n(λ, μ)(q) = #(N(F)x_{λ} ∩ Kx_{μ})

as a function of q. In fact we will show that n(λ, μ)(q) is a polynomial in q with degree bounded above by ⟨ρ, μ + λ⟩ − 2⟨ρ_{M}, λ⟩. This is enough to prove the upper bound on the dimension of N(L)x_{λ} ∩ G(𝔬)x_{μ}.

To prove this we will calculate n(λ, μ)(q) in terms of values of constant terms of standard spherical functions, and we will estimate those values by manipulating the Kato-Lusztig formula [Lu, Ka82]. An exposition of this key ingredient, as well as the Satake isomorphism used below, can be found in [HKP].

Let H_{K}(G) = C_{c}(K\G/K) and H_{K}(M) = C_{c}(K_{M}\M/K_{M}) denote the spherical Hecke algebras of G and M respectively. Convolution is defined using the Haar measures giving K respectively K_{M} volume 1. For a parabolic subgroup P = MN of G, the constant term homomorphism c_{M}^{G} : H_{K}(G) → H_{K}(M) is defined by the formula

c_{M}^{G}(f)(m) = δ_{P} (m)^{−1/2} ∫_{N} f(nm)dn.

Here δ_{P} (m) := |det(Ad(m); Lie(N))|, and the Haar measure on N is such that N ∩ K has volume 1. We define in a similar way δ_{B}, δ_{B_{M}}, c_{A}^{G}, and c_{A}^{M}. Recall that U = U_{M} N, and so

δ_{B}(t) = δ_{P} (t)δ_{B_{M}} (t)

for t ∈ A, and

c_{A}^{G}(f)(t) = (c_{A}^{M} ∘ c_{M}^{G})(f)(t).

Also, c_{A}^{G} (resp. c_{A}^{M}) is the Satake isomorphism S^{G} for G (resp. S^{M} for M). Thus, the following diagram commutes:

\xymatrix{R(G^{∨}) \ar[r]^{≅} \ar[d]_{rest.} & ℂ[X_{∗}(A)]^{W} \ar[d]_{incl.} & H_{K}(G) \ar[l]_{S^{G}} \ar[d]_{c_{M}^{G}} \\ R(M^{∨}) \ar[r]^{≅} & ℂ[X_{∗}(A)]^{W_{M}} & H_{K}(M), \ar[l]_{S^{M}}}

where R(G^{∨}) denotes the representation ring for G^{∨}.

For a G-dominant coweight μ, let f_{μ}^{G} = char(KμK), and let f_{λ}^{M} have the analogous meaning. For any M-dominant coweight λ of M, we define numbers a_{λμ} and b_{λμ}(q) by the equalities

rest.(χ_{μ}^{G}) = ∑_{λ} a_{λμ} χ_{λ}^{M},
c_{M}^{G}(f_{μ}^{G}) = ∑_{λ} b_{λμ}(q) f_{λ}^{M}.

Lemma 5.4.3. Assume that λ ∈ S_{M}(μ).

(1) We have b_{λμ}(q) ∈ ℤ[q^{1/2}, q^{−1/2}], and q^{−⟨ρ,μ⟩+⟨ρ_{M} ,λ⟩}b_{λμ}(q) ∈ ℤ[q^{−1}].

(2) There is an equality

b_{λμ}(q) = a_{λμ} q^{⟨ρ,μ⟩−⟨ρ_{M} ,λ⟩} + {terms with lower q^{1/2}-degree}.

Proof. Recall the Kato-Lusztig formula

χ_{μ}^{G} = ∑_{μ′≤μ} q^{−⟨ρ,μ⟩}P_{w_{μ′} ,w_{μ}}(q) S^{G}(f_{μ′}^{G}),

where μ′ ranges over G-dominant elements in Σ(μ), t_{μ′} ∈ W̃ denotes the corresponding translation element, and w_{μ′} denotes the longest element in the double coset W t_{μ′}W. Of course P_{w_{μ′} ,w_{μ}}(q) is the Kazhdan-Lusztig polynomial, and hence its q-degree is strictly bounded above by ⟨ρ, μ − μ′⟩ if μ ≠ μ′. Also, P_{w_{μ},w_{μ}}(q) = 1.

The Kato-Lusztig formula for the group M can be written

χ_{λ′}^{M} = ∑_{λ} p_{λλ′}^{M}(q^{−1}) \frac{S^{M}(f_{λ}^{M})}{q^{⟨ρ_{M} ,λ⟩}},

for some polynomials p_{λ,λ′}^{M} ∈ ℤ[q^{−1}] which vanish unless λ ≤_{M} λ′. This expresses

the relation between two bases of R(M^{∨}) given by an “upper triangular” invertible matrix. Inverting the corresponding relation for the group G gives the formula

\frac{S^{G}(f_{μ}^{G})}{q^{⟨ρ,μ⟩}} = ∑_{μ′} q_{μ′μ}^{G}(q^{−1}) χ_{μ′}^{G},

for some q^{G} ∈ ℤ[q^{−1}]. We find that

\frac{S(f_{μ}^{G})}{q^{⟨ρ,μ⟩}} = ∑_{μ′} q_{μ′μ}^{G} χ_{μ′}^{G}

= ∑_{μ′} ∑_{λ′} a_{λ′μ′} q_{μ′μ}^{G} χ_{λ′}^{M}

= ∑_{μ′,λ′,λ} a_{λ′μ′} q_{μ′μ}^{G} p_{λλ′}^{M} \frac{S(f_{λ}^{M})}{q^{⟨ρ_{M} ,λ⟩}} .

We also have the relation

\frac{S(f_{μ}^{G})}{q^{⟨ρ,μ⟩}} = ∑_{λ} \frac{b_{λμ}(q)}{q^{⟨ρ,μ⟩−⟨ρ_{M} ,λ⟩}} \frac{S(f_{λ}^{M})}{q^{⟨ρ_{M} ,λ⟩}} .

Thus we see that for each λ ∈ S_{M}(μ),

\frac{b_{λμ}(q)}{q^{⟨ρ,μ⟩−⟨ρ_{M} ,λ⟩}} = ∑_{μ′,λ′} a_{λ′μ′} q_{μ′μ}^{G} p_{λλ′}^{M}.

Part (1) follows from this equality, as does (2) once one notes that q_{μ′,μ}^{G} and p_{λ,λ′}^{M} as polynomials in ℤ[q^{−1}] have non-zero constant terms if and only if μ′ = μ and λ = λ′ respectively. □

Now we continue with the proof of the proposition. Using the integration formula from the Iwasawa decomposition G = KNM (y = knm), we see that the number of points

n(λ, μ)(q) = #(Nϵ^{λ}K/K ∩ Kϵ^{μ}K/K)

is given by

n(λ, μ)(q) = ∫_{G} 1_{NK}(ϵ^{−λ}y^{−1}) 1_{Kϵ^{−μ}K}(y) dy
= ∫_{M} ∫_{N} ∫_{K} 1_{NK}(ϵ^{−λ}m^{−1}n^{−1}k^{−1}) 1_{Kϵ^{−μ}K}(knm) dk dn dm
= ∫_{M} ∫_{N} 1_{NK}(m^{−1}) 1_{Kϵ^{−μ}K}(nmϵ^{−λ}) dn dm
= ∫_{N} 1_{Kϵ^{−μ}K}(nϵ^{−λ}) dn
= δ_{P} (ϵ^{−λ})^{1/2} c_{M}^{G}(1_{Kϵ^{−w^{G}_{0}μ}K})(ϵ^{−λ})
= q^{⟨ρ_{N} ,λ⟩} c_{M}^{G}(1_{Kϵ^{−w^{G}_{0}μ}K})(ϵ^{−w_{0}^{M} λ}).

Here ρ_{N} is the half-sum of the positive roots occurring in Lie(N), and w_{0}^{G} (resp. w_{0}^{M}) is the longest element in the Weyl group W (resp. W_{M}).

Using Lemma 5.4.3, we see that

dim(N(L)x_{λ} ∩ G(𝔬)x_{μ}) ≤ ⟨ρ_{N}, λ⟩ + ⟨ρ, −w_{0}^{G}μ⟩ − ⟨ρ_{M}, −w_{0}^{M}λ⟩
= ⟨ρ − ρ_{M}, λ⟩ + ⟨ρ, μ⟩ − ⟨ρ_{M}, λ⟩
= ⟨ρ, λ + μ⟩ − 2⟨ρ_{M}, λ⟩,

and equality holds if and only if χ_{λ}^{M} is contained in χ_{μ}^{G}. Moreover, in that case, the number of irreducible components of the intersection having top dimension is the multiplicity a_{λμ}. □

Corollary 5.4.4. Let μ be a G-dominant coweight. Then for all μ_{M} ∈ S_{M}(μ) there is an inequality

(5.4.5)  d(μ, μ_{M}) ≤ ⟨ρ, μ + μ_{M}⟩ − 2⟨ρ_{M}, μ_{M}⟩,

and when μ_{M} ∈ Σ(μ)_{M- max}, this inequality is an equality.

Proof. If μ_{M} ∈ Σ(μ)_{M- max}, then it is easy to see that χ_{μ_{M}}^{M} appears in the restriction of χ_{μ}^{G} to M^{∨}. □

5.5. Remark. Taking M = A in Proposition 5.4.2, we recover the dimension formula of Mirković-Vilonen, Proposition 2.14.1.

5.6. Reduction step. Let b ∈ M(L) and assume that b is basic in M(L). The map η_{M} : M(L) → Λ_{M} is constant on σ-conjugacy classes and induces a bijection between the set of basic σ-conjugacy classes in M(L) and the group Λ_{M}. Put ν := η_{M}(b) ∈ Λ_{M}. We denote by ν̄ ∈ 𝔞_{M} the image of ν under the canonical homomorphism Λ_{M} → 𝔞_{M} (see (5.1.1)); since b is basic, ν̄ is the Newton point of b.

For any G-dominant coweight μ we have the affine Deligne-Lusztig variety

X_{μ}^{G}(b) = {x ∈ G(L)/K : x^{−1}bσ(x) ∈ Kϵ^{μ}K},

which may or may not be non-empty. Moreover, for any M-dominant coweight μ_{M} we have the affine Deligne-Lusztig variety

X_{μ_{M}}^{M} (b) = {x ∈ M(L)/K_{M} : x^{−1}bσ(x) ∈ K_{M}ϵ^{μ_{M}} K_{M}}.

Since b is basic in M(L) the affine Deligne-Lusztig variety X_{μ_{M}}^{M} (b) is non-empty (see [KR]) if and only if p_{M}(μ_{M}) = ν.

Our goal is to express dim X_{μ}^{G}(b) in terms of the numbers dim X_{μ_{M}}^{M} (b) and d(μ, μ_{M}) for the various coweights μ_{M} ∈ S_{M}(μ, ν), and then to use this expression to reduce Rapoport’s dimension conjecture to the basic case.

We will use the following map α : X^{G} → X^{M}. By the Iwasawa decomposition we have G(L) = P (L)K. Therefore there is an obvious bijection

P (L)/K_{P} ≃ G(L)/K,

where K_{P} := P (𝔬). The canonical surjective homomorphism P (L) ↠ M(L) induces a surjection

P (L)/K_{P} ↠ M(L)/K_{M}.

The map α is defined as the composition

α : X^{G} = G(L)/K ≃ P (L)/K_{P} ↠ M(L)/K_{M} = X^{M}.

We hasten to warn the reader that α is not a morphism of ind-schemes. However, for any connected component Y of X^{M}, the inverse image α^{−1}Y is a locally closed subset of X^{G}, and the map α^{−1}Y → Y obtained by restriction from α is a morphism of ind-schemes. For m ∈ M(L) we can identify the fiber α^{−1}(mx_{0}) with N(L)/N(𝔬) via

N(L)/N(𝔬) ∋ n ↦ mnx_{0} ∈ α^{−1}(mx_{0}).

Proposition 5.6.1. Assume that the σ-conjugacy class of b in M(L) is basic, and let ν be the element ν = η_{M}(b) ∈ Λ_{M}.

(1) The space X_{μ}^{G}(b) is non-empty if and only if ν lies in the image of the set Σ(μ)_{M- dom}.

(2) The image of X_{μ}^{G}(b) under α is contained in

⋃_{μ_{M} ∈S_{M} (μ,ν)} X_{μ_{M}}^{M} (b).

Denote by β the map

β : X_{μ}^{G}(b) → ⋃_{μ_{M} ∈S_{M} (μ,ν)} X_{μ_{M}}^{M} (b)

obtained by restriction from α.

(3) For μ_{M} ∈ S_{M}(μ, ν) and x ∈ X_{μ_{M}}^{M} (b) the fiber β^{−1}(x) is non-empty and

dim β^{−1}(x) = d(μ, μ_{M}) + ⟨ρ, ν̄ − ν̄_{dom}⟩ − ⟨2ρ_{N}, ν̄⟩.

In particular β is surjective.

(4) For all μ_{M} ∈ S_{M}(μ, ν) and all λ ∈ Λ_{M} the set β^{−1}(X_{μ_{M}}^{M} (b) ∩ η_{M}^{−1}(λ)) is locally closed in X_{μ}^{G}(b), and

dim β^{−1}(X_{μ_{M}}^{M} (b) ∩ η_{M}^{−1}(λ)) = dim X_{μ_{M}}^{M} (b) + d(μ, μ_{M}) + ⟨ρ, ν̄ − ν̄_{dom}⟩ − ⟨2ρ_{N}, ν̄⟩.

(5) If X_{μ}^{G}(b) is non-empty, its dimension is given by

sup{dim X_{μ_{M}}^{M} (b) + d(μ, μ_{M}) : μ_{M} ∈ S_{M}(μ, ν)} + ⟨ρ, ν̄ − ν̄_{dom}⟩ − ⟨2ρ_{N}, ν̄⟩.

Proof. We begin by proving (2). Let x ∈ X_{μ}^{G}(b). Write x = mnx_{0} with m ∈ M(L), n ∈ N(L). Then

n^{−1}m^{−1}bσ(m)σ(n) ∈ Kϵ^{μ}K,

from which it follows that m^{−1}bσ(m) lies in K_{M}ϵ^{μ_{M}} K_{M} for a unique μ_{M} ∈ S_{M}(μ). Thus α(x) = mx_{0} ∈ X_{μ_{M}}^{M} (b), showing that X_{μ_{M}}^{M} (b) is non-empty and hence that μ_{M} ∈ S_{M}(μ, ν). This proves (2).

Next we prove (3). Let μ_{M} ∈ S_{M}(μ, ν) and x ∈ X_{μ_{M}}^{M} (b). Choose m ∈ M(L) such that x = mx_{0}. Then b′ := m^{−1}bσ(m) ∈ K_{M}ϵ^{μ_{M}} K_{M} and

β^{−1}(x) = f_{b′}^{−1}(Kϵ^{μ}Kb^{′−1} ∩ N(L))/N(𝔬),

where f_{b′} is the morphism N(L) → N(L) defined by

f_{b′}(n) = n^{−1}b′σ(n)b^{′−1}.

Since f_{b′} is surjective, the fiber β^{−1}(x) is non-empty.

Since b′ is σ-conjugate to b in M(L), its Newton point is also ν̄. Therefore Proposition 5.3.1 tells us that

dim β^{−1}(x) = dim(Kϵ^{μ}Kb^{′−1} ∩ N(L)) + ⟨ρ, ν̄ − ν̄_{dom}⟩.

Furthermore

N(L)b′x_{0} ∩ Kx_{μ} = (Kϵ^{μ}Kb^{′−1} ∩ N(L))/b′N(𝔬)b^{′−1},

from which it follows that

dim(Kϵ^{μ}Kb^{′−1} ∩ N(L)) = d(μ, μ_{M}) + dim(b′N(𝔬)b^{′−1}),

because b′ ∈ K_{M}ϵ^{μ_{M}} K_{M}. Since dim b′N(𝔬)b^{′−1} is obviously equal to −⟨2ρ_{N}, ν̄⟩, we conclude that

dim β^{−1}(x) = d(μ, μ_{M}) + ⟨ρ, ν̄ − ν̄_{dom}⟩ − ⟨2ρ_{N}, ν̄⟩.

This proves (3), and (1) follows from (2), (3), Lemma 5.4.1 and the fact that X_{μ_{M}}^{M} (b) is non-empty for all μ_{M} ∈ S_{M}(μ, ν).

Since dim β^{−1}(x) is independent of x ∈ X_{μ_{M}}^{M} (b), part (4) follows from part (3) and the fact (noted already in the introduction) that

dim(X_{μ_{M}}^{M} (b) ∩ η_{M}^{−1}(λ)) = dim X_{μ_{M}}^{M} (b)

for all λ ∈ Λ_{M}.

Finally we prove (5). Assume that X_{μ}^{G}(b) is non-empty, so that S_{M}(μ, ν) is non-empty as well. Then X_{μ}^{G}(b) is set-theoretically the union of the locally closed pieces

(5.6.1)  β^{−1}(X_{μ_{M}}^{M} (b) ∩ η_{M}^{−1}(λ)),

where μ_{M} ranges over the finite set S_{M}(μ, ν) and λ ranges through Λ_{M}. The space X_{μ}^{G}(b) is preserved by the action of A_{M}(F), and the action of a ∈ A_{M}(F) on X_{μ}^{G}(b) sends β^{−1}(X_{μ_{M}}^{M} (b) ∩ η_{M}^{−1}(λ)) to β^{−1}(X_{μ_{M}}^{M} (b) ∩ η_{M}^{−1}(λ + λ_{a})), where λ_{a} := η_{M}(a). Since η_{M}(A_{M}(F)) has finite index in Λ_{M}, the action of A_{M}(F) has finitely many orbits on the set of pieces in our decomposition. Reasoning as in 2.5, we conclude that dim X_{μ}^{G}(b) is equal to the supremum of the dimensions of these pieces, and so we obtain (5) from (4). □

5.7. Remark. Part (1) of the proposition we just proved is not new; it was proved in [KR]. Moreover the strategy used to prove the proposition is simply a refinement of the method used to prove non-emptiness in [KR].

5.8. Reduction of Rapoport’s dimension conjecture to the basic case. We continue with G-dominant μ and a basic element b ∈ M(L), as above. We again use b to get ν ∈ Λ_{M} and the Newton point ν̄ ∈ 𝔞_{M}.

We write def_{M}(b) for the defect of b, a non-negative integer no greater than the semisimple rank of M. By definition def_{M}(b) is simply the F -rank of M minus the F -rank of the inner form J of M obtained by twisting M by b. (The group J(F) is the σ-centralizer of b in M(L).) Rapoport’s conjecture (applied to M and the basic element b) states that for all M-dominant μ_{M} with p_{M}(μ_{M}) = ν the dimension of X_{μ_{M}}^{M} (b) is given by

(5.8.1)  dim X_{μ_{M}}^{M} (b) = ⟨ρ_{M}, μ_{M}⟩ − \frac{1}{2} def_{M}(b).

Note that ⟨ρ_{M}, μ_{M}⟩ and def_{M}(b)/2 lie in \frac{1}{2}ℤ, but that their difference is always an integer.

Let’s assume this formula is true and combine it with Proposition 5.6.1 in order to calculate dim X_{μ}^{G}(b). Recall that X_{μ}^{G}(b) is non-empty if and only if ν lies in the image of Σ(μ)_{M- dom}.

Theorem 5.8.1. Suppose that X_{μ}^{G}(b) is non-empty. Assume that (5.8.1) is true for all μ_{M} ∈ S_{M}(μ, ν). Then the dimension of X_{μ}^{G}(b) is given by

dim X_{μ}^{G}(b) = ⟨ρ, μ − ν̄_{dom}⟩ − \frac{1}{2} def_{G}(b).

Proof. By Proposition 5.6.1 dim X_{μ}^{G}(b) is the sum of

⟨ρ, ν̄ − ν̄_{dom}⟩ − ⟨2ρ_{N}, ν̄⟩

and the supremum of the numbers

(5.8.2)  dim X_{μ_{M}}^{M} (b) + d(μ, μ_{M})

as μ_{M} ranges through S_{M}(μ, ν).

Combining Rapoport’s dimension conjecture (5.8.1) with Corollary 5.4.4, we see that the number (5.8.2) is bounded above by

⟨ρ_{M}, μ_{M}⟩ − def_{M}(b)/2 + ⟨ρ, μ + μ_{M}⟩ − 2⟨ρ_{M}, μ_{M}⟩

with equality whenever μ_{M} ∈ Σ(μ)_{M- max}. Using the obvious equality ⟨ρ_{N}, μ_{M}⟩ = ⟨ρ_{N}, ν̄⟩, the displayed expression above simplifies to

⟨ρ, μ⟩ + ⟨ρ_{N}, ν̄⟩ − def_{M}(b)/2,

a number which is independent of μ_{M}. Since we have supposed that X_{μ}^{G}(b) is non-empty, ν lies in the image of Σ(μ)_{M- dom}. Therefore by Lemma 5.4.1 ν is in the image of Σ(μ)_{M- max}, which shows that the supremum of the numbers (5.8.2) is equal to the last displayed expression. Adding ⟨ρ, ν̄ − ν̄_{dom}⟩ − ⟨2ρ_{N}, ν̄⟩ to this, and using that ⟨ρ_{M}, ν̄⟩ = 0 since ν̄ ∈ 𝔞_{M}, we find that

dim X_{μ}^{G}(b) = ⟨ρ, μ − ν̄_{dom}⟩ − def_{M}(b)/2.

Now it remains only to observe that def_{M}(b) = def_{G}(b). Indeed, the F -rank of a reductive group is the same as that of any of its Levi subgroups, so the F -ranks of G and M are the same. Moreover, the group J attached to (M, b) is a Levi subgroup of the group J_{G} attached to (G, b), so their F -ranks are the same. □

5.9. Reduction to the superbasic case. Let b ∈ G(L). As in the introduction b determines a homomorphism ν_{b} : 𝔻 → G over L. As in [Kot85, (4.4.3)] we then have

(5.9.1)  Int(b) ∘ σ(ν_{b}) = ν_{b}.

Replacing b by a σ-conjugate, we again assume that ν_{b} factors through A, which guarantees that σ(ν_{b}) = ν_{b}. Equation (5.9.1) then says that b ∈ M(L), where M is the Levi subgroup of G over F obtained as the centralizer of ν_{b} in G. Since ν_{b} is central in M, the element b is basic in M(L). We have seen that if Rapoport’s dimension conjecture is true for (M, b), then it is true for (G, b).

Thus it is enough to prove Rapoport’s conjecture when b is basic. Of course it may still happen that a basic element b is contained in a proper Levi subgroup of G. For example this happens when b = 1, as long as G is not a torus. We say that an element b ∈ G(L) is superbasic if no σ-conjugate of b lies in a proper Levi subgroup of G over F. Clearly superbasic elements are basic.

Suppose that b ∈ G(L) is basic. Then the group J discussed in the introduction is the inner form of G obtained by twisting the action of σ by b, so that

J(F) = {g ∈ G(L) : g^{−1}bσ(g) = b},

or in other words J(F) is the centralizer of bσ in G(L).

Lemma 5.9.1. The basic element b is superbasic if and only if J is anisotropic modulo the center Z(G) of G.

Proof. The group J/Z(G) is not anisotropic if and only if there exists a non-central homomorphism 𝔾_{m} → J which is defined over F. This happens if and only if there exist non-central μ ∈ X_{∗}(A) and an element g ∈ G(L) such that bσ and Int(g) ∘ μ commute. Now bσ and Int(g) ∘ μ commute if and only if g^{−1}(bσ)g and μ commute, and this happens if and only if g^{−1}bσ(g) and μ commute (since σ and μ commute), which is the same as saying that g^{−1}bσ(g) lies in the Levi subgroup of G obtained as the centralizer of μ. □

It follows from Theorem 5.8.1 that it is enough to prove Rapoport’s conjecture when b is superbasic. In the introduction we observed that the intersections of X_{μ}(b) with the various connected components of the affine Grassmannian are all isomorphic to each other. Let us denote by X_{μ}(b)_{λ} the intersection of X_{μ}(b) with the connected component of the affine Grassmannian indexed by λ ∈ Λ_{G}. It is easy to see that

X_{μ}(b)_{λ} = X_{μ_{ad}}(b_{ad})_{λ_{ad}},

where the subscript “ad” is being used to indicate the corresponding objects for the adjoint group G/Z(G) of G. Thus it is enough to prove Rapoport’s conjecture for superbasic elements in adjoint groups.

Any adjoint group is a direct product of simple groups. Therefore it is enough to prove Rapoport’s conjecture for superbasic elements in simple (split) groups. From Lemma 5.9.1 we know that a basic element is superbasic if and only if the inner form J of our simple group G is anisotropic. This can happen only when G is of type A_{n}, and then it is easy to see which elements are superbasic. Thus we obtain

Proposition 5.9.2. If Rapoport’s conjecture is true for all basic elements b in GL_{n}(L) such that the valuation of det(b) is relatively prime to n, then Rapoport’s conjecture is true in general.

Proof. Observe that GL_{n}(L) → P GL_{n}(L) is surjective, and that a basic element b ∈ GL_{n}(L) is superbasic if and only if val(det b) is relatively prime to n. □

6. Affine Deligne-Lusztig varieties inside the affine flag manifold

6.1. Statement of the problem. We continue with G, A, W as before, but make a few changes in notation. We write B(A) for the set of Borel subgroups B = AU containing A. We fix an alcove in the apartment 𝔞 := X_{∗}(A)_{ℝ} associated to A, and we denote by I the corresponding Iwahori subgroup of G(L). Note that A(𝔬) ⊂ I. We write W̃ for the affine Weyl group W ⋉ X_{∗}(A) = N_{G(L)}(A)/A(𝔬).

We will now let X denote the affine flag manifold G(L)/I and a_{1} its base point. For x ∈ W̃ we put a_{x} := xa_{1} ∈ X. Recall that

(6.1.1)  X = ∐_{x∈W̃} Ia_{x}

and that for any B = AU ∈ B(A)

(6.1.2)  X = ∐_{x∈W̃} U(L)a_{x}.

Given b ∈ G(L) and x ∈ W̃, we get the affine Deligne-Lusztig variety

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

a locally closed subset of X, possibly empty, whose dimension we are going to compute for all b of the form b = ϵ^{ν} for some ν ∈ X_{∗}(A). However the calculation will in fact only express dim X_{x}(ϵ^{ν}) in terms of the (unknown) dimensions of intersections of I- and U(L)-orbits in X. This is at least satisfactory for computer experiments, since the dimensions of these intersections can be determined by an algorithm involving only W̃ and the set of affine roots.

Here is the algorithm (see [Da] and [BT72, Prop. 2.3.12]). Consider the I-orbit of a_{x}, which is an affine space 𝔸 of dimension ℓ(x). Choose a reduced expression for x. Equivalently, choose a minimal gallery G joining a_{1} to a_{x}. (Here a_{1} is the alcove fixed by our Iwahori I, and the notions of length and reduced expression are defined using the simple reflections through the walls of a_{1}.) The affine space 𝔸 is the product of a succession of affine lines, one for each wall between two alcoves of G. We will now decompose this affine space into finitely many pieces, each of which is contained in a single U(L)-orbit, by specifying which U(L)-orbit a point y ∈ 𝔸 is in. To the Borel B = AU corresponds a Weyl chamber C_{U}, which is the unique one with the property that A(𝔬)U(L) is the union of the fixers in G(L) of all ”quartiers” of the form x + C_{U}, for x ∈ X_{∗}(A)_{ℝ} (see [BT72] 4.1.5, 4.4.4). We can use any alcove in C_{U} to retract y into the standard apartment. The result depends on the choice of alcove, but for alcoves deep inside C_{U}, all retractions of y are the same (cf. loc. cit. 2.9.1). The alcove obtained as this common retraction of y lies in the U(L)-orbit of y, telling us which U(L)-orbit y belongs to. We work our way up the minimal gallery G, simultaneously considering the retractions of all galleries IG relative to an alcove sufficiently deep in C_{U}. At the step corresponding to the edge between the i^{th} and i + 1^{st} alcoves of G, there are two possibilities: first, it may happen that the whole affine line at this step folds away from the alcove we used to get the retraction; second, it may happen that one point in the affine line folds towards this retraction alcove, and the rest fold away from it. Keeping track of all these folding possibilities gives a decomposition of 𝔸 into finitely many pieces, each of which is contained in a single U(L)-orbit and is a product of affine lines and affine lines minus a point. Taking the supremum of the dimensions of the pieces lying in a given U(L)-orbit, we obtain the dimension of the intersection of 𝔸 with that U(L)-orbit.

6.2. Intersections of I- and U(L)-orbits. Let B = AU ∈ B(A) and let x, y ∈ W̃. Define a non-negative integer d(x, y, B) by

(6.2.1)  d(x, y, B) := dim Ia_{x} ∩ U(L)a_{y}.

(We set d(x, y, B) = −∞ if this intersection is empty.) It is clear that

(6.2.2)  Ia_{x} ∩ U(L)a_{y} = (IxIy^{−1} ∩ U(L))/(U(L) ∩ yIy^{−1}),

and hence that

(6.2.3)  d(x, y, B) = dim IxIy^{−1} ∩ U(L) − dim U(L) ∩ yIy^{−1}.

Here we are again using our dimension theory for U(L), it being obvious that both IxIy^{−1} ∩ U(L) and U(L) ∩ yIy^{−1} are admissible subsets of U(L).

In the special case y = ϵ^{ν} with ν ∈ X_{∗}(A), we have

U(L) ∩ yIy^{−1} = ϵ^{ν}(U(L) ∩ I)ϵ^{−ν}

and hence

dim U(L) ∩ yIy^{−1} = dim U(L) ∩ I − 2⟨ρ_{B}, ν⟩,

where ρ_{B} is the half-sum of the roots of A that are positive for B. Therefore we conclude that

(6.2.4)  dim IxIϵ^{−ν} ∩ U(L) − dim U(L) ∩ I = d(x, ϵ^{ν}, B) − 2⟨ρ_{B}, ν⟩.

6.3. Computation of dim X_{x}(ϵ^{ν}). Pick B = AU ∈ B(A). As before we use the fact that X_{x}(ϵ^{ν}) is A(F)-stable to compute its dimension. The only difference is that A(F) does not permute the U(L)-orbits U(L)a_{y} transitively. Nevertheless we have the only slightly more complicated formula

(6.3.1)  dim X_{x}(ϵ^{ν}) = sup_{w∈W} dim X_{x}(ϵ^{ν}) ∩ U(L)a_{w}.

Clearly left multiplication by w^{−1} gives an isomorphism from

X_{x}(ϵ^{ν}) ∩ U(L)a_{w}

to

X_{x}(ϵ^{w^{−1}ν}) ∩ w^{−1}U(L)wa_{1},

and hence

(6.3.2)  dim X_{x}(ϵ^{ν}) = sup_{w∈W} dim X_{x}(ϵ^{wν}) ∩ wUw^{−1}(L)a_{1}.

So we need to understand dim X_{x}(ϵ^{ν}) ∩ U(L)a_{1} for all ν ∈ X_{∗}(A) and all B = AU ∈ B(A). Again writing f_{ν} for the map u ↦ u^{−1}ϵ^{ν}σ(u)ϵ^{−ν} from U(L) to itself, we have

X_{x}(ϵ^{ν}) ∩ U(L)a_{1} = f_{ν}^{−1}(IxIϵ^{−ν} ∩ U(L))/U(L) ∩ I

and hence

(6.3.3)  dim X_{x}(ϵ^{ν}) ∩ U(L)a_{1} = dim f_{ν}^{−1}(IxIϵ^{−ν} ∩ U(L)) − dim U(L) ∩ I

Using Proposition 2.12.1, Lemma 2.13.1 and (6.2.4), we see that

(6.3.4) dim f_{ν}^{−1}(IxIϵ^{−ν} ∩ U(L)) − dim U(L) ∩ I = d(x, ϵ^{ν}, B) − ⟨ρ_{B}, ν + ν_{B}⟩,

where ν_{B} is the unique element in the W -orbit of ν that is dominant for B. Thus we have proved

Theorem 6.3.1. For any x ∈ W̃, ν ∈ X_{∗}(A) and B ∈ B(A) there is an equality

dim X_{x}(ϵ^{ν}) = sup_{w∈W}{d(x, ϵ^{wν}, wBw^{−1}) − ⟨ρ_{B}, ν + ν_{B}⟩}.

In particular for ν = 0 we obtain

dim X_{x}(1) = sup_{B∈B(A)} d(x, 1, B).

This means in particular that X_{x}(ϵ^{ν}) ≠ ∅ if and only if there exists w ∈ W such that d(x, ϵ^{wν}, wBw^{−1}) ≥ 0.

7. Computations

7.1. The algorithm. In this section we present computational results on the nonemptiness and dimension of affine Deligne-Lusztig varieties in the affine flag manifold for a few groups of low rank. These results are computed according to Theorem 6.3.1 of Section 6.3, using the algorithm of Section 6.1 to compute the dimensions d(x, ϵ^{wν}, wBw^{−1}).

Computations such as the ones presented here can in principle be done by hand, but their complexity leads us to use a computer program. The program is capable of representing all elements of the affine Weyl group of a root system up to a given length. It can also multiply these elements. The program can represent affine weights and coweights, as well as the action of the affine Weyl group on these weights and coweights.

Even using a computer, the computations presented here took some time. For instance, computing which alcoves correspond to non-empty affine Deligne-Lusztig varieties together with their dimensions in the G_{2} case up to length 56 took several days on a medium-fast PC. To some extent, the program makes use of the automatical structure of the affine Weyl group (see [Ca], for instance). However, multiplication is carried out in the spirit of the method explained on p. 96 in loc. cit., not by using a multiplier automaton (see section 3 in loc. cit.). Rewriting this part of the program would certainly lead to a significant speed-up. Memory consumption was never a problem.

7.2. Reuman’s conjecture. We fix a Borel subgroup B ⊂ G containing A, and let α_{i} denote the simple B-positive roots of A, i.e. the simple roots in Lie(B). Let C_{0} denote the dominant Weyl chamber, which by definition is the set {x ∈ X_{∗}(A)_{ℝ} | ⟨α_{i}, x⟩ > 0, ∀i}. We call the unique alcove in the dominant Weyl chamber whose closure contains the origin the base alcove. As Iwahori subgroup I we choose the Iwahori fixing this base alcove; I is the inverse image of the opposite Borel group of B under the projection K ⟶ G(k̄). We recall Reuman’s conjecture from [Reu04], starting with a definition.

Definition 7.2.1. For w ∈ W let w(C_{0}) denote the Weyl chamber corresponding to w. For each simple positive root α_{i} of our root system, let ϖ_{i}^{∨} denote the corresponding fundamental coweight. Then we define the corresponding shrunken Weyl chamber w(C_{0})^{shr} ⊂ w(C_{0}) by

w(C_{0})^{shr} = w(C_{0}) + ∑_{{i \atop wα_{i}>0}} wϖ_{i}^{∨}.

The union of the shrunken Weyl chambers is the set of all alcoves a in the standard apartment such that for each positive finite root α, either the base alcove and a lie on opposite sides of the hyperplane {α = 0}, or they lie on opposite sides of the hyperplane {α = 1}.

We denote by η_{1} : W_{a} ⟶ W the usual surjective homomorphism from the affine Weyl group to the finite Weyl group, and by η_{2} : W_{a} ⟶ W the map which associates to each alcove the finite Weyl chamber in which it lies.

Conjecture 7.2.2. If x lies in the shrunken Weyl chambers, then X_{x}(1) ≠ ∅ if and only if

(7.2.1)  η_{2}(x)^{−1}η_{1}(x)η_{2}(x) ∈ W \ ⋃_{T ⊊S} W_{T} ,

and in this case the dimension is given by

\frac{1}{2}(ℓ(x) + ℓ(η_{2}(x)^{−1}η_{1}(x)η_{2}(x))).

Here S denotes the set of simple reflections, and for T ⊂ S, W_{T} denotes the subgroup of W generated by T. The set W \ ⋃_{T ⊊S} W_{T} is all elements of W for which any reduced expression contains all simple reflections.

In loc. cit. the conjecture was proved for the root systems A_{2} and C_{2}. It is also true in all cases that were checked computationally: G_{2} up to length 56, A_{3} up to length 38, C_{3} up to length 31, A_{4} up to length 15, C_{4} up to length 16. However, in the rank 4 case ‘most’ of the alcoves of ‘small’ length lie outside the shrunken Weyl chambers: it would be desirable to expand the range of our computations to encompass greater lengths.

For root systems of rank 2, results can be represented by a picture of the standard apartment (figure 1 for A_{2}, figure 2 for C_{2}, and figure 3 for G_{2}). Alcoves corresponding to non-empty affine Deligne-Lusztig varieties are gray; those corresponding to empty affine Deligne-Lusztig varieties white; the base alcove is black. There is a dot at the origin. Non-empty affine Deligne-Lusztig varieties have their dimension on the corresponding alcove.

Outside the shrunken Weyl chambers, (7.2.1) is not equivalent to X_{x}(1) ≠ ∅. For instance, all X_{x}(1) with x ∈ W are non-empty, even if x ∈ ⋃_{T ⊊S} W_{T}. However, for x ∉ W, (7.2.1) is a necessary criterion for non-emptiness in all the cases we checked. In figures 4, 5, and 6 we show the standard apartments of the root systems of type A_{2}, C_{2} and G_{2}. The alcoves with non-empty affine Deligne-Lusztig varieties are light gray, and those for which (7.2.1) wrongly predicts non-emptiness are dark gray. All dark gray alcoves lie outside the shrunken Weyl chambers. Except in the A_{2} case, even if the affine Deligne-Lusztig variety for x is non-empty, the predicted dimension may be wrong.

One can try to describe the set of alcoves for which the criterion (7.2.1) is not equivalent to the non-emptiness of the associated affine Deligne-Lusztig variety. For instance, in the A_{2} case, for x ∉ W, (7.2.1) wrongly predicts non-emptiness of X_{x}(1) if and only if the alcove corresponding to x lies outside the shrunken Weyl chambers and does not have the same number of vertices as the base alcove lying on the wall through the origin that it touches. We are not aware of a description of this kind that holds for more general root systems.

The criterion (7.2.1) is not invariant under the symmetry of root systems of type A and C. It follows from [Reu04] that for A_{2}, this realization leads to a complete description of which affine Deligne-Lusztig varieties are non-empty: X_{x}(1) ≠ ∅ if and only if (7.2.1) holds for x and its images under rotation by 120 and 240 degrees about the center of the base alcove. This does not work in the C_{2} case, however.

7.3. Lau’s observation. The following proposition is a direct consequence of [Kot85], Prop. 5.4.

Proposition 7.3.1. Let G be semisimple and simply connected. Let x ∈ W_{a}, and let \underline{x} ∈ N_{G(L)}(A) be a representative of x which lies in G(F). Then x has finite order if and only if there exists an y ∈ G(L) with \underline{x} = y^{−1}σ(y). In particular, in this case X_{x}(1) is non-empty.

Starting from this proposition and analyzing Reuman’s results in the A_{2} case, Eike Lau observed that one can get another complete characterization of the nonempty affine Deligne-Lusztig varieties for A_{2}:

Proposition 7.3.2. Let G = SL_{3}, and let x ∈ W_{a}, ℓ(x) > 1. Then X_{x}(1) ≠ ∅ if and only if there exists n ≥ 1 with ℓ(x^{n}) < ℓ(x) − 1.

One can prove this proposition by systematically analyzing Reuman’s result for SL_{3} ([Reu04] or figure 1). It would be interesting to have a more conceptual proof. For other root systems, a similar criterion does not seem to hold (at least it is not sufficient to replace the offset −1 in the proposition by another integer). Nevertheless, this criterion still seems to give a reasonable approximation to the truth. It is possible that a minor modification of this criterion might hold more generally.

7.4. Partial folding sets. Another approach to understanding the set of x with non-empty affine Deligne-Lusztig variety is to consider the folding results separately for each direction. More precisely, for fixed w ∈ W, consider the set

{x ∈ W̃; d(x, 1, wB^{−}w^{−1}) ≥ 0},

where B^{−} denotes the Borel group opposite to B. We illustrate the partial results in the A_{2} case in figures 7 (w = id), 8 (w = s_{1}), 9 (w = s_{1}s_{2}) and 10 (w = s_{1}s_{2}s_{1}). In these figures, the alcove corresponding to x ∈ W̃ is colored

white if X_{x}(1) = ∅
light gray if X_{x}(1) ≠ ∅, but d(x, 1, wB^{−}w^{−1}) = −∞

medium gray

if d(x, 1, wB^{−}w^{−1}) ≥ 0, but there exists w′ ∈ W
such that d(x, 1, w′B^{−}w^{′−1}) > d(x, 1, wB^{−}w^{−1})

dark gray if d(x, 1, wB^{−}w^{−1}) ≥ 0, and for all w′ ∈ W, d(x, 1, w′B^{−}w^{′−1}) ≤ d(x, 1, wB^{−}w^{−1})

It is unclear whether these separate pieces are easier to understand than the entire result.

7.5. The case b ≠ 1. We give two examples with b ≠ 1: figure 11 shows results for A_{2}, b = ϵ^{(1,0,−1)}; figure 12 shows results for C_{2}, b = ϵ^{(1,0)}.

The following conjecture describes the dimension of X_{x}(ϵ^{ν}) for x of sufficient length.

Conjecture 7.5.1. Let G be one of the groups SL_{2}, SL_{3}, Sp_{4}. Let b = ϵ^{ν}, and write ℓ(b) for the length of the translation in W̃ determined by ν. Then there exists n_{0} such that for all x ∈ W̃ with ℓ(x) ≥ n_{0}, X_{x}(b) ≠ ∅ if and only if X_{x}(1) ≠ ∅. In this case,

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

This conjecture describes X_{x}(b) for x in a larger region of the standard apartment than Reuman’s Conjecture 7.1 in [Reu04], but it is less precise because n_{0} is not specified. The results in [Reu02] support the new conjecture. It is possible that the conjecture holds for other groups, too. Various examples we have examined seem to indicate that there is a way to extend Reuman’s conjecture to general b, but it is still premature to formulate a precise statement.

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[Reu04] D. Reuman, Formulas for the dimensions of some affine Deligne-Lusztig varieties, Michigan Math. J. 52 (2004), no. 2, 435–451.

Ulrich Görtz, Mathematisches Institut, Universität Bonn, Beringstr. 1, 53115 Bonn, Germany

   E-mail address: ugoertz@math.uni-bonn.de

Thomas J. Haines, Mathematics Department, University of Maryland, College Park, MD 20742-4015

   E-mail address: tjh@math.umd.edu

Robert E. Kottwitz, Department of Mathematics, University of Chicago, 5734 University Avenue, Chicago, Illinois 60637

   E-mail address: kottwitz@math.uchicago.edu

Daniel C. Reuman, Laboratory of Populations, Rockefeller University, 1230 York Ave., New York, NY 10021

   E-mail address: reumand@rockefeller.edu

40 U. GÖRTZ, T. J. HAINES, R. E. KOTTWITZ, AND D. C. REUMAN

Figure 1. Non-empty affine Deligne-Lusztig varieties and their dimensions, type A_{2}, b = 1

DIMENSIONS OF SOME AFFINE DELIGNE-LUSZTIG VARIETIES 41

Figure 2. Non-empty affine Deligne-Lusztig varieties and their dimensions, type C_{2}, b = 1

42 U. GÖRTZ, T. J. HAINES, R. E. KOTTWITZ, AND D. C. REUMAN

Figure 3. Non-empty affine Deligne-Lusztig varieties and their dimensions, type G_{2}, b = 1

DIMENSIONS OF SOME AFFINE DELIGNE-LUSZTIG VARIETIES 43

Figure 4. Comparison with Reuman’s criterion, type A_{2}, b = 1

44 U. GÖRTZ, T. J. HAINES, R. E. KOTTWITZ, AND D. C. REUMAN

Figure 5. Comparison with Reuman’s criterion, type C_{2}, b = 1

DIMENSIONS OF SOME AFFINE DELIGNE-LUSZTIG VARIETIES 45

Figure 6. Comparison with Reuman’s criterion, type G_{2}, b = 1

46 U. GÖRTZ, T. J. HAINES, R. E. KOTTWITZ, AND D. C. REUMAN

Figure 7. Partial folding results, type A_{2}, b = 1, w = id

DIMENSIONS OF SOME AFFINE DELIGNE-LUSZTIG VARIETIES 47

Figure 8. Partial folding results, type A_{2}, b = 1, w = s_{1} Figure 9. Partial folding results, type A_{2}, b = 1, w = s_{1}s_{2}

DIMENSIONS OF SOME AFFINE DELIGNE-LUSZTIG VARIETIES 49

Figure 10. Partial folding results, type A_{2}, b = 1, w = s_{1}s_{2}s_{1}

50 U. GÖRTZ, T. J. HAINES, R. E. KOTTWITZ, AND D. C. REUMAN

Figure 11. Non-empty affine Deligne-Lusztig varieties and their dimensions, type A_{2}, b = ϵ^{(1,0,−1)}

DIMENSIONS OF SOME AFFINE DELIGNE-LUSZTIG VARIETIES 51

Figure 12. Non-empty affine Deligne-Lusztig varieties and their dimensions, type C_{2}, b = ϵ^{(1,0)}