Algebraic Zip Data

Richard Pink, Torsten Wedhorn, and Paul Ziegler

Received: November 11, 2010

Communicated by Peter Schneider

Abstract. An algebraic zip datum is a tuple Z = (G, P, Q, φ) consisting of a reductive group G together with parabolic subgroups P and Q and an isogeny φ: P/R_{u}P → Q/R_{u}Q. We study the action of the group E_{Z} := {(p, q) ∈ P ×Q | φ(π_{P} (p)) = π_{Q}(q)} on G given by ((p, q), g) ↦ pgq^{−1}. We define certain smooth E_{Z}-invariant subvarieties of G, show that they define a stratification of G. We determine their dimensions and their closures and give a description of the stabilizers of the E_{Z}-action on G. We also generalize all results to non-connected groups.

We show that for special choices of Z the algebraic quotient stack [E_{Z}\G] is isomorphic to [G\Z] or to [G\Z′], where Z is a G-variety studied by Lusztig and He in the theory of character sheaves on spherical compactifications of G and where Z′ has been defined by Moonen and the second author in their classification of F -zips. In these cases the E_{Z}-invariant subvarieties correspond to the so-called “G-stable pieces” of Z defined by Lusztig (resp. the G-orbits of Z′).

2010 Mathematics Subject Classification: 14L30 (20G15, 20F55, 20G40) Keywords and Phrases: Generalized G-stable pieces, zip data, F -zips

1 Introduction

1.1 Background

Let G be a connected reductive linear algebraic group over an algebraically closed field k. Then G × G acts on G via simultaneous left and right translation ((g_{1}, g_{2}), g) ↦ g_{1}gg_{2}^{−1}. In a series of papers, Lusztig ([Lus1], [Lus2]), He ([He2], [He1], [He3]), and Springer ([Spr3]) studied a certain spherical G × Gequivariant smooth compactification Ḡ of G. For G semi-simple adjoint this is the so-called wonderful compactification from [DCP]. In general the G × Gorbits Z_{I} ⊂ Ḡ are in natural bijection to the subsets I of the set of simple reflections in the Weyl group of G. Lusztig and He defined and studied socalled G-stable pieces in Z_{I}, which are certain subvarieties that are invariant under the diagonally embedded subgroup G ⊂ G × G. These G-stable pieces play an important role in their study of character sheaves on Ḡ. Lusztig and He also consider non-connected groups, corresponding to twisted group actions. Other generalizations of these varieties have been considered by Lu and Yakimow ([LY2]). A further motivation to study G-stable pieces comes from Poisson geometry: It was proved by Evens and Lu ([EL]), that for certain Poisson structure, each G-orbit on Z_{I} is a Poisson submanifold. In [MW] Moonen and the second author studied the De Rham cohomology H_{DR}^{•}(X/k) of a smooth proper scheme X with degenerating Hodge spectral sequence over an algebraically closed field k of positive characteristic. They showed that H_{DR}^{•}(X/k) carries the structure of a so-called F -zip, namely: it is a finite-dimensional k-vector space together with two filtrations (the “Hodge” and the “conjugate” filtration) and a Frobenius linear isomorphism between the associated graded vector spaces (the “Cartier isomorphism”). They showed that the isomorphism classes of F -zips of fixed dimension n and with fixed type of Hodge filtration are in natural bijection with the orbits under G := GL_{n,k} of a variant Z′_{I} of the G×G-orbit Z_{I} studied by Lusztig. They studied the varieties Z′_{I} for arbitrary reductive groups G and determined the G-orbits in them as analogues of the G-stable pieces in Z_{I}. By specializing G to classical groups they deduce from this a classification of F -zips with additional structure, e.g., with a non-degenerate symmetric or alternating form. They also consider nonconnected groups. Moreover, the automorphism group of an F -zip is isomorphic to the stabilizer in G of any corresponding point in Z′_{I}. When X varies in a smooth family over a base S, its relative De Rham cohomology forms a family of F -zips over S. The set of points of S where the F -zip lies in a given isomorphism class is a natural generalization of an Ekedahl-Oort stratum in the Siegel moduli space. Information about the closure of such a stratum corresponds to information about how the isomorphism class of an F -zip can vary in a family, and that in turn is equivalent to determining which G-orbits in Z′_{I} are contained in the closure of a given G-orbit. In each of these cases one is interested in the classification of the G-stable pieces, the description of their closures, and the stabilizers of points in G. In this article we give a uniform approach to these questions that generalizes all the above situations.

1.2 Main results

The central definition in this article is the following:

Definition 1.1. A connected algebraic zip datum is a tuple Z = (G, P, Q, φ) consisting of a connected reductive linear algebraic group G over k together with parabolic subgroups P and Q and an isogeny φ: P/R_{u}P → Q/R_{u}Q. The group

E_{Z} := {(p, q) ∈ P ×Q | φ(π_{P} (p)) = π_{Q}(q)}

is called the zip group associated to Z. It acts on G through the map ((p, q), g) ↦ pgq^{−1}. The union of the E_{Z}-orbits of all elements of a subset X ⊂ G is denoted by o_{Z}(X).

Fix such data Z = (G, P, Q, φ). To apply the machinery of Weyl groups to Z we choose a Borel subgroup B of G, a maximal torus T of B, and an element g of G such that B ⊂ Q, ^{g}B ⊂ P, φ(π_{P} (^{g}B)) = π_{Q}(B), and φ(π_{P} (^{g}T)) = π_{Q}(T). Let W denote the Weyl group of G with respect to T, and S ⊂ W the set of simple reflections corresponding to B. Let I ⊂ S be the type of the parabolic P and W_{I} ⊂ W its Weyl group. Let ^{I}W be the set of all w ∈ W that have minimal length in their coset W_{I}w. To each w ∈ ^{I}W we associate the E_{Z}-invariant subset

G^{w} = o_{Z}(gBẇB)  (1.2)

and prove (Theorems 5.10, 5.11 and 5.14):

Theorem 1.3. The E_{Z}-invariant subsets G^{w} form a pairwise disjoint decomposition of G into locally closed smooth subvarieties. The dimension of G^{w} is dim P + ℓ(w).

Next the isogeny φ induces an isomorphism of Coxeter system ψ: (W_{I}, I) →^{∼} (W_{J}, J) (see (3.11) for its precise definition), where W_{J} ⊂ W and J ⊂ S are the Weyl group and the type of the parablic subgroup Q. Let ≤ denote the Bruhat order on W. We prove (Theorem 6.2):

Theorem 1.4. The closure of G^{w} is the union of G^{w′} for all w′ ∈ ^{I}W such that there exists y ∈ W_{I} with yw′ψ(y)^{−1} ≤ w.

We call Z orbitally finite if the number of E_{Z}-orbits in G is finite. We give a necessary and sufficient criterion for this to happen (Proposition 7.1). In particular it happens when the differential of φ at 1 vanishes, for instance if φ is a Frobenius isogeny (Proposition 7.3). We prove (Theorem 7.5):

Theorem 1.5. If Z is orbitally finite, then each G^{w} is a single E_{Z}-orbit, and so the set {gẇ | w ∈ ^{I}W } is a set of representatives for the E_{Z}-orbits in G.

One can also consider the E_{Z}-orbit of gẇ for any element w ∈ W instead of just those in ^{I}W. It is then natural to ask when two such orbits lie in the same E_{Z}-invariant piece G^{w}. (For orbitally finite Z this is equivalent to asking when the orbits are equal.) We give an explicit description of this equivalence relation on W that depends only on the subgroup W_{I} and the homomorphism ψ (Theorem 9.17). We prove that all equivalence classes have the same cardinality #W_{I}, although they are in general no cosets of W_{I} and we do not know a simple combinatorial description for them. It is intriguing that we obtain analogous results for an abstract zip datum based on an arbitrary finitely generated Coxeter group (Theorem 9.11) or even an arbitrary abstract group (Theorem 9.6) in place of W. Other results include information on point stabilizers and infinitesimal stabilizers (Section 8), the generalization of the main results to non-connected groups (Section 10), a dual parametrization by a set W^{J} in place of ^{I}W (Section 11) and the relation with the varieties Z_{I} studied by Lusztig and He and their generalizations Z′_{I} (Section 12).

1.3 Applications

Let us explain why this theory of algebraic zip data is a generalization of the situations described in Subsection 1.1. In Section 12 we consider a connected reductive algebraic group G over k, an isogeny φ: G → G, a subset I of the set of simple reflections associated to G, and an element x in the Weyl group of G satisfying certain technical conditions. To such data we associate a certain algebraic variety X_{I,φ,x} with an action of G, a certain connected algebraic zip datum Z with underlying group G, and morphisms

G ⟵^{λ} G × G ⟶^{ρ} X_{I,φ,x}  (1.6)

In Theorem 12.8 we show that there is a closure preserving bijection between the E_{Z}-invariant subsets of A ⊂ G and the G-invariant subsets of B ⊂ X_{I,φ,x} given by λ^{−1}(A) = ρ^{−1}(B). We also prove that the stabilizer in E_{Z} of g ∈ G is isomorphic to the stabilizer in G of any point of the G-orbits in X_{I,φ,x} corresponding to the orbit of g. These results can also be phrased in the language of algebraic stacks, see Theorem 12.7.

In the special case φ = id_{G} the above X_{I,φ,x} is the variety Z_{I} defined by Lusztig. In Theorem 12.19 we verify that the subsets G^{w} ⊂ G correspond to the G-stable pieces defined by him. Thus Theorem 1.4 translates to a description of the closure relation between these G-stable pieces, which had been proved before by He [He2].

If char(k) is positive and φ is the Frobenius isogeny associated to a model of G over a finite field, the above X_{I,φ,x} is the variety Z′_{I} defined by Moonen and the second author. In this case the zip datum Z is orbitally finite, and so we obtain the main classification result for the G-orbits in Z′_{I} from [MW], the closure relation between these G-orbits, and the description of the stabilizers in G of points in Z′_{I}. In this case the closure relation had been determined in the unpublished note [Wed], the ideas of which are reused in the present article. Meanwhile Viehmann [Vie] has given a different proof of the closure relation in this case using the theory of loop groups. For those cases which pertain to the study of modulo p reductions of F -crystals with additional structure that show up in the study of special fibers of good integral models of Shimura varieties of Hodge type Moonen ([Moo]) and, more generally, Vasiu ([Vas]) have obtained similar classification results. In these cases Vasiu (loc. cit.) has also shown that the connected component of the stabilizers are unipotent.

For G = GL_{n} (resp. a classical group) we therefore obtain a new proof of the classification of F -zips (resp. of F -zips with additional structure) from [MW]. We can also deduce how F -zips (possibly with additional structure) behave in families, and can describe their automorphism groups as the stabilizers in E_{Z} of the corresponding points of G. This is applied in [VW] to the study of Ekedahl-Oort strata for Shimura varieties of PEL type.

1.4 Contents of the paper

In Section 2 we collect some results on algebraic groups and Coxeter systems that are used in the sequel. Algebraic zip data Z are defined in Section 3, where we also establish basic properties of the triple (B, T, g), called a frame of Z. Section 4 is based on the observation that every E_{Z}-orbit is contained in the double coset P gẋQ for some x ∈ W and meets the subset gẋM, where M is a Levi subgroup of Q. In it we define another zip datum Zẋ with underlying reductive group M and establish a number of results relating the E_{Z}-orbits in P gẋQ to the E_{Zẋ}-orbits in M. This is the main induction step used in most of our results. In Section 5 we give different descriptions of the E_{Z}-invariant subsets G^{w} for w ∈ ^{I}W and prove Theorem 1.3. In Section 6 we determine the closure of G^{w} and prove Theorem 1.4. Orbitally finite zip data are studied in Section 7, proving Theorem 1.5. Section 8 contains some results on point stabilizers and infinitesmial stabilizers. Abstract zip data are defined and studied in Section 9. In Section 10 our main results are generalized to algebraic zip data based on non-connected groups. In Section 11 we discuss a dual parametrization of the subsets G^{w} by a subset W^{J} of W in place of ^{I}W. Finally, in Section 12 we prove the results described in Subsection 1.3 above.

The paper is based on parts of the unpublished note [Wed] by the second author and the master thesis [Zie] by the third author, but goes beyond both.

After the referee pointed out to us the references [LY1] and [He3], we became aware that there Lu, Yakimov and He study a class of group actions which contains ours when φ is an isomorphism. In this case, Theorems 1.3 and 1.4 were already proven in [loc. cit]. Also, many of the ideas we have used to study the decomposition of G into E_{Z}-stable pieces are already present there.

Acknowledgements. We thank the referee for pointing out some references. The second author was partially supported by the SPP 1388 “Representation theory” of the DFG.

2 Preliminaries on algebraic groups and Coxeter groups

Throughout, the inner automorphism associated to an element h of a group G will be denoted int(h): G → G, g ↦ ^{h}g := hgh^{−1}. Similarly, for any subset X ⊂ G we set ^{h}X := hXh^{−1}.

2.1 General facts about linear algebraic groups

Throughout, we use the language of algebraic varieties over a fixed algebraically closed field k. By an algebraic group G we always mean a linear algebraic group over k. We let R_{u}G denote the unipotent radical of the identity component of G and π_{G}: G ↠ G/R_{u}G the canonical projection. An isogeny between two connected algebraic groups is a surjective homomorphism with finite kernel. Consider an algebraic group G, an algebraic subgroup H of G, and a quasiprojective variety X with a left action of H. Then we denote by G ×^{H} X the quotient of G×X under the left action of H defined by h·(g, x) = (gh^{−1}, h·x), which exists by [Ser], Section 3.2. The action of G on G×X by left multplication on the first factor induces a left action of G on G ×^{H} X. This is the pushout of X with respect to the inclusion H ↪ G.

Lemma 2.1. For G, H, and X as above, the morphism X → G ×^{H} X which sends x ∈ X to the class of (1, x) induces a closure-preserving bijection between the H-invariant subsets of X and the G-invariant subsets of G×^{H} X. If Y ⊂ X is an H-invariant subvariety of X, then the corresponding G-invariant subset of G ×^{H} X is the subvariety G ×^{H} Y of G ×^{H} X.

Proof. The morphism in question is the composite of the inclusion i: X → G×X, x ↦ (1, x) and the projection pr: G×X → G×^{H} X. Let (g, h) ∈ G×H act on G × X from the left by (g′, x) ↦ (gg′h^{−1}, h · x). Then the G × Hinvariant subsets of G×X are the sets of the form G×A for H-invariant subsets A ⊂ X. Therefore i induces a closure-preserving bijection between the Hinvariant subsets of X and the G×H-invariant subsets of G×X. Furthermore, since G ×^{H} X carries the quotient topology with respect to pr, the morphism pr induces a closure-preserving bijection between the G × H-invariant subsets of G×X and the G-invariant of G×^{H} X. Altogether this proves the claim.

Lemma 2.2 (see [Slo], Lemma 3.7.4). Let G be an algebraic group with an algebraic subgroup H. Let X be a variety with a left action of G. Let f: X → G/H be a G-equivariant morphism, and let Y ⊂ X be the fiber f^{−1}(H). Then Y is stabilized by H, and the map G ×^{H} Y → X sending the equivalence class of (g, y) to g · y defines an isomorphism of G-varieties.

Lemma 2.3. Let G be an algebraic group acting on an algebraic variety Z and let P ⊂ G be an algebraic subgroup such that G/P is proper. Then for any P -invariant subvariety Y ⊂ Z one has

G · \overline{Y} = \overline{G · Y} .

Proof. Clearly we have

G · Y ⊂ G · \overline{Y} ⊂ \overline{G · Y}

and therefore it suffices to show that G·\overline{Y} is closed in Z. The action π: G×Z → Z of G on Z induces a morphism π̄: G ×^{P} Z → Z which can be written as the composition

G ×^{P} Z ⟶^{∼} G/P × Z ⟶ Z.

Here the first morphism is the isomorphism given by [g, z] ↦ (gP, g · z) and the second morphism is the projection. As G/P is proper, we deduce that the morphism π̄ is closed. Now \overline{Y} is P -invariant and therefore G ×^{P} \overline{Y} is defined, and it is a closed subscheme of G ×^{P} Z. Therefore π̄(G ×^{P} \overline{Y}) = G · \overline{Y} is closed in Z.

The following statements concern images under twisted conjugation:

Theorem 2.4 (Lang-Steinberg, see [Ste], Theorem 10.1). Let G be a connected algebraic group and φ: G → G an isogeny with only a finite number of fixed points. Then the morphism G → G, g ↦ gφ(g)^{−1} is surjective.

Proposition 2.5. Let G be a connected reductive algebraic group with a Borel subgroup B and a maximal torus T ⊂ B. Let φ: G → G be an isogeny with φ(B) = B. In (b) also assume that φ(T) = T.

(a) The morphism G × B → G, (g, b) ↦ gbφ(g)^{−1} is surjective.

(b) The morphism G × T → G, (g, t) ↦ gtφ(g)^{−1} has dense image.

If G is simply connected semisimple and φ is an automorphism of G, (b) has been shown by Springer ([Spr2] Lemma 4).

Proof. For (a) see [Ste], Lemma 7.3. Part (b) and its proof are a slight modification of this. Equivalently we may show that for some t_{0} ∈ T, the image of the morphism α̃: G × T → G, (g, t) ↦ gtt_{0}φ(g)^{−1}t_{0}^{−1} is dense. For this it will suffice to show that the differential of α̃ at 1 is surjective. This differential is the linear map

Lie(G) × Lie(T) → Lie(G)
(X, Y) ↦ X + Y − Lie(φ_{t_{0}})(X),

where φ_{t_{0}} := int(t_{0}) ∘ φ. This linear map has image

Lie(T) + (1 − Lie(φ_{t_{0}})) Lie(G).

Let B^{−} be the Borel subgroup opposite to B with respect to T. Since φ(B) = B and φ(T) = T, the differential of φ_{t_{0}} at 1 preserves Lie(R_{u}B) and Lie(R_{u}B^{−}). If we find a t_{0} such that Lie(φ_{t_{0}}) has no fixed points on Lie(R_{u}B) and Lie(R_{u}B^{−}) we will be done.

Let Φ be the set of roots of G with respect to T. For each α ∈ Φ, let x_{α} be a basis vector of Lie(U_{α}), where U_{α} is the unipotent root subgroup of G associated to α. As the isogeny φ sends T to itself, it induces a bijection φ̃: Φ → Φ such that φ(U_{α}) = U_{φ̃(α)}. For each α ∈ Φ there exists a c(α) ∈ k such that Lie(φ)(x_{α}) = c(α)x_{φ̃(α)}. This implies Lie(φ_{t_{0}})(x_{α}) = α(t_{0})c(α)x_{φ̃(α)}. Since φ_{t_{0}} fixes R_{u}B and R_{u}B^{−}, its differential permutes Φ^{+} and Φ^{−}, where Φ^{+} (resp. Φ^{−}) is the set of roots that are positive (resp. negative) with respect to B. Hence Lie(φ_{t_{0}}) can only have a fixed point in Lie(R_{u}B) or Lie(R_{u}B^{−}) if there exists a cycle (α_{1}, ⋯, α_{n}) of the permutation φ̃ in Φ^{+} or Φ^{−} such that

∏_{i=1}^{n} α_{i}(t_{0})c(α_{i}) = 1.

This shows that for t_{0} in some non-empty open subset of T, the differential Lie(φ_{t_{0}}) has no fixed points on Lie(R_{u}B) and Lie(R_{u}B^{−}).

2.2 Coset Representatives in Coxeter Groups

Here we collect some facts about Coxeter groups and root systems which we shall need in the sequel. Let W be a Coxeter group and S its generating set of simple reflections. Let ℓ denote the length function on W; thus ℓ(w) is the smallest integer n ⩾ 0 such that w = s_{1} ⋯ s_{n} for suitable s_{i} ∈ S. Any such product with ℓ(w) = n is called a reduced expression for w. Let I be a subset of S. We denote by W_{I} the subgroup of W generated by I, which is a Coxeter group with set of simple reflections I. Also, we denote by W^{I} (respectively^{I}W) the set of elements w of W which have minimal length in their coset wW_{I} (respectively W_{I}w). Then every w ∈ W can be written uniquely as w = w^{I} · w_{I} = w̃_{I} · ^{I}w with w_{I}, w̃_{I} ∈ W_{I} and w^{I} ∈ W^{I} and ^{I}w ∈ ^{I}W. Moreover, these decompositions satisfy ℓ(w) = ℓ(w_{I}) + ℓ(w^{I}) = ℓ(w̃_{I}) + ℓ(^{I}w) (see [DDPW], Proposition 4.16). In particular, W^{I} and ^{I}W are systems of representatives for the quotients W/W_{I} and W_{I}\W, respectively. The fact that ℓ(w) = ℓ(w^{−1}) for all w ∈ W implies that W^{I} = (^{I}W)^{−1}. Furthermore, if J is a second subset of S, we let ^{I}W^{J} denote the set of x ∈ W which have minimal length in the double coset W_{I}xW_{J}. Then^{I}W^{J} = ^{I}W ∩W^{J}, and it is a system of representatives for W_{I}\W/W_{J} (see [DDPW] (4.3.2)). In the next propositions we take an element x ∈ ^{I}W^{J}, consider the conjugate subset ^{x^{−1}}I ⊂ W, and abbreviate I_{x} := J ∩^{x^{−1}}I ⊂ J. Then ^{I_{x}}W_{J} is the set of elements w_{J} of W_{J} which have minimal length in their coset W_{I_{x}}w_{J}. Likewise W_{I}^{I∩^{x}J} is the set of elements w_{I} of W_{I} which have minimal length in their coset w_{I}W_{I∩^{x}J}.

Proposition 2.6 (Kilmoyer, [DDPW], Proposition 4.17). For x ∈ ^{I}W^{J} we have

W_{I} ∩^{x}W_{J} = W_{I∩^{x}J}  and  W_{J} ∩^{x^{−1}}W_{I} = W_{J∩^{x^{−1}}I} = W_{I_{x}}.

Proposition 2.7 (Howlett, [DDPW], Proposition 4.18). For any x ∈ ^{I}W^{J}, every element w of the double coset W_{I}xW_{J} is uniquely expressible in the form w = w_{I}xw_{J} with w_{I} ∈ W_{I} and w_{J} ∈ ^{I_{x}}W_{J}. Moreover, this decomposition satisfies

ℓ(w) = ℓ(w_{I}xw_{J}) = ℓ(w_{I}) + ℓ(x) + ℓ(w_{J}).

Proposition 2.8. The set ^{I}W is the set of all xw_{J} for x ∈ ^{I}W^{J} and w_{J} ∈ I_{x}W_{J}.

Proof. Take x ∈ ^{I}W^{J} and w_{J} ∈ ^{I_{x}}W_{J}. Then for any w_{I} ∈ ^{I}W, Proposition 2.7 applied to the product w_{I}xw_{J} implies that ℓ(w_{I}xw_{J}) = ℓ(w_{I}) + ℓ(x) + ℓ(w_{J}) ≥ ℓ(x) + ℓ(w_{J}) = ℓ(xw_{J}). This proves that xw_{J} ∈ ^{I}W. Conversely take w ∈ ^{I}W and let w = w_{I}xw_{J} be its decomposition from Proposition 2.7. Then by the first part of the proof we have xw_{J} ∈ ^{I}W. Since W_{I}w = W_{I}xw_{J}, this implies that w = xw_{J}.

Proposition 2.9. The set W^{J} is the set of all w_{I}x for x ∈ ^{I}W^{J} and w_{I} ∈ W_{I}^{I∩^{x}J}.

Proof. Apply Proposition 2.8 with I and J interchanged and invert all elements of W^{J}.

Next we recall the Bruhat order on W, which we denote by ≤ and <. This natural partial order is characterized by the following property: For x, w ∈ W we have x ≤ w if and only if for some (or, equivalently, any) reduced expression w = s_{1} ⋯ s_{n} as a product of simple reflections s_{i} ∈ S, one gets a reduced expression for x by removing certain s_{i} from this product. More information about the Bruhat order can be found in [BB], Chapter 2. Using this order, the set ^{I}W can be described as

IW = {w ∈ W | w < sw for all s ∈ I}  (2.10)

(see [BB], Definition 2.4.2 and Corollary 2.4.5). Assume in addition that W is the Weyl group of a root system Φ, with S corresponding to a basis of Φ. Denote the set of positive roots with respect to the given basis by Φ^{+} and the set of negative roots by Φ^{−}. For I ⊂ S, let Φ_{I} be the root system spanned by the basis elements corresponding to I, and set Φ_{I}^{±} := Φ_{I} ∩ Φ^{±}. Then by [Car], Proposition 2.3.3 we have

W^{I} = {w ∈ W | wΦ_{I}^{+} ⊂ Φ^{+}} = {w ∈ W | wΦ_{I}^{−} ⊂ Φ^{−}}.  (2.11)

Also, by [Car], Proposition 2.2.7, the length of any w ∈ W is

ℓ(w) = #{α ∈ Φ^{+} | wα ∈ Φ^{−}}.  (2.12)

Lemma 2.13. Let w ∈ ^{I}W and write w = xw_{J} with x ∈ ^{I}W^{J} and w_{J} ∈ W_{J}. Then

ℓ(x) = #{α ∈ Φ^{+} ∖ Φ_{J} | wα ∈ Φ^{−} ∖ Φ_{I}}.

Proof. First note that α ∈ Φ^{+} and wα ∈ Φ^{−} already imply wα ∉ Φ_{I}, because otherwise we would have α ∈ w^{−1}Φ_{I}^{−}, which by (2.11) is contained in Φ^{−} because w^{−1} ∈ W^{I}. Thus the right hand side of the claim is #{α ∈ Φ^{+} ∖ Φ_{J} | wα ∈ Φ^{−}}. Secondly, if α ∈ Φ_{J}^{+}, using again (2.11) and x ∈ W^{J} we find that wα ∈ Φ^{−} if and only if w_{J}α ∈ Φ_{J}^{−}. Thus with (2.12) we obtain

#{α ∈ Φ^{+} ∖ Φ_{J} | wα ∈ Φ^{−}} = #{α ∈ Φ^{+} | wα ∈ Φ^{−}} − #{α ∈ Φ_{J}^{+} | w_{J}α ∈ Φ_{J}^{−}}
= ℓ(w) − ℓ(w_{J}) = ℓ(x).

2.3 Reductive groups, Weyl groups, and parabolics

Let G be a connected reductive algebraic group, let B be a Borel subgroup of G, and let T be a maximal torus of B. Let Φ(G, T) denote the root system of G with respect to T, let W (G, T) := Norm_{G}(T)/T denote the associated Weyl group, and let S(G, B, T) ⊂ W (G, T) denote the set of simple reflections defined by B. Then W (G, T) is a Coxeter group with respect to the subset S(G, B, T). A priori this data depends on the pair (B, T). However, any other such pair (B′, T′) is obtained on conjugating (B, T) by some element g ∈ G which is unique up to right multiplication by T. Thus conjugation by g induces isomorphisms Φ(G, T) →^{∼} Φ(G, T′) and W (G, T) →^{∼} W (G, T′) and S(G, B, T) →^{∼} S(G, B′, T′) that are independent of g. Moreover, the isomorphisms associated to any three such pairs are compatible with each other. Thus Φ := Φ(G, T) and W := W (G, T) and S := S(G, B, T) for any choice of (B, T) can be viewed as instances of ‘the’ root system and ‘the’ Weyl group and ‘the’ set of simple reflections of G, in the sense that up to unique isomorphisms they depend only on G. It then also makes sense to say that the result of a construction (as in Subsection 5.2 below) depending on an element of W is independent of (B, T). For any w ∈ W (G, T) we fix a representative ẇ ∈ Norm_{G}(T). By choosing representatives attached to a Chevalley system (see [DG] Exp. XXIII, §6) for all w_{1}, w_{2} ∈ W with ℓ(w_{1}w_{2}) = ℓ(w_{1}) + ℓ(w_{2}) we obtain

ẇ_{1}ẇ_{2} = (w_{1}w_{2})˙ .  (2.14)

In particular the identity element 1 ∈ W is represented by the identity element 1 ∈ G. A parabolic subgroup of G that contains B is called a standard parabolic of G. Any standard parabolic possesses a unique Levi decomposition P = R_{u}P ⋊ L with T ⊂ L. Any such L is called a standard Levi subgroup of G, and the set I of simple reflections in the Weyl group of L is called the type of L or of P. In this way there is a unique standard parabolic P_{I} of type I for every subset I ⊂ S, and vice versa. The type of a general parabolic P is by definition the type of the unique standard parabolic conjugate to P; it is independent of (B, T) in the above sense. Any conjugate of a standard Levi subgroup of G is called a Levi subgroup of G.

For any subset I ⊂ S let Par_{I} denote the set of all parabolics of G of type I. Then there is a natural bijection G/P_{I} →^{∼} Par_{I}, gP_{I} ↦ ^{g}P_{I}. For any two subsets I, J ⊂ S we let G act by simultaneous conjugation on Par_{I} × Par_{J}. As a consequence of the Bruhat decomposition (see [Spr1] 8.4.6 (3)), the G-orbit of any pair (P, Q) ∈ Par_{I} × Par_{J} contains a unique pair of the form (P_{I}, ẋP_{J}) with x ∈ ^{I}W^{J}. This element x is called the relative position of P and Q and is denoted by relpos(P, Q). We will also use several standard facts about intersections of parabolics and/or Levi subgroups, for instance (see [Car], Proposition 2.8.9):

Proposition 2.15. Let L be a Levi subgroup of G and T a maximal torus of L. Let P be a parabolic subgroup of G containing T and P = R_{u}P ⋊ H its Levi decomposition with T ⊂ H. Then L ∩ P is a parabolic subgroup of L with Levi decomposition

L ∩ P = (L ∩ R_{u}P) ⋊ (L ∩ H).

If P is a Borel subgroup of G, then L ∩ P is a Borel subgroup of L.

3 Connected algebraic zip data

We now define the central technical notions of this article.

Definition 3.1. A connected algebraic zip datum is a tuple Z = (G, P, Q, φ) consisting of a connected reductive group G with parabolic subgroups P and Q and an isogeny φ: P/R_{u}P → Q/R_{u}Q. The group

E_{Z} := {(p, q) ∈ P ×Q | φ(π_{P} (p)) = π_{Q}(q)}  (3.2)

is called the zip group associated to Z. It acts on G by restriction of the left action

(P ×Q) × G → G, ((p, q), g) ↦ pgq^{−1}.  (3.3)

For any subset X ⊂ G we denote the union of the E_{Z}-orbits of all elements of X by

o_{Z}(X).  (3.4)

Note that if X is a constructible subset of G, then so is o_{Z}(X). Throughout the following sections we fix a connected algebraic zip datum Z = (G, P, Q, φ). We also abbreviate U := R_{u}P and V := R_{u}Q, so that φ is an isogeny P/U → Q/V. Our aim is to study the orbit structure of the action of E_{Z} on G.

Example 3.5. For dimension reasons we have P = G if and only if Q = G. In that case the action of E_{Z} = graph(φ) is equivalent to the action of G on itself by twisted conjugation (h, g) ↦ hgφ(h)^{−1}.

In order to work with Z it is convenient to fix the following data.

Definition 3.6. A frame of Z is a tuple (B, T, g) consisting of a Borel subgroup B of G, a maximal torus T of B, and an element g ∈ G, such that

(a) B ⊂ Q,

(b) ^{g}B ⊂ P,

(c) φ(π_{P} (^{g}B)) = π_{Q}(B), and

(d) φ(π_{P} (^{g}T)) = π_{Q}(T).

Proposition 3.7. Every connected algebraic zip datum possesses a frame.

Proof. Choose a Borel subgroup B of Q and a maximal torus T of B. Let T̄′ ⊂ B̄′ ⊂ P/U denote the respective identity components of φ^{−1}(π_{Q}(T)) ⊂ φ^{−1}(π_{Q}(B)) ⊂ P/U. Then B̄′ is a Borel subgroup of P/U, and T̄′ is a maximal torus of B̄′. Thus we have B̄′ = π_{P} (B′) for a Borel subgroup B′ of P, and T̄′ = π_{P} (T′) for some maximal torus T′ of B′. Finally take g ∈ G such that B′ = ^{g}B and T′ = ^{g}T. Then (B, T, g) is a frame of Z.

Proposition 3.8. Let (B, T, g) be a frame of Z. Then every frame of Z has the form (^{q}B,^{q}T, pgtq^{−1}) for (p, q) ∈ E_{Z} and t ∈ T, and every tuple of this form is a frame of Z.

Proof. Let (B′, T′, g′) be another frame of Z. Since all Borel subgroups of Q are conjugate, we have B′ = ^{q}B for some element q ∈ Q. Since all maximal tori of B′ are conjugate, after multiplying q on the left by an element of B′ we may in addition assume that T′ = ^{q}T. Similarly we can find an element p ∈ P such that ^{g′}B′ = ^{pg}B and ^{g′}T′ = ^{pg}T. Combining these equations with the defining properties of frames we find that

φ(π_{P} (p))π_{Q}(B) = ^{φ(π_{P} (p))}φ(π_{P} (^{g}B)) = φ(π_{P} (^{pg}B)) = φ(π_{P} (^{g′}B′)) =
= π_{Q}(B′) = π_{Q}(^{q}B) = ^{π_{Q}(q)}π_{Q}(B),

and similarly ^{φ(π_{P} (p))}π_{Q}(T) = ^{π_{Q}(q)}π_{Q}(T). Thus φ(π_{P} (p)) = π_{Q}(q) · π_{Q}(t′) for some element t′ ∈ T. Since we may still replace q by qt′ without changing the above equations, we may without loss of generality assume that φ(π_{P} (p)) = π_{Q}(q), so that (p, q) ∈ E_{Z}. On the other hand, the above equations imply that B = ^{g^{−1}p^{−1}g′q}B and T = ^{g^{−1}p^{−1}g′q}T, so that t := g^{−1}p^{−1}g′q ∈ T and hence g′ = pgtq^{−1}. This proves the first assertion. The second involves a straightforward calculation that is left to the conscientious reader.

Throughout the following sections we fix a frame (B, T, g) of Z. It determines unique Levi components ^{g}T ⊂ L ⊂ P and T ⊂ M ⊂ Q. Via the isomorphisms L →^{∼} P/U and M →^{∼} Q/V we can then identify φ with an isogeny φ: L → M. The zip group then becomes

E_{Z} = {(uℓ, vφ(ℓ)) | u ∈ U, v ∈ V, ℓ ∈ L}  (3.9)

and acts on G by ((uℓ, vφ(ℓ)), g) ↦ uℓgφ(ℓ)^{−1}v^{−1}. Moreover, conditions 3.6 (c) and (d) are then equivalent to

φ(^{g}B ∩ L) = B ∩ M,  and  φ(^{g}T) = T,  (3.10)

which are a Borel subgroup and a maximal torus of M, respectively. Let Φ be the root system, W the Weyl group, and S ⊂ W the set of simple reflections of G with respect to (B, T). Let I ⊂ S be the type of^{g^{−1}}P and J ⊂ S the type of Q. Then M has root system Φ_{J}, Weyl group W_{J}, and set of simple reflections J ⊂ W_{J}. Similarly ^{g^{−1}}L has root system Φ_{I}, Weyl group W_{I}, and set of simple reflections I ⊂ W_{I}, and the inner automorphism int(g) identifies these with the corresponding objects associated to L. Moreover, the equations (3.10) imply that φ ∘ int(g) induces an isomorphism of Coxeter systems

ψ: (W_{I}, I) →^{∼} (W_{J}, J).  (3.11)

Recall that Φ, W, and S can be viewed as independent of the chosen frame, as explained in Subsection 2.3.

Proposition 3.11. The subsets I, J and the isomorphism ψ are independent of the frame.

Proof. Consider another frame (^{q}B,^{q}T, pgtq^{−1}) with (p, q) ∈ E_{Z} and t ∈ T, as in Proposition 3.8. Then we have a commutative diagram

\xymatrix{(^{g^{−1}}L, B, T) \ar[d]_{int(qt^{−1})}^{≀} \ar[r]^{int(g)}_{∼} & (L,^{g}B,^{g}T) \ar[d]_{int(p)}^{≀} \ar[r]^{φ} & (M, B, T) \ar[d]_{int(q)}^{≀} \\ (^{qg^{−1}}L,^{q}B,^{q}T) \ar[r]^{int(pgtq^{−1})}_{∼} & (^{p}L,^{pg}B,^{pg}T) \ar[r]^{φ} & (^{q}M,^{q}B,^{q}T),}

whose upper row contains the data inducing ψ for the old frame and whose lower row is the analogue for the new frame. Since the vertical arrows are inner automorphisms, they induce the identity on the abstract Coxeter system (W, S) of G as explained in Subsection 2.3. Everything follows from this.

4 Induction step

We keep the notations of the preceding section. Since ^{g^{−1}}P and Q are parabolic subgroups containing the same Borel B, by Bruhat (see [Spr1] 8.4.6 (3)) we have a disjoint decomposition

G = ∐_{x∈^{I}W^{J}} ^{g^{−1}}P ẋQ.

Left translation by g turns this into a disjoint decomposition

G = ∐_{x∈^{I}W^{J}} P gẋQ.  (4.1)

Here each component P gẋQ is an irreducible locally closed subvariety of G that is invariant under the action of E_{Z}. In this section we fix an element x ∈ ^{I}W^{J} and establish a bijection between the E_{Z}-orbits in P gẋQ and the orbits of another zip datum constructed from Z and ẋ. This will allow us to prove facts about the orbits inductively. The base case of the induction occurs when the decomposition possesses just one piece, i.e., when P = Q = G.

Lemma 4.2. The stabilizer of gẋQ ⊂ P gẋQ in E_{Z} is the subgroup

E_{Z,ẋ} := {(p, q) ∈ E_{Z} | p ∈ P ∩^{gẋ}Q},

and the action of E_{Z} induces an E_{Z}-equivariant isomorphism

E_{Z} ×^{E_{Z,ẋ}} gẋQ ⟶^{∼} P gẋQ, [((p, q), h)] ↦ phq^{−1}.

Proof. The action (3.3) of (p, q) ∈ E_{Z} on P gẋQ induces the action on the quotient P gẋQ/Q by left multiplication with p. From (3.2) we see that the latter action is transitive, and the stabilizer of the point gẋQ is E_{Z,ẋ}; hence there is an E_{Z}-equivariant isomorphism P gẋQ/Q ≅ E_{Z}/E_{Z,ẋ}. Thus everything follows by applying Lemma 2.2 to the projection morphism P gẋQ ↠ P gẋQ/Q ≅ E_{Z}/E_{Z,ẋ}.

Construction 4.3. Consider the following subgroups of the connected reductive algebraic group M (which are independent of the representative ẋ of x):

\begin{matrix}P_{x} := M ∩^{ẋ^{−1}g^{−1}}P, & Q_{x} := φ(L ∩^{gẋ}Q), \\ U_{x} := M ∩^{ẋ^{−1}g^{−1}}U, & V_{x} := φ(L ∩^{gẋ}V), \\ L_{x} := M ∩^{ẋ^{−1}g^{−1}}L, & M_{x} := φ(L ∩^{gẋ}M). \end{matrix}

Proposition 2.15 shows that P_{x} is a parabolic with unipotent radical U_{x} and Levi component L_{x}, and that Q_{x} is a parabolic with unipotent radical V_{x} and Levi component M_{x}. Moreover, φ ∘ int(gẋ) induces an isogeny φẋ: L_{x} → M_{x}, or equivalently P_{x}/U_{x} → Q_{x}/V_{x}. Thus we obtain a connected algebraic zip datum Zẋ := (M, P_{x}, Q_{x}, φẋ). By (3.9) its zip group is

E_{Zẋ} = {(u′ℓ′, v′φẋ(ℓ′)) | u′ ∈ U_{x}, v′ ∈ V_{x}, ℓ′ ∈ L_{x}}.  (4.4)

Lemma 4.5. There is a surjective homomorphism

E_{Z,ẋ} ↠ E_{Zẋ}, (p, q) ↦ (m, φ(ℓ)),

where p = uℓ for u ∈ U and ℓ ∈ L, and ^{ẋ^{−1}g^{−1}}p = vm for v ∈ V and m ∈ M.

Proof. For ease of notation abbreviate h := gẋ, so that ^{h}T = ^{g}T ⊂ L and therefore T ⊂ ^{h^{−1}}L ⊂ ^{h^{−1}}P. Thus ^{h^{−1}}P and Q are parabolics of G with T - invariant Levi decompositions ^{h^{−1}}P = ^{h^{−1}}U ⋊^{h^{−1}}L and Q = V ⋊ M. It follows (see [Car] Thm. 2.8.7) that any element of^{h^{−1}}P ∩Q can be written as a product abu′ℓ′ with unique

\begin{matrix}a ∈ ^{h^{−1}}U ∩ V, & u′ ∈ ^{h^{−1}}U ∩ M = U_{x}, \\ b ∈ ^{h^{−1}}L ∩ V, & ℓ′ ∈ ^{h^{−1}}L ∩ M = L_{x}. \end{matrix}

Consider (p, q) ∈ E_{Z,ẋ} with p = uℓ and ^{h^{−1}}p = vm as in the lemma. Then we can write the element ^{h^{−1}}p = abu′ℓ′ ∈ ^{h^{−1}}P ∩ Q in the indicated fashion. Comparing the different factorizations yields the equations v = ab, m = u′ℓ′, u = ^{h}(abu′b^{−1}), and ℓ = ^{h}(bℓ′). Thus φ(ℓ) = φ(^{h}b)φ(^{h}ℓ′) = v′φẋ(ℓ′) with v′ := φ(^{h}b) ∈ φ(L ∩ ^{h}V) = V_{x}. In view of (4.4) it follows that (m, φ(ℓ)) = (u′ℓ′, v′φẋ(ℓ′)) lies in E_{Zẋ}, and so the map in question is well-defined. Since m and ℓ are obtained from p by projection to Levi components, the map is a homomorphism. Conversely, every element of E_{Zẋ} can be obtained in this way from some element p ∈ P ∩ ^{h}Q. By (3.9) we can then also find q ∈ Q with (p, q) ∈ E_{Z,ẋ}. Thus the map is surjective, and we are done.

Lemma 4.6. The surjective morphism

π: gẋQ ↠ M, gẋm̃ṽ ↦ m̃

for m̃ ∈ M and ṽ ∈ V is equivariant under the group E_{Z,ẋ}, which acts on gẋQ as in Lemma 4.2 and on M through the homomorphism from Lemma 4.5.

Proof. Take (p, q) ∈ E_{Z,ẋ} with p = uℓ and ^{ẋ^{−1}g^{−1}}p = vm as in Lemma 4.5. Then (3.9) implies that q = v_{1}φ(ℓ) for some v_{1} ∈ V. Thus the action of (p, q) sends gẋm̃ṽ ∈ gẋQ to the element

pgẋ · m̃ṽ · q^{−1} = gẋvm · m̃ṽ · φ(ℓ)^{−1}v_{1}^{−1} = gẋ · mm̃φ(ℓ)^{−1} · (an element of V ).

The morphism π maps this element to mm̃φ(ℓ)^{−1} ∈ M. But this is also the image of m̃ = π(gẋm̃ṽ) under the action of (m, φ(ℓ)) ∈ E_{Zẋ}. Thus the morphism is equivariant.

Proposition 4.7. There is a closure-preserving bijection between E_{Zẋ}- invariant subsets Y ⊂ M and E_{Z}-invariant subsets X ⊂ P gẋQ, defined by Y = M ∩ ẋ^{−1}g^{−1}X and X = o_{Z}(gẋY). Moreover, Y is a subvariety if and only if X is one, and in that case X ≅ E_{Z} ×^{E_{Z,ẋ}} π^{−1}(Y).

Proof. From (3.2) and (3.3) we see that the subgroup V ≅ {(1, v) | v ∈ V } ⊂ E_{Z,ẋ} acts by right translation on gẋQ. Thus every E_{Z,ẋ}-invariant subset of gẋQ is a union of cosets of V and therefore of the form Z = gẋY V = π^{−1}(Y) for a subset Y ⊂ M, which moreover satisfies Y = M ∩ ẋ^{−1}g^{−1}Z. By Lemmas 4.5 and 4.6 this defines a bijection between E_{Zẋ}-invariant subsets Y ⊂ M and E_{Z,ẋ}-invariant subsets Z ⊂ gẋQ. On the other hand, Lemmas 2.1 and 4.2 yield a bijection between E_{Z,ẋ}-invariant subsets Z ⊂ gẋQ and E_{Z}-invariant subsets X ⊂ P gẋQ that is characterized by Z = gẋQ∩X and X = o_{Z}(Z). Together we obtain the desired bijection with Y = M ∩ ẋ^{−1}g^{−1}(gẋQ ∩ X) = M ∩ ẋ^{−1}g^{−1}X and X = o_{Z}(gẋY V) = o_{Z}(gẋY). The equations Z = π^{−1}(Y) and Y = M ∩ ẋ^{−1}g^{−1}Z imply that the bijection between Y and Z preserves closures and maps subvarieties to subvarieties. The corresponding facts for the bijection between Z and X follow from Lemma 2.1, which also implies the last statement.

Proposition 4.8. If X and Y in Proposition 4.7 are subvarieties, then

dim X = dim Y + dim P − dim P_{x} + ℓ(x).

Proof. By the definition of E_{Z,ẋ} we have

dim E_{Z} −dim E_{Z,ẋ} = dim P −dim(P ∩^{gẋ}Q) = dim P −dim P_{x}−dim(P ∩^{gẋ}V).

With the last statement of Proposition 4.7 this implies that

dim X = dim Y + dim V + dim P − dim P_{x} − dim(P ∩^{gẋ}V).

From the decomposition of V into root subgroups it follows that dim V − dim(P ∩^{gẋ}V) = dim V − dim(V ∩^{ẋ^{−1}g^{−1}}P) is the cardinality of the set

{α ∈ Φ^{+} ∖ Φ_{J} | xα ∈ Φ^{−} ∖ Φ_{I}}.

By Lemma 2.13 for w_{J} = 1 this cardinality is ℓ(x).

Lemma 4.9. For any subset Y ⊂ M we have o_{Z}(gẋ o_{Zẋ}(Y)) = o_{Z}(gẋY).

Proof. It suffices to show that gẋ o_{Zẋ}(Y) ⊂ o_{Z}(gẋY), which follows from a straightforward calculation that is left to the reader. Alternatively the formula can be deduced from the formal properties stated in Proposition 4.7.

We can also give an inductive description of the stabilizers of points in P gẋQ. However, this does not give the scheme-theoretic stabilizers, which may in fact be non-reduced. Likewise, the following lemma does not describe the schemetheoretic kernel:

Lemma 4.10. The kernel of the homomorphism from Lemma 4.5 is (U ∩^{gẋ}V)× V.

Proof. Let p = uℓ and ^{ẋ^{−1}g^{−1}}p = vm be as in Lemma 4.5. Then (p, q) is in the kernel if and only if m = 1 and φ(ℓ) = 1. The first equation is equivalent to p = ^{gẋ}v ∈ ^{gẋ}V, which implies that ℓ is unipotent. Being in the kernel of the isogeny φ is then equivalent to ℓ = 1. Thus the second equation is equivalent to p ∈ U, and the two together are equivalent to p ∈ U ∩ ^{gẋ}V. By (3.9) we then have q ∈ V, and so we are done.

Proposition 4.11. For any m ∈ M there is a short exact sequence

1 ⟶ U ∩^{gẋ}V ⟶ Stab_{E_{Z}} (gẋm) ⟶^{4.5} Stab_{E_{Zẋ}} (m) ⟶ 1.

Proof. The second half of Lemma 4.2 and Lemma 4.6 imply that we have an equality, respectively a homomorphism

Stab_{E_{Z}} (gẋm) = Stab_{E_{Z,ẋ}}(gẋm) ⟶^{4.5} Stab_{E_{Zẋ}} (m).

This homomorphism is surjective, because the subgroup V ≅ {(1, v) | v ∈ V } contained in the kernel of the surjection E_{Z,ẋ} ↠ E_{Zẋ} acts transitively on the fibers of π. By Lemma 4.10 the kernel is the stabilizer of gẋm in the group (U ∩^{gẋ}V) × V acting by left and right translation. This stabilizer consists of (u,^{(gẋm)^{−1}}u) for all u ∈ U ∩^{gẋ}V, and we are done.

Finally, the assumption x ∈ ^{I}W^{J} allows us to construct a frame of Zẋ:

Proposition 4.12. The tuple (M ∩B, T, 1) is a frame of Zẋ, and the associated Levi components of P_{x} and Q_{x} are L_{x} and M_{x}, respectively.

Proof. First, the assumptions T ⊂ M and ^{gẋ}T = ^{g}T ⊂ L imply that T ⊂ M ∩ ẋ^{−1}g^{−1}L, the latter being L_{x} by Construction 4.3. Together with the equation φ(^{g}T) = T from (3.10) they also imply that T = φ(^{gẋ}T) ⊂ φ(L ∩^{gẋ}M), the latter being M_{x} by Construction 4.3. This proves the statement about the Levi components. We can also directly deduce that φẋ(T) = φ(^{gẋ}T) = T. Next, as T is a common maximal torus of M and B, Proposition 2.15 implies that M ∩ B is a Borel subgroup of M. Recall that M has the root system Φ_{J}, so that M ∩ B corresponds to the subset Φ_{J}^{+} = Φ_{J} ∩ Φ^{+}. For the same reasons M ∩^{ẋ^{−1}}B is a Borel subgroup of M corresponding to the subset Φ_{J} ∩ x^{−1}Φ^{+}. But with (2.11) the assumption x ∈ ^{I}W^{J} ⊂ W^{J} implies that xΦ_{J}^{+} ⊂ Φ^{+}, and hence Φ_{J}^{+} ⊂ Φ_{J} ∩ x^{−1}Φ^{+}. Since both subsets correspond to Borel subgroups, they must then coincide, and therefore M ∩ B = M ∩^{ẋ^{−1}}B. With the inclusion gB ⊂ P from (3.6) we deduce that

M ∩ B = M ∩^{ẋ^{−1}}B ⊂ M ∩^{ẋ^{−1}g^{−1}}P =^{4.3} P_{x}.

In the same way one shows that L ∩^{g}B = L ∩^{gẋ}B, which together with B ⊂ Q implies that

M ∩ B =^{(3.10)} φ(L ∩^{g}B) = φ(L ∩^{gẋ}B) ⊂ φ(L ∩^{gẋ}Q) =^{4.3} Q_{x}.

The equation M ∩ B = φ(L ∩^{gẋ}B) and Construction 4.3 also imply that

φẋ((M ∩ B) ∩ L_{x}) = φ(^{gẋ}M ∩^{gẋ}B ∩ L) ⊂

⊂ φ(L ∩^{gẋ}M) ∩ φ(L ∩^{gẋ}B) = (M ∩ B) ∩ M_{x}.

As both sides of this inclusion are Borel subgroups of M_{x}, they must be equal. Thus (M ∩ B, T, 1) satisfies Definition 3.6 in the variant (3.10), as desired.

Recall that M has the Weyl group W_{J} with the set of simple reflections J, and that ψ: W_{I} →^{∼} W_{J} is the isomorphism induced by φ ∘ int(g).

Proposition 4.13. (a) The type of the parabolic P_{x} of M is I_{x} := J ∩^{x^{−1}}I.

(b) The type of the parabolic Q_{x} of M is J_{x} := ψ(I ∩^{x}J).

(c) The isomomorphism ψ_{x}: W_{I_{x}} →^{∼} W_{J_{x}} induced by φẋ is the restriction of ψ ∘ int(x).

Proof. Proposition 2.6 implies that L_{x} = M ∩ ^{ẋ^{−1}g^{−1}}L has the Weyl group W_{J} ∩ ^{x^{−1}}W_{I} = W_{I_{x}}, which shows (a). Likewise M_{x} = φ(L ∩ ^{gẋ}M) has the Weyl group ψ(W_{I} ∩^{x}W_{J}) = W_{J_{x}}, which implies (b). Finally, (c) follows from φẋ = φ ∘ int(gẋ).

5 Decomposition of G

In this section we construct a natural decomposition of G into finitely many E_{Z}-invariant subvarieties G^{w}.

5.1 The Levi subgroup H_{w}

Fix an element w ∈ ^{I}W. Note that we can compare any subgroup H of^{˙w^{−1}g^{−1}}L with its image φ∘int(gẇ)(H) in M, because both are subgroups of G. Moreover, the collection of all such H satisfying φ ∘ int(gẇ)(H) = H possesses a unique largest element, namely the subgroup generated by all such subgroups.

Definition 5.1. We let H_{w} denote the unique largest subgroup of ^{˙w^{−1}g^{−1}}L satisfying φ ∘ int(gẇ)(H_{w}) = H_{w}. We let φẇ: H_{w} → H_{w} denote the isogeny induced by φ ∘ int(gẇ), and let H_{w} act on itself from the left by the twisted conjugation (h, h′) ↦ hh′φẇ(h)^{−1}.

Remark 5.2. Since φ ∘ int(gẇ)(T) = φ(^{g}T) = T by (3.10), the defining property of H_{w} implies that T ⊂ H_{w}. Thus H_{w} does not depend on the choice of representative ẇ of w, justifying the notation H_{w}. Also, in the case that w = x ∈ ^{I}W^{J} observe that the φẇ defined here is the restriction to H_{w} of the isogeny φẋ from Construction 4.3. Using the same notation for both is therefore only mildly abusive.

Example 5.3. In the case P = Q = G from Example 3.5 we have M = L = G and I = J = ψ(J) = S and hence ^{I}W = {1} and H_{1} = G.

To analyze H_{w} in the general case we apply the induction step from Section 4. Let w = xw_{J} be the decomposition from Proposition 2.8 with x ∈ ^{I}W^{J} and w_{J} ∈ ^{I_{x}}W_{J} for I_{x} = J ∩ ^{x^{−1}}I. Since W_{J} is the Weyl group of M, and I_{x} is the type of the parabolic P_{x} ⊂ M by Proposition 4.13 (a), we can also apply Definition 5.1 to the pair (Zẋ, w_{J}) in place of (Z, w).

Lemma 5.4. The subgroup H_{w} and the isogeny φẇ associated to (Z, w) in Definition 5.1 are equal to those associated to (Zẋ, w_{J}).

Proof. Since ẇ_{J} ∈ M = φ(L), Definition 5.1 and Construction 4.3 imply that

H_{w} ⊂ M ∩^{˙w_{J}^{−1}ẋ^{−1}g^{−1}}L = ^{˙w_{J}^{−1}}(M ∩^{ẋ^{−1}g^{−1}}L) = ^{˙w_{J}^{−1}}L_{x}

and that φẋ ∘ int(ẇ_{J})(H_{w}) = φ ∘ int(gẇ)(H_{w}) = H_{w}. Since H_{w} is the largest subgroup of ^{˙w^{−1}g^{−1}}L with this property, it is also the largest in ^{˙w_{J}^{−1}}L_{x}.

Remark 5.5. The preceding lemma implies that H_{w} and φẇ also remain the same if we repeat the induction step with (Zẋ, w_{J}) in place of (Z, w), and so on. When the process becomes stationary, we have reached a pair consisting of a zip datum as in Example 5.3 and the Weyl group element 1, whose underlying connected reductive group and isogeny are H_{w} and φẇ. This induction process is the idea underlying many proofs throughout this section.

Proposition 5.6. The subgroup H_{w} is the standard Levi subgroup of G containing T whose set of simple reflections is the unique largest subset K_{w} of ^{w^{−1}}I satisfying ψ ∘ int(w)(K_{w}) = K_{w}.

Proof. For any subset K of ^{w^{−1}}I the equality ψ ∘ int(w)(K) = K makes sense, because both sides are subsets of W. The collection of all such K satisfying that equality possesses a unique largest element K_{w}, namely the union of all of them. Then K_{w} = ψ ∘ int(w)(K_{w}) ⊂ ψ(I) = J ⊂ S, and so K_{w} consists of simple reflections. Let H denote the standard Levi subgroup of G containing T with the set of simple reflections K_{w}. Then the isogeny φ ∘ int(gẇ): ^{˙w^{−1}g^{−1}}L → M sends T to itself by Remark 5.2, and the associated isomorphism of Weyl groups ψ ∘ int(w): ^{w^{−1}}W_{I} → W_{J} sends K_{w} to itself by construction. Together this implies that φ ∘ int(gẇ)(H) = H and hence H ⊂ H_{w}. We now prove the equality H_{w} = H by induction on dim G. In the base case M = G we have I = J = S and w = 1 and thus K_{1} = S and H = G, while H_{1} = G by Example 5.3; hence we are done. Otherwise write w = xw_{J} as above. Then Lemma 5.4 and the induction hypothesis show that H_{w} is a Levi subgroup of M containing T with a set of simple reflections K ⊂ ^{w_{J}^{−1}}I_{x} satisfying ψ_{x} ∘ int(w_{J})(K) = K. But ^{w_{J}^{−1}}I_{x} = ^{w_{J}^{−1}}(J ∩ ^{x^{−1}}I) ⊂ ^{w^{−1}}I and ψ_{x} ∘ int(w_{J}) is the restriction of ψ ∘ int(x) ∘ int(w_{J}) = ψ ∘ int(w). By the maximality of K_{w} we thus have K ⊂ K_{w} and therefore H_{w} ⊂ H. Together with the earlier inequality H ⊂ H_{w} we deduce that H_{w} = H, as desired.

5.2 First description of G^{w}

Definition 5.7. For any w ∈ ^{I}W we set G^{w} := o_{Z}(gẇH_{w}).

Proposition 5.8. The set G^{w} does not depend on the representative ẇ of w or the frame.

Proof. The independence of ẇ follows from the inclusion T ⊂ H_{w}. For the rest note first that by Propositions 3.11 and 5.6 the set K_{w} is independent of the frame. Consider another frame (^{q}B,^{q}T, pgtq^{−1}) for (p, q) ∈ E_{Z} and t ∈ T, as in Proposition 3.8. Recall from Subsection 2.3 that the isomorphism W (G, T) →^{∼} W (G,^{q}T) is induced by int(q) : Norm_{G}(T) →^{∼} Norm_{G}(^{q}T). It follows that w ∈ ^{I}W as an element of the abstract Weyl group of G is represented by qẇq^{−1} ∈ Norm_{G}(^{q}T), and with Proposition 5.6 it follows that the Levi subgroup associated to w and the new frame is ^{q}H_{w}. Thus the right hand side in Definition 5.7 associated to the new frame is

o_{Z}((pgtq^{−1})(qẇq^{−1})^{q}H_{w}) = o_{Z}(pgtẇH_{w}q^{−1}) = o_{Z}(gtẇH_{w}) = o_{Z}(gẇH_{w}),

where the second equation follows from (p, q) ∈ E_{Z} and the third from ẇ^{−1}tẇ ∈ T ⊂ H_{w}. Thus G^{w} is independent of the frame.

In Example 5.3 we have H_{1} = G and hence G^{1} = G. Otherwise recall from Proposition 4.12 that Zẋ has the frame (M ∩ B, T, 1). Thus by Lemma 5.4, the subset associated to (Zẋ, w_{J}) by Definition 5.7 is M^{w_{J}} := o_{Zẋ}(ẇ_{J}H_{w}).

Lemma 5.9. Under the bijection of Proposition 4.7, the subset M^{w_{J}} ⊂ M corresponds to the subset G^{w} ⊂ P gẋQ. In particular G^{w} = o_{Z}(gẋM^{w_{J}}). Also, there is a bijection between the E_{Zẋ}-orbits X′ ⊂ M^{w_{J}} and the E_{Z}-orbits X ⊂ G^{w}, defined by X = o_{Z}(gẋX′).

Proof. Using, in this order, the definition of G^{w}, the equation (2.14), Lemma 4.9, and the definition of M^{w_{J}} we find that

G^{w} = o_{Z}(gẇH_{w}) = o_{Z}(gẋẇ_{J}H_{w}) = o_{Z}(gẋ o_{Zẋ}(ẇ_{J}H_{w})) = o_{Z}(gẋM^{w_{J}}).

The other assertions follow from Proposition 4.7.

5.3 Main properties of G^{w}

Theorem 5.10. The G^{w} for all w ∈ ^{I}W form a disjoint decomposition of G.

Proof. We show this by induction on dim G. In the base case M = G we have IW = {1} and H_{1} = G = G^{1} by Example 5.3; hence the theorem is trivially true. Otherwise take an element x ∈ ^{I}W^{J}. By the induction hypothesis applied to the zip datum E_{Zẋ} the subsets M^{w_{J}} for w_{J} ∈ ^{I_{x}}W_{J} form a disjoint decomposition of M. Thus by Proposition 4.7 and Lemma 5.9, the subsets G^{xw_{J}} for w_{J} ∈ ^{I_{x}}W_{J} form a disjoint decomposition of P gẋQ. Combining this with the Bruhat decomposition (4.1) it follows that the subsets G^{xw_{J}} for all x and w_{J} form a disjoint decomposition of G. But by Proposition 2.8 these are precisely the subsets G^{w} for w ∈ ^{I}W, as desired.

Theorem 5.11. For any w ∈ ^{I}W the subset G^{w} is a nonsingular subvariety of G of dimension dim P + ℓ(w).

Proof. Again we proceed by induction on dim G. If M = G, there is only one piece G^{1} = G = P associated to w = 1, and the assertion is clear. Otherwise write w = xw_{J} as in Proposition 2.8. By the induction hypothesis the subset M^{w_{J}} is a nonsingular subvariety of M of dimension dim P_{x} + ℓ(w_{J}). Thus by Propositions 4.7 and 4.8 and Lemma 5.9 the subset G^{w} is a nonsingular subvariety of dimension

[dim P_{x} + ℓ(w_{J})] + dim P − dim P_{x} + ℓ(x) = dim P + ℓ(x) + ℓ(w_{J}).

By Proposition 2.7 the last expression is equal to dim P + ℓ(w), as desired.

Theorem 5.12. For any w ∈ ^{I}W, there is a bijection between the H_{w}-orbits Y ⊂ H_{w} and the E_{Z}-orbits X ⊂ G^{w}, defined by X = o_{Z}(gẇY) and satisfying

codim(X ⊂ G^{w}) = codim(Y ⊂ H_{w}).

Proof. If M = G, we have w = 1 and G = G^{1} = H_{1}, and E_{Z} ≅ G acts on itself by the twisted conjugation (h, h′) ↦ h · h′ · φ(h)^{−1}. Thus the E_{Z}-orbits X ⊂ G are precisely the cosets gY for H_{1}-orbits Y according to Definition 5.1, which finishes that case. If M ≠ G write w = xw_{J} as in Proposition 2.8. Then Zẋ has the frame (M ∩ B, T, 1) by Proposition 4.12, and so by Lemma 5.4 and the induction hypothesis there is a bijection between the H_{w}-orbits Y ⊂ H_{w} and the E_{Zẋ}-orbits X′ ⊂ M^{w_{J}}, defined by X′ = o_{Zẋ}(ẇ_{J}Y) and satisfying codim(X′ ⊂ M^{w_{J}}) = codim(Y ⊂ H_{w}). By Proposition 4.7 and Lemma 5.9 there is a bijection between these X′ and the E_{Z}-orbits X ⊂ G^{w}, defined by X = o_{Z}(gẋX′). Moreover, since pushout and flat pullback preserve codimensions, the last statement in Proposition 4.7 implies that

codim(X ⊂ G^{w}) = codim(o_{Z}(gẋX′) ⊂ o_{Z}(gẋM^{w_{J}})) = codim(X′ ⊂ M^{w_{J}}).

Finally, since ẇ = ẋẇ_{J} by (2.14), Lemma 4.9 shows that X = o_{Z}(gẋ o_{Zẋ}(ẇ_{J}Y)) = o_{Z}(gẋẇ_{J}Y) = o_{Z}(gẇY), finishing the induction step.

5.4 Other descriptions of G^{w}

Lemma 5.13. For any element g′ ∈ G we have

o_{Z}(gBg′B) = o_{Z}(gg′B) = o_{Z}(gBg′).

Proof. Take any element b ∈ B. Then the condition 3.6 (b) implies that p := gbg^{−1} ∈ P, and so there exists q ∈ Q such that (p, q) ∈ E_{Z}. By the condition 3.6 (c) we then have q ∈ B. It follows that gbg′B = pgg′Bq^{−1} ⊂ o_{Z}(gg′B). Since b was arbitrary, this shows that gBg′B ⊂ o_{Z}(gg′B), whence the first equality. A similar argument proves the second equality.

Theorem 5.14. For any w ∈ ^{I}W we have

G^{w} = o_{Z}(gẇH_{w}) = o_{Z}(gẇ(H_{w} ∩ B)) = o_{Z}(gẇB) = o_{Z}(gBẇ) = o_{Z}(gBẇB).

Proof. The first equation is Definition 5.7 of G^{w}, and the last two equations are cases of Lemma 5.13. The remaining two equations are proved by induction on dim G. In the base case M = G we have w = 1 and H_{1} = G; hence the second term is o_{Z}(gG) = G, and the third and fourth terms are both equal to o_{Z}(gB). By Proposition 2.5 applied to the isogeny φ ∘ int(g) the latter is equal to G, as desired. In the case M ≠ G write w = xw_{J} as in Proposition 2.8. Then Zẋ has the frame (M ∩B, T, 1) by Proposition 4.12, and so by Lemma 5.4 and the induction hypothesis we have

o_{Zẋ}(ẇ_{J}H_{w}) = o_{Zẋ}(ẇ_{J}(H_{w} ∩ B)) = o_{Zẋ}(ẇ_{J}(M ∩ B)).

Using Lemma 4.9 this implies that

o_{Z}(gẋẇ_{J}H_{w}) = o_{Z}(gẋẇ_{J}(H_{w} ∩ B)) = o_{Z}(gẋẇ_{J}(M ∩ B)).

By (2.14) we may replace ẋẇ_{J} by ẇ in these equations. Moreover, (3.3) and (3.9) show that gẇB = gẇ(M ∩ B)V ⊂ o_{Z}(gẇ(M ∩ B)) and so o_{Z}(gẇB) = o_{Z}(gẇ(M ∩ B)). Thus both equations follow.

Example 5.15. If P is a Borel subgroup, then so is Q, and we have ^{I}W = W. The last equation in Theorem 5.14 then implies that G^{w} = gBẇB for all w ∈ W.

For a further equivalent description of G^{w} see Subsection 11.1.

6 Closure relation

In this section, we determine the closure of G^{w} in G for any w ∈ ^{I}W. To formulate a precise result recall that ≤ denotes the Bruhat order on W.

Definition 6.1. For w, w′ ∈ ^{I}W we write w′ ≼ w if and only if there exists y ∈ W_{I} such that yw′ψ(y)^{−1} ≤ w.

Theorem 6.2. For any w ∈ ^{I}W we have

\overline{G^{w}} = ∐_{\begin{matrix}w′∈^{I}W \\ w′≼w \end{matrix}} G^{w′}.

A direct consequence of this is:

Corollary 6.3. The relation ≼ is a partial order on ^{I}W.

Remark 6.4. The relation ≼ has been introduced by He in [He2] for a somewhat more special class of isomorphisms ψ: W_{I} →^{∼} W_{J}. He gives a direct combinatorial proof that ≼ is a partial order (Proposition 3.13 of loc. cit.), which can be adapted to our more general setting (see [Wed], Section 4).

The rest of this section is devoted to proving Theorem 6.2. We will exploit the fact that the closure relation for the Bruhat decomposition of G is known. Namely, for any w ∈ W we have by [Spr1], Proposition 8.5.5:

\overline{BẇB} = ∐_{\begin{matrix}w′∈W \\ w′≤w \end{matrix}} Bẇ′B.  (6.5)

Lemma 6.6. For any w ∈ W we have

\overline{o_{Z}(gBẇB)} = ⋃_{\begin{matrix}w′∈W \\ w′≤w \end{matrix}} o_{Z}(gẇ′B).

Proof. Let B_{Z} ⊂ E_{Z} denote the subgroup of all elements (uℓ, vφ(ℓ)) with u ∈ U, v ∈ V, and ℓ ∈ L ∩ ^{g}B. Then E_{Z}/B_{Z} ≅ L/(L ∩ ^{g}B) is proper, and gBẇB ⊂ G is a B_{Z}-invariant subvariety. Thus Lemma 2.3 and (6.5) imply that

\overline{o_{Z}(gBẇB)} = o_{Z}( \overline{gBẇB} ) = ⋃_{w′≤w} o_{Z}(gBẇ′B).

The desired equality then follows from Lemma 5.13.

Lemma 6.7. For any w, v ∈ W and b ∈ B there exists u ∈ W such that u ≤ v and ẇbv̇ ∈ Bẇu̇B.

Proof. We prove the statement by induction on ℓ(v). If v = 1, we may take u = 1. For the induction step write v = v′s for some simple reflection s such that ℓ(v′) = ℓ(v) − 1. By the induction hypothesis there exists u′ ≤ v′ such that ẇbv̇′ ∈ Bẇu̇′B. Hence ẇbv̇ ∈ Bẇu̇′Bṡ ⊂ Bẇu̇′ṡB ∪ Bẇu̇′B, so either u = u′s or u = u′ will have the required property.

Lemma 6.8. For any z ∈ W and w ∈ ^{I}W and v ∈ W_{I} such that z ≤ wψ(v), there exists y ∈ W_{I} such that yzψ(y)^{−1} ≤ vw.

Proof. Choose reduced expressions for w and v as products of simple reflections. Since ψ(I) = J, this also yields a reduced expression for ψ(v). Together this yields an expression for wψ(v) as a product of simple reflections, which is not necessarily reduced. However, by [BB], Theorem 2.2.2 a reduced expression for wψ(v) can be obtained from the given one by possibly deleting some factors. By the definition of the Bruhat order, the assumption z ≤ wψ(v) means that a reduced expression for z is obtained from this by deleting further factors, if any. Let y′ denote the product of all factors remaining from w. Since all factors in the reduced expression for v lie in I, the product of all factors remaining from ψ(v) is equal to ψ(y) for some y ∈ W_{I}. By construction we then have z = y′ψ(y), and so yzψ(y)^{−1} = yy′. But the assumptions on w and v imply that ℓ(vw) = ℓ(v)+ℓ(w); hence the product of the given reduced expressions for v and w is a reduced expression for vw. By construction yy′ is obtained from that product by possibly deleting some factors, so we deduce that yy′ ≤ vw, as desired.

Lemma 6.9. For any w ∈ ^{I}W and w′ ∈ W and b, b′ ∈ B such that o_{Z}(gẇb) = o_{Z}(gẇ′b′) there exists y ∈ W_{I} such that ywψ(y)^{−1} ≤ w′.

Proof. We proceed by induction on dim G. In the base case M = G we have w = 1 and may take y = 1. So assume that M ≠ G. Write w = xw_{J} as in Proposition 2.8 with x ∈ ^{I}W^{J} and w_{J} ∈ ^{I_{x}}W_{J}. From o_{Z}(gẇb) = o_{Z}(gẇ′b′) we deduce that P gẋQ = P gẇQ = P gẇ′Q, which in view of (4.1) implies that w′ ∈ W_{I}xW_{J}. Write w′ = v′xw′_{J} for v′ ∈ W_{I} and w′_{J} ∈ ^{I_{x}}W_{J}, as in Proposition 2.7. Recall that φ(gv̇′g^{−1}) ∈ Norm_{M}(T) is a representative of ψ(v′) ∈ W_{J}. Thus by Lemma 6.7, there exists u ∈ W such that u ≤ ψ(v′) and ẋẇ′_{J}b′φ(gv̇′g^{−1}) ∈ Bẋẇ′_{J} u̇B. The first condition implies that u ∈ W_{J}, the Weyl group of M. The action of E_{Z} and the second condition imply

o_{Z}(gẇ′b′) = o_{Z}(gv̇′ẋẇ′_{J}b′) = o_{Z}(gẋẇ′_{J}b′φ(gv̇′g^{−1})) ⊂ o_{Z}(gBẋẇ′_{J} u̇B).

Here the last term is equal to o_{Z}(gẋẇ′_{J} u̇B) by Lemma 5.13. Thus there exists b′′ ∈ B such that

o_{Z}(gẋẇ_{J}b) = o_{Z}(gẇb) = o_{Z}(gẇ′b′) = o_{Z}(gẋẇ′_{J} u̇b′′).

By the action of E_{Z} we may and do assume that b, b′′ ∈ M ∩ B. Then ẇ_{J}b and ẇ′_{J} u̇b′′ lie in M, and so Proposition 4.7 implies that o_{Zẋ}(ẇ_{J}b) = o_{Zẋ}(ẇ′_{J} u̇b′′). By the induction hypothesis there therefore exists y_{x} ∈ W_{I_{x}} such that

y_{x}w_{J}ψ_{x}(y_{x})^{−1} ≤ w′_{J}u.

Now we work our way back up. Since both sides of the last relation lie in W_{J}, and since x ∈ W^{J}, we deduce that

z := xy_{x}w_{J}ψ_{x}(y_{x})^{−1} ≤ xw′_{J}u.

Recall that u ≤ ψ(v′), which implies that u = ψ(u′) for some u′ ∈ W_{I} satisfying u′ ≤ v′. Also, note that xw′_{J} ∈ ^{I}W by Proposition 2.8. Thus by Lemma 6.8 there exists y′ ∈ W_{I} such that

y′zψ(y′)^{−1} ≤ u′xw′_{J}.

As u′ and v′ lie in W_{I}, and xw′_{J} ∈ ^{I}W, we deduce that

y′zψ(y′)^{−1} ≤ u′xw′_{J} ≤ v′xw′_{J} = w′.

Finally, since ψ_{x} = ψ ∘ int(x), we have

y′zψ(y′)^{−1} = y′xy_{x}w_{J}ψ(xy_{x}x^{−1})^{−1}ψ(y′)^{−1} = = (y′xy_{x}x^{−1})xw_{J}ψ(y′xy_{x}x^{−1})^{−1} = ywψ(y)^{−1}

with y := y′xy_{x}x^{−1} ∈ W_{I}. Thus ywψ(y)^{−1} ≤ w′, as desired.

Lemma 6.10. For any w ∈ ^{I}W, the set o_{Z}(gẇT) is dense in G^{w}.

Proof. Theorem 5.12 implies that o_{Z}(gẇT) = o_{Z}(gẇY), where Y ⊂ H_{w} is the orbit of T under twisted conjugation by H_{w}. But Proposition 2.5 (b) asserts that Y is dense in H_{w}. Thus o_{Z}(gẇY) is dense in o_{Z}( \overline{gẇY} ) = o_{Z}(gẇH_{w}) = G^{w}, as desired.

Proof of Theorem 6.2. Consider w′ ∈ ^{I}W such that G^{w′} ∩ \overline{G^{w}} ≠ ∅. Then by Theorem 5.14 and Lemma 6.6 there exist b, b′ ∈ B and w′′ ∈ W such that w′′ ≤ w and o_{Z}(gw′b) = o_{Z}(gw′′b′). Lemma 6.9 then implies that yw′ψ(y)^{−1} ≤ w′′ for some y ∈ W_{I}. Together it follows that yw′ψ(y)^{−1} ≤ w, and hence w′ ≼ w, proving “⊂”. To prove “⊃” consider w′ ∈ ^{I}W with w′ ≼ w. By definition there exists y ∈ W_{I} such that w′′ := yw′ψ(y)^{−1} ≤ w. Lemma 6.6 and Theorem 5.14 then show that o_{Z}(gẇ′′T) ⊂ \overline{G^{w}}. Therefore

o_{Z}(gẇ′T) = o_{Z}(gẏẇ′T φ(gẏg^{−1})^{−1}) =

= o_{Z}(gẏẇ′φ(gẏg^{−1})^{−1}T ) = o_{Z}(gẇ′′T) ⊂ \overline{G^{w}}.

With Lemma 6.10 for o_{Z}(gẇ′T) we conclude that G^{w′} ⊂ \overline{G^{w}}, as desired.

7 Orbitally finite zip data

Proposition 7.1. The following assertions are equivalent:

(a) For any w ∈ ^{I}W, the number of fixed points of the endomorphism φẇ = φ ∘ int(gẇ) of H_{w} from Definition 5.1 is finite.

(b) For any w ∈ ^{I}W the E_{Z}-invariant subvariety G^{w} is a single orbit under E_{Z}.

(c) The number of orbits of E_{Z} on G is finite.

Proof. If (a) holds, the Lang-Steinberg Theorem 2.4 shows that the orbit of 1 ∈ H_{w} under twisted conjugation is all of H_{w}, and by Theorem 5.12 this implies (b). The implication (b)⇒(c) is trivial. So assume (c). Then again by Theorem 5.12, the number of orbits in H_{w} under twisted conjugation by φẇ is finite for any w ∈ ^{I}W. In particular there exists an open orbit; let h be an element thereof. Then for dimension reasons its stabilizer is finite. But

Stab_{H_{w}}(h) = {h′ ∈ H_{w} | h′hφẇ(h′)^{−1} = h} = {h′ ∈ H_{w} | h′ = hφẇ(h′)h^{−1}}

is also the set of fixed points of the endomorphism int(h) ∘ φẇ of H_{w}. Thus the Lang-Steinberg Theorem 2.4 implies that {h′hφẇ(h′)^{−1}h^{−1} | h′ ∈ H_{w}} = H_{w}. After right multiplication by h this shows that the orbit of h is all of H_{w}. We may thus repeat the argument with the identity element in place of h, and deduce that the set of fixed points of φẇ on H_{w} is finite, proving (a).

Definition 7.2. We call Z orbitally finite if the conditions in Proposition 7.1 are met.

Proposition 7.3. If the differential of φ at 1 vanishes, then Z is orbitally finite.

Proof. If the differential of φ vanishes, then so does the differential of φẇ = φ ∘ int(gẇ)|H_{w} for any w ∈ ^{I}W. Let H_{w}^{f} denote the fixed point locus of φẇ, which is a closed algebraic subgroup. Then the restriction φẇ|H_{w}^{f} is the identity and its differential is zero. This is possible only when dim H_{w}^{f} = 0, that is, when H_{w}^{f} is finite.

Remark 7.4. In particular Proposition 7.3 applies when the base field has characteristic p > 0 and the isogeny φ is a relative Frobenius L → L^{(p^{r})} ≅ M.

Since gẇ ∈ G^{w} by Definition 5.7, we can now rephrase condition 7.1 (b) and Theorems 5.10, 5.11, and 6.2 as follows:

Theorem 7.5. Assume that Z is orbitally finite. Then:

(a) For any w ∈ ^{I}W we have G^{w} = o_{Z}(gẇ).

(b) The elements gẇ for w ∈ ^{I}W form a set of representatives for the E_{Z}- orbits in G.

(c) For any w ∈ ^{I}W the orbit o_{Z}(gẇ) has dimension dim P + ℓ(w).

(d) For any w ∈ ^{I}W the closure of o_{Z}(gẇ) is the union of o_{Z}(gẇ′) for all w′ ∈ ^{I}W with w′ ≼ w.

8 Point stabilizers

In this section we study the stabilizer in E_{Z} of an arbitrary element g′ ∈ G. Take w ∈ ^{I}W such that g′ ∈ G^{w}. Then Theorem 5.12 shows that g′ is conjugate to gẇh for some h ∈ H_{w}. Thus it suffices to consider the stabilizer of gẇh. Recall from Definition 5.1 that H_{w} acts on itself by twisted conjugation with the isogeny φẇ, which is defined as the restriction of φ ∘ int(gẇ).

Theorem 8.1. For any w ∈ ^{I}W and h ∈ H_{w} the stabilizer Stab_{E_{Z}} (gẇh) is the semi-direct product of a connected unipotent normal subgroup with the subgroup

{(int(gẇ)(h′), φ(int(gẇ)(h′))) | h′ ∈ Stab_{H_{w}}(h)}.  (8.2)

Proof. For any h′ ∈ H_{w} we have int(gẇ)(h′) · gẇh · φ(int(gẇ)(h′))^{−1} = gẇh if and only if h′hφ(int(gẇ)(h′))^{−1} = h if and only if h′ ∈ Stab_{H_{w}}(h). This implies that (8.2) is a subgroup of Stab_{E_{Z}} (gẇh). For the rest we proceed by induction on dim G. If M = G, we have w = 1 and ẇ = 1 and G = H_{1}, and (g′, φ(g′)) ∈ E_{Z} acts on G by the twisted conjugation g′′ ↦ g′g′′φ(g′)^{−1}. Under left translation by g this corresponds to the action of H_{w} on itself, so that Stab_{E_{Z}} (gẇh) is precisely the subgroup (8.2) and the normal subgroup is trivial. If M ≠ G write w = xw_{J} as in Proposition 2.8. Then ẇ = ẋẇ_{J} and Zẋ has the frame (M ∩ B, T, 1), and Proposition 4.11 shows that Stab_{E_{Z}} (gẇh) is an extension of Stab_{E_{Zẋ}} (ẇ_{J}h) by a connected unipotent normal subgroup. Moreover, by the induction hypothesis Stab_{E_{Zẋ}} (ẇ_{J}h) is the semi-direct product of a connected unipotent normal subgroup with the subgroup

{(int(ẇ_{J})(h′), φẋ(int(ẇ_{J})(h′))) | h′ ∈ Stab_{H_{w}}(h)}.  (8.3)

Furthermore a direct calculation shows that the projection in Proposition 4.11 sends the subgroup (8.2) isomorphically to the subgroup (8.3). Since any extension of connected unipotent groups is again connected unipotent, the theorem follows.

Remark 8.4. For the stabilizer, a similar result was obtained by Evens and Lu ([EL] Theorem 3.13).

If the differential of φ at 1 vanishes, we can also describe the infinitesimal stabilizer in the Lie algebra. Since in that case the zip datum is orbitally finite by Proposition 7.3, it suffices to consider the stabilizer of gẇ.

Theorem 8.5. Assume that the differential of φ at 1 vanishes. For any w ∈ ^{I}W let w = xw_{J} be the decomposition from Proposition 2.8. Then the infinitesimal stabilizer of gẇ in the Lie algebra of E_{Z} has dimension dim V − ℓ(x).

Proof. Since dφ = 0, we have Lie E_{Z} = Lie P × Lie V ⊂ Lie(P × Q). Thus an arbitrary tangent vector of E_{Z} at 1 has the form (1+dp, 1+dv) for dp ∈ Lie P and dv ∈ Lie V, viewed as infinitesimal elements of P and V in Leibniz’s sense. That element stabilizes gẇ if and only if (1+dp)gẇ(1+dv)^{−1} = gẇ. This condition is equivalent to dp·gẇ −gẇ ·dv = 0, or again to dp = Ad_{gẇ}(dv). The dimension is therefore dim(Lie P ∩ Ad_{gẇ}(Lie V)) = dim(Lie^{g^{−1}}P ∩ Lie ẇV). As both ^{g^{−1}}P and ẇV are normalized by T, the dimension is just the number of root spaces in the last intersection. This number is

#[(Φ^{+} ∪ Φ_{I}) ∩ w(Φ^{+} ∖ Φ_{J})] = #{α ∈ Φ^{+} ∖ Φ_{J} | wα ∈ Φ^{+} ∪ Φ_{I}}
= dim V − #{α ∈ Φ^{+} ∖ Φ_{J} | wα ∈ Φ^{−} ∖ Φ_{I}}.

By Lemma 2.13 it is therefore dim V − ℓ(x), as desired.

Remark 8.6. The dimension in Theorem 8.5 depends only on the first factor of w = xw_{J} and thus only on the Bruhat cell P gẇQ. Since that Bruhat cell is an irreducible variety and in general composed of more than one E_{Z}-orbit, these orbits have different dimensions. Thus the corresponding point stabilizers in E_{Z} have different dimension, while the dimension of their Lie algebra stabilizer is constant. Therefore the scheme-theoretic stabilizer of gẇ is in general not reduced.

9 Abstract zip data

By Theorem 5.10 the subsets G^{w} for all w ∈ ^{I}W form a disjoint decomposition of G satisfying gẇ ∈ G^{w}. It is natural to ask which other elements of the form gẇ′ for w′ ∈ W are contained in a given G^{w}. When Z is orbitally finite, by Theorem 7.5 this question is equivalent to asking which elements gẇ′ for w′ ∈ W lie in the same E_{Z}-orbit. This problem turns out to depend only on the groups W_{I} ⊂ W and the homomorphism ψ and can therefore be studied abstractly. We return to this situation at the end of this section.

9.1 Abstract groups

Definition 9.1. An abstract zip datum is a tuple A = (Γ, Δ, ψ) consisting of a group Γ, a subgroup Δ, and a homomorphism ψ: Δ → Γ.

Fix such an abstract zip datum A. For any γ ∈ Γ, the collection of subgroups E of ^{γ^{−1}}Δ satisfying ψ ∘ int(γ)(E) = E possesses a unique largest element, namely the subgroup generated by all such subgroups.

Definition 9.2. For any γ ∈ Γ we let E_{γ} denote the unique largest subgroup of ^{γ^{−1}}Δ satisfying ψ ∘ int(γ)(E_{γ}) = E_{γ}.

Lemma 9.3. For any γ ∈ Γ and δ ∈ Δ and ε ∈ E_{γ}, we have E_{δγεψ(δ)^{−1}} = ψ(δ)E_{γ}.

Proof. Abbreviate γ′ := δγεψ(δ)^{−1}. Then the calculation ψ(^{γ′}(^{ψ(δ)}E_{γ})) = ψ(^{δγε}E_{γ}) = ^{ψ(δ)}ψ(^{γ}E_{γ}) = ^{ψ(δ)}E_{γ} and the definition of E_{γ′} imply that ψ(δ)E_{γ} ⊂ E_{γ′}. In particular, ε′ := ^{ψ(δ)}ε^{−1} is an element of E_{γ′}. Since γ = δ′γ′ε′ψ(δ′)^{−1} with δ′ := δ^{−1} ∈ Δ, a calculation like the first shows that ψ(δ′)E_{γ′} ⊂ E_{γ}. Together it follows that E_{γ′} = ^{ψ(δ)}E_{γ}, as desired.

Definition 9.4. For any γ, γ′ ∈ Γ we write γ′ ∼ γ if and only if there exist δ ∈ Δ and ε ∈ E_{γ} such that γ′ = δγεψ(δ)^{−1}. For any γ ∈ Γ we abbreviate o_{A}(γ) := {γ′ ∈ Γ | γ′ ∼ γ}.

Lemma 9.5. This is an equivalence relation.

Proof. Reflexivity is clear, and symmetry was shown already in the proof of Lemma 9.3. To prove transitivity, suppose that γ′ = δγεψ(δ)^{−1} for δ ∈ Δ and ε ∈ E_{γ} and γ′′ = δ′γ′ε′ψ(δ′)^{−1} for δ′ ∈ Δ and ε′ ∈ E_{γ′}. Then ^{ψ(δ)^{−1}}ε′ ∈ E_{γ} by Lemma 9.3, and so γ′′ = δ′δγεψ(δ)^{−1}ε′ψ(δ′)^{−1} = δ′′γε′′ψ(δ′′)^{−1} for δ′′ := δ′δ ∈ Δ and ε′′ := ε ^{ψ(δ)^{−1}}ε′ ∈ E_{γ}, as desired.

Theorem 9.6. If Δ is finite, each equivalence class in Γ has cardinality #Δ and the number of equivalence classes is [Γ : Δ].

Proof. Take any γ ∈ Γ; then the group E_{γ} ⊂ ^{γ^{−1}}Δ is finite, too. Consider the surjective map Δ × E_{γ} ↠ o_{A}(γ), (δ, ε) ↦ δγεψ(δ)^{−1}. Two elements (δ, ε), (δ′, ε′) ∈ Δ × E_{γ} lie in the same fiber if and only if δγεψ(δ)^{−1} = δ′γε′ψ(δ′)^{−1} if and only if εψ(δ^{−1}δ′) = γ^{−1}(δ^{−1}δ′)γε′. With ε′′ := γ^{−1}(δ^{−1}δ′)γ ∈ ^{γ^{−1}}Δ this is equivalent to εψ(^{γ}ε′′) = ε′′ε′. Since ε, ε′ ∈ E_{γ}, this equation implies that the subgroup generated by E_{γ} and ε′′ is mapped onto itself under ψ ∘ int(γ). By maximality it is therefore equal to E_{γ}, and so ε′′ ∈ E_{γ}. Together we find that the elements in the same fiber as (δ, ε) are precisely the elements (δ′, ε′) with δ′ = δ^{γ}ε′′ and ε′ = (ε′′)^{−1}εψ(^{γ}ε′′) for some ε′′ ∈ E_{γ}. Thus each fiber has cardinality #E_{γ}, and so the image has cardinality #Δ, proving the first assertion. The second assertion is a direct consequence of the first.

We can also perform an induction step as in Section 4 for abstract zip data, obtaining analogues of Lemma 5.4 and Proposition 4.7. For this fix an element ξ ∈ Γ, say in a set of representatives for the double quotient Δ\Γ/ψ(Δ). Then Definitions 9.2 and 9.4 imply that the equivalence class of any γ ∈ Δξψ(Δ) is again contained in Δξψ(Δ).

Construction 9.7. Set Γ_{ξ} := ψ(Δ) and Δ_{ξ} := ψ(Δ)∩^{ξ^{−1}}Δ, and let ψ_{ξ}: Δ_{ξ} → Γ_{ξ} denote the restriction of ψ ∘ int(ξ). This defines a new, possibly smaller, abstract zip datum

A_{ξ} := (Γ_{ξ}, Δ_{ξ}, ψ_{ξ}).

Lemma 9.8. For any γ ∈ Γ_{ξ}, the group E_{ξγ} associated by Definition 9.2 to the pair (A, ξγ) is equal to the group associated to the pair (A_{ξ}, γ).

Proof. Since γ ∈ Γ_{ξ} = ψ(Δ), Definition 9.2 implies that

E_{ξγ} ⊂ ψ(Δ) ∩^{γ^{−1}ξ^{−1}}Δ = ^{γ^{−1}}(ψ(Δ) ∩^{ξ^{−1}}Δ) = ^{γ^{−1}}Δ_{ξ}

and that E_{ξγ} = ψ ∘ int(ξγ)(E_{ξγ}) = ψ_{ξ} ∘ int(γ)(E_{ξγ}). Since E_{ξγ} is the largest subgroup of ^{γ^{−1}ξ^{−1}}Δ with this property, it is also the largest in ^{γ^{−1}}Δ_{ξ}.

Proposition 9.9. There is a bijection between A_{ξ}-equivalence classes in Γ_{ξ} and A-equivalence classes in Δξψ(Δ), defined by o_{A_{ξ}}(γ) ↦ o_{A}(ξγ) and o_{A_{ξ}}(γ) = Γ_{ξ} ∩ ξ^{−1}o_{A}(ξγ).

Proof. Take any γ, γ′ ∈ Γ_{ξ}. Then γ′ ∈ ξ^{−1}o_{A}(ξγ) if and only if ξγ′ = δξγεψ(δ)^{−1} for some δ ∈ Δ and ε ∈ E_{ξγ}. Writing δ = ^{ξ}δ′ this is equivalent to γ′ = δ′γεψ(^{ξ}δ′)^{−1} for δ′ ∈ ^{ξ^{−1}}Δ and ε ∈ E_{ξγ}. In this equation γ and γ′ and ψ(^{ξ}δ′) lie in Γ_{ξ} = ψ(Δ) by assumption, and so does ε ∈ E_{ξγ} ⊂ ψ(Δ) by Definition 9.2. Thus the equation requires that δ′ lies in ψ(Δ), and so a fortiori in ψ(Δ) ∩^{ξ^{−1}}Δ = Δ_{ξ}. In view of Lemma 9.8 the condition is thus equivalent to γ′ ∈ o_{A_{ξ}}(γ), proving the equation at the end of the proposition. That equation implies that the map o_{A_{ξ}}(γ) ↦ o_{A}(ξγ) from A_{ξ}-equivalence classes in Γ_{ξ} to A-equivalence classes in Δξψ(Δ) is well-defined and injective.

But any element of Δξψ(Δ) has the form δξγ for δ ∈ Δ and γ ∈ Γ_{ξ} and is therefore equivalent to ξγψ(δ) ∈ ξΓ_{ξ}. Thus the map is also surjective, and we are done.

9.2 Coxeter groups

Definition 9.10. Let W be a Coxeter group with a finite set of simple reflections S. Let ψ: W_{I} →^{∼} W_{J} ⊂ W be an isomorphism of Coxeter groups with ψ(I) = J for subsets I, J ⊂ S. Then A := (W, W_{I}, ψ) is an abstract zip datum that we call of Coxeter type.

Fix such an abstract zip datum of Coxeter type A. Recall that ^{I}W^{J} is a set of representatives for the double quotient W_{I}\W/W_{J}. We will apply the induction step from Proposition 9.9 to x ∈ ^{I}W^{J}. As in Proposition 4.13 set I_{x} := J ∩^{x^{−1}}I and J_{x} := ψ(I ∩ ^{x}J), which are both subsets of J. Then W_{J} = ψ(W_{I}), and W_{I_{x}} = ψ(W_{I}) ∩ ^{x^{−1}}W_{I} by Proposition 2.6, and ψ_{x} := ψ ∘ int(x) induces an isomorphism ψ_{x}: W_{I_{x}} →^{∼} W_{J_{x}} such that ψ_{x}(I_{x}) = J_{x}. Thus the new abstract zip datum from Construction 9.7 is A_{x} := (W_{J}, W_{I_{x}}, ψ_{x}) and hence again of Coxeter type. Using this we obtain the following analogue of Theorem 5.10, which also has been previously proved by He ([He3] Corollary 2.6).

Theorem 9.11. For A of Coxeter type ^{I}W is a set of representatives for the equivalence classes in W.

Proof. We prove this by induction on #S. If I = S, we have W_{I} = W_{J} = W and so E_{w} = W for every w ∈ W. Then there is exactly one equivalence class, represented by the unique element of ^{I}W = {1}, and the assertion holds. Otherwise we have #I < #S. Take any x ∈ ^{I}W^{J}. Then by the induction hypothesis^{I_{x}}W_{J} is a set of representatives for the A_{x}-equivalence classes in W_{J}. Thus Proposition 9.9 implies that x^{I_{x}}W_{J} is a set of representatives for the Aequivalence classes in W_{I}xW_{J}. Varying x, Proposition 2.8 implies that ^{I}W is a set of representatives for the equivalence classes in W, as desired.

For use in Section 11 we include the following results.

Lemma 9.12. (a) For any w ∈ ^{I}W there exists y ∈ W_{I} such that w′ := ywψ(y)^{−1} ∈ W^{J}.

(b) The element w′ in (a) is independent of y.

Proof. For (a) we use induction on #S. If I = S, we have ^{I}W = {1} and w = 1, and so y = 1 does the job. Otherwise #I < #S. Write w = xw_{J} as in Proposition 2.8 with x ∈ ^{I}W^{J} and w_{J} ∈ ^{I_{x}}W_{J}. Then by the induction hypothesis applied to A_{x} there exists y′ ∈ W_{I_{x}} such that

w′_{J} := y′w_{J}ψ_{x}(y′)^{−1} ∈ W_{J}^{J_{x}} = W_{ψ(I)}^{ψ(I∩^{x}J)} = ψ(W_{I}^{I∩^{x}J} ).

Setting y := ψ^{−1}(y′w_{J}) ∈ W_{I} and using the definition of ψ_{x} we deduce that

w′ := ywψ(y)^{−1} = ψ^{−1}(y′w_{J}) · xw_{J} · (y′w_{J})^{−1}
= ψ^{−1}(y′w_{J}) · xy^{′−1}x^{−1} · x
= ψ^{−1}(y′w_{J}ψ_{x}(y′)^{−1}) · x
= ψ^{−1}(w′_{J}) · x ∈ W_{I}^{I∩^{x}J} · x.

By Proposition 2.9 the right hand side is contained in W^{J}, showing (a). To prove (b) consider another element y′ ∈ W_{I} such that w′′ := y′wψ(y′)^{−1} ∈ W^{J}. Then with ỹ := ψ(y′y^{−1}) ∈ W_{J} we have w′′ = y′y^{−1}w′ψ(y)ψ(y′)^{−1} = ψ^{−1}(ỹ)w′ỹ^{−1} and hence w^{′′−1} = ỹw^{′−1}ψ^{−1}(ỹ)^{−1}. Now observe that on replacing (I, J, ψ) by (J, I, ψ^{−1}) we obtain another abstract zip datum A′ := (W, W_{J}, ψ^{−1}) dual to A. The last equality then shows that w^{′′−1} and w^{′−1} are equivalent according to Definition 9.4 for A′. Since these elements also lie in ^{J}W, Theorem 9.11 applied to A′ shows that they are equal. Therefore w′′ = w′, as desired.

Proposition 9.13. There exists a unique bijection σ: ^{I}W → W^{J} with the property that for any w ∈ ^{I}W there exists y ∈ W_{I} such that σ(w) = ywψ(y)^{−1}.

Proof. The existence of a unique map σ: ^{I}W → W^{J} with the stated property is equivalent to Lemma 9.12. By applying the same lemma to the abstract zip datum A′ := (W, W_{J}, ψ^{−1}) in place of A we find that for any w′ ∈ ^{J}W there exists y′ ∈ W_{J} such that w := y′w′ψ^{−1}(y′)^{−1} ∈ W^{I}, and the element w is independent of y′. After replacing (w′, w) by (w^{′−1}, w^{−1}) this means that for any w′ ∈ W^{J} there exists y′ ∈ W_{J} such that w := ψ^{−1}(y′)w′y^{′−1} ∈ ^{I}W, and the element w is independent of y′. But with y := ψ^{−1}(y′)^{−1} ∈ W_{I} the last equation is equivalent to w′ = ywψ(y)^{−1}, and so for any w′ ∈ W^{J} there exists a unique w ∈ ^{I}W with w′ = σ(w). In other words the map is bijective, as desired.

Proposition 9.14. The bijection in Proposition 9.13 satisfies ℓ(w) = ℓ(σ(w)) for all w ∈ ^{I}W.

Proof. Write the defining relation in the form yw = σ(w)ψ(y). Here y ∈ W_{I} and w ∈ ^{I}W imply that ℓ(yw) = ℓ(y) + ℓ(w), and similarly σ(w) ∈ W^{J} and ψ(y) ∈ W_{J} imply that ℓ(σ(w)ψ(y)) = ℓ(σ(w)) + ℓ(ψ(y)). Moreover, since ψ sends simple reflections to simple reflections, it satisfies ℓ(ψ(y)) = ℓ(y). Together it follows that ℓ(w) = ℓ(σ(w)).

Lemma 9.15. Let σ: ^{I}W → W^{J} be the bijection from Proposition 9.13. For any x ∈ ^{I}W^{J} let σ_{x}: ^{I_{x}}W_{J} → W_{J}^{J_{x}} denote the bijection obtained by applying Proposition 9.13 to A_{x}. Then for all w_{J} ∈ ^{I_{x}}W_{J} we have σ(xw_{J}) = ψ^{−1}(σ_{x}(w_{J})) · x.

Proof. The proof of Lemma 9.12 (a) shows that σ(w) = w′ = ψ^{−1}(w′_{J})·x where w′_{J} = σ_{x}(w_{J}), as desired.

Remark 9.16. Propositions 9.13 and 9.14 can also be deduced from more general results of He ([He3] Proposition 4.3).

9.3 Back to algebraic groups

Now we return to the situation and the notations of the preceding sections. Clearly the connected algebraic zip datum Z gives rise to an abstract zip datum of Coxeter type A := (W, W_{I}, ψ), which by Proposition 3.11 is independent of the frame, up to unique isomorphism. Theorem 5.10 implies that for any w′ ∈ W the element gẇ′ lies in G^{w} for a unique w ∈ ^{I}W.

Theorem 9.17. For any w′ ∈ W and w ∈ ^{I}W we have gẇ′ ∈ G^{w} if and only if w′ ∼ w with respect to A.

Proof. We prove this by induction on #S. If J = S, there is exactly one G^{w} for w = 1 and exactly one A-equivalence class in W, so the assertion holds. Otherwise we have #J < #S. Write w = xw_{J} with x ∈ ^{I}W^{J} and w_{J} ∈ ^{I_{x}}W_{J}, as in Proposition 2.8. Then by (4.1) and Lemma 5.9 the condition gẇ′ ∈ G^{w} requires that w′ ∈ W_{I}xW_{J}, and so does the condition w′ ∼ w by the remarks in Subsection 9.2. It therefore suffices to consider w′ = yxw′_{J} with y ∈ W_{I} and w′_{J} ∈ ^{I_{x}}W_{J}, as in Proposition 2.7. But then w′ ∼ xw′_{J}ψ(y) with respect to A, and gẇ′ = gẏẋẇ′_{J} is in the same E_{Z}-orbit as gẋẇ′_{J}φ(^{g}ẏ). After replacing w′ by xw′_{J}ψ(y) we may thus assume that w′ = xw′_{J} for some w′_{J} ∈ W_{J}. Then Proposition 4.7 and Lemma 5.9 show that gẇ′ ∈ G^{w} if and only if ẇ′_{J} ∈ M^{w_{J}}. By the induction hypothesis this is equivalent to w′_{J} ∼ w_{J} with respect to A_{x}. By Proposition 9.9 this in turn is equivalent to w′ ∼ w with respect to A, as desired.

Combining Theorems 7.5 and 9.17 we deduce:

Corollary 9.18. If Z is orbitally finite, then for any w, w′ ∈ W the elements gẇ and gẇ′ lie in the same E_{Z}-orbit if and only if w ∼ w′ with respect to A.

10 Non-connected algebraic zip data

In this section we generalize the main results of Sections 5 and 6 to nonconnected groups. Throughout we denote a not necessarily connected linear algebraic group by Ĝ, its identity component by G, and its finite group of connected components by π_{0}(Ĝ) := Ĝ/G; and similarly for other letters of the alphabet. Note that the unipotent radical R_{u}G is a normal subgroup of Ĝ. Any homomorphism φ̂: Ĝ → Ĥ restricts to a homomorphism φ: G → H.

Definition 10.1. An algebraic zip datum is a tuple Ẑ = (Ĝ, P̂, Q̂, φ̂) consisting of a linear algebraic group Ĝ with subgroups P̂ and Q̂ and a homomorphism φ̂: P̂/R_{u}P → Q̂/R_{u}Q, such that Z := (G, P, Q, φ) is a connected algebraic zip datum. The zip group EẐ ⊂ P̂ × Q̂, its action on Ĝ, and the orbit oẐ(X) of a subset X ⊂ Ĝ are defined in exact analogy to (3.2), (3.3), and (3.4).

Throughout this section we fix an algebraic zip datum Ẑ = (Ĝ, P̂, Q̂, φ̂) with associated connected algebraic zip datum Z = (G, P, Q, φ). We fix a frame (B, T, g) of Z and use the other pertaining notations from Sections 3 through 5. We also define

Ŵ := NormĜ(T)/T  and  Ω := (NormĜ(B) ∩ NormĜ(T))/T,

so that Ω ≅ π_{0}(Ĝ) and Ŵ = W ⋊ Ω. For each ω ∈ Ω we fix a representative ω̇ ∈ NormĜ(B) ∩ NormĜ(T), and for ŵ = wω ∈ Ŵ with w ∈ W and ω ∈ Ω we set ˙ ŵ := ẇω̇ ∈ NormĜ(T). Note that by definition E_{Z} is the identity component of EẐ. Thus to study the EẐ-orbits in Ĝ, we first study the orbits under E_{Z} and then the action of EẐ/E_{Z} on them.

Lemma 10.2. For any ω ∈ Ω the conjugate connected algebraic zip datum

ω̇Z := (G, P, ω̇Q, int(ω̇) ∘ φ)

has zip group E ω̇_{Z} = {(p, ω̇q) | (p, q) ∈ E_{Z}} and frame (B, T, g), and the isomorphism of varieties G → Gω̇, g′ ↦ g′ω̇ induces a bijection from the E ω̇_{Z}- orbits in G to the E_{Z}-orbits in Gω̇.

Proof. Direct calculation.

Lemma 10.3. The subsets o_{Z}(gB ˙ŵB) for all ŵ ∈ ^{I}W Ω form a disjoint decomposition of Ĝ.

Proof. Take any ω ∈ Ω. Then by Theorems 5.10 and 5.14 the subsets o ω̇_{Z}(gBẇB) for all w ∈ ^{I}W form a disjoint decomposition of G. Thus by Lemma 10.2 the subsets o_{Z}(gBẇBω̇) for all w ∈ ^{I}W form a disjoint decomposition of Gω̇. Since ω̇ ∈ NormĜ(B) by assumption, the latter subset is equal to o_{Z}(gBẇω̇B). By varying ω the proposition follows.

Next define L̂ := NormP̂(L) and M̂ := NormQ̂(M), so that P̂ = U ⋊ L̂ and Q̂ = V ⋊ M̂, and φ̂ can be identified with a homomorphism L̂ → M̂. Set

Ŵ_{I} := Norm_{g^{−1}}_{L̂}(T)/T,  Ω_{I} := (Norm_{g^{−1}}_{L̂}(B) ∩ Norm_{g^{−1}}_{L̂}(T))/T,
Ŵ_{J} := NormM̂(T)/T,  Ω_{J} := (NormM̂(B) ∩ NormM̂(T))/T.

These groups are subgroups of Ŵ and satisfy

\begin{matrix}Ŵ_{I} = W_{I} ⋊ Ω_{I}, & Ω_{I} ≅ π_{0}(L̂) ≅ π_{0}(P̂), \\ Ŵ_{J} = W_{J} ⋊ Ω_{J}, & Ω_{J} ≅ π_{0}(M̂) ≅ π_{0}(Q̂). \end{matrix}

Also φ̂∘int(g) induces a homomorphism ψ̂: Ŵ_{I} → Ŵ_{J} extending ψ: W_{I} → W_{J} and sending Ω_{I} to Ω_{J}. Moreover, the elements (^{g}ω̇, φ̂(^{g}ω̇)) for all ω ∈ Ω_{I} are representatives of the connected components of EẐ.

Lemma 10.4. (a) The map (υ, ŵ) ↦ υŵψ̂(υ)^{−1} defines a left action of Ω_{I} on ^{I}W Ω.

(b) Take any υ ∈ Ω_{I} and ŵ ∈ ^{I}W Ω and abbreviate ŵ′ := υŵψ̂(υ)^{−1} ∈ ^{I}W Ω. Then the element (^{g}υ̇, φ̂(^{g}υ̇)) ∈ EẐ sends o_{Z}(gB ˙ŵB) to o_{Z}(gB ˙ŵ′B).

Proof. Conjugation by Ω_{I} preserves the set of simple reflections I and thus the subset ^{I}W ⊂ W. In (a) we therefore have υŵψ̂(υ)^{−1} = ^{υ}ŵ · υψ̂(υ)^{−1} ∈ IW Ω · Ω = ^{I}W Ω, as desired. In (b) the elements υ̇ and φ̂(^{g}υ̇) normalize B; hence the image is

gυ̇ · o_{Z}(gB ˙ŵB) · φ̂(^{g}υ̇)^{−1} = o_{Z}(^{g}υ̇gB ˙ŵBφ̂(^{g}υ̇)^{−1}) = o_{Z}(gBυ̇ ˙ŵφ̂(^{g}υ̇)^{−1}B).

As υ̇ ˙ŵφ̂(^{g}υ̇)^{−1} differs from ˙ ŵ′ by an element of T, this proves (b).

For any ŵ ∈ ^{I}W Ω we now define

Ĝŵ := oẐ(gB ˙ŵB),  (10.5)

which is independent of the representative ˙ ŵ. Lemma 10.4 implies that Ĝŵ is the union of o_{Z}(gB ˙ŵ′B) for all ŵ′ in the Ω_{I}-orbit of ŵ under the action in 10.4 (a). Thus Ĝŵ depends only on ŵ modulo Ω_{I}, and with Lemma 10.3 we conclude:

Theorem 10.6. The subsets Ĝŵ for all ŵ ∈ ^{I}W Ω modulo the action of Ω_{I} from 10.4 (a) form a disjoint decomposition of Ĝ.

To describe the closure relation between the subsets Ĝŵ we define analogues of the Bruhat order ≤ on Ŵ = W Ω and of the relation ≼ from Definition 6.1 on IW Ω:

Definition 10.7. For ŵ = wω and ŵ′ = w′ω′ with w, w′ ∈ W and ω, ω′ ∈ Ω we write ŵ′ ≤ ŵ if and only if w′ ≤ w and ω′ = ω.

Definition 10.8. For ŵ, ŵ′ ∈ ^{I}W Ω we write ŵ′ ≼ ŵ if and only if there exists ŷ ∈ Ŵ_{I} such that ŷŵ′ψ̂(ŷ)^{−1} ≤ ŵ.

Theorem 10.9. For any ŵ ∈ ^{I}W Ω we have

\overline{Ĝŵ} = ⋃_{\begin{matrix}ŵ′∈^{I}W Ω \\ ŵ′≼ŵ \end{matrix}} Ĝ^{ŵ′}.

Proof. Write ŵ = wω with w ∈ ^{I}W and ω ∈ Ω. Then the conjugate zip datum ω̇Z has the isogeny int(ω̇) ∘ φ: L → ω̇M and hence the induced isomorphism of Weyl groups int(ω) ∘ ψ: W_{I} →^{∼} ^{ω}W_{J} = W^{ω}_{J}. Thus Theorems 5.14 and 6.2 and Definition 6.1 imply that

\overline{oω̇_{Z}(gBẇB)} = ⋃_{w′} oω̇_{Z}(gBẇ′B),

where the union ranges over all w′ ∈ ^{I}W such that yw′ ^{ω}ψ(y)^{−1} ≤ w for some y ∈ W_{I}. Note that this inequality is equivalent to yw′ωψ(y)^{−1} ≤ wω by Definition 10.7. Thus with Lemma 10.2 we deduce that

\overline{o_{Z}(gB ˙ŵB)} = \overline{o_{Z}(gBẇω̇B)} = ⋃_{w′} o_{Z}(gBẇ′ω̇B) = ⋃_{ŵ′} o_{Z}(gB ˙ŵ′B),

where the last union ranges over all ŵ′ ∈ ^{I}W Ω such that yŵ′ψ(y)^{−1} ≤ ŵ for some y ∈ W_{I}. By taking the union of conjugates of this under (^{g}υ̇, φ̂(^{g}υ̇)) ∈ EẐ for all υ ∈ Ω_{I} we obtain the closure of Ĝŵ. By Lemma 10.4 the right hand side then yields the union of o_{Z}(gB ˙ŵ′′B) for all ŵ′′ = υŵ′ψ̂(υ)^{−1} with yŵ′ψ(y)^{−1} ≤ ŵ for some υ ∈ Ω_{I} and y ∈ W_{I}. But here ŷ:= yυ^{−1} runs through the group W_{I}Ω_{I} = Ŵ_{I} and the inequality is equivalent to

ŷŵ′′ψ̂(ŷ)^{−1} = yυ^{−1}ŵ′ψ̂(υ)ψ(y)^{−1} ≤ ŵ.

By Definition 10.8 these ŵ′′ are precisely the elements of ^{I}W Ω satisfying ŵ′′ ≼ ŵ.

Finally, let us call Ẑ orbitally finite if the conjugates ω̇Z are orbitally finite for all ω ∈ Ω. This holds in particular when the differential of φ̂ at 1 vanishes, because then we can apply Proposition 7.3 to ω̇Z. Combining Theorem 7.5 with the remarks leading up to Theorem 10.6 we deduce:

Theorem 10.10. Assume that Ẑ is orbitally finite. Then:

(a) For any ŵ ∈ ^{I}W Ω we have Ĝŵ = oẐ(g ˙ŵ).

(b) If ŵ ∈ ^{I}W Ω runs through a system of representatives for the action of Ω_{I} from 10.4 (a), then g ˙ŵ runs through a set of representatives for the EẐ-orbits in Ĝ.

11 Dual parametrization

The decomposition of G from Theorem 5.10 is parametrized in a natural way by elements of ^{I}W. In this section we translate that parametrization into an equally natural parametrization by elements of W^{J}, which was used by Lusztig and He (see Section 12). We also carry out the corresponding translation in the non-connected case.

11.1 The connected case

For any w ∈ W^{J} we set

G^{w} := o_{Z}(gBẇB).  (11.1)

Note that this does not depend on the representative ẇ of w and conforms to Definition 5.7 by Theorem 5.14. In Proposition 9.13 we have already established a natural bijection σ: ^{I}W → W^{J}.

Theorem 11.2. For any w ∈ ^{I}W we have G^{w} = G^{σ(w)}.

Proof. If I = J = S, we have ^{I}W = W^{J} = {1} and so w = σ(w) = 1; hence the assertion holds trivially. Otherwise #I < #S. Write w = xw_{J} as in Proposition 2.8 with x ∈ ^{I}W^{J} and w_{J} ∈ ^{I_{x}}W_{J}, and let σ_{x}: ^{I_{x}}W_{J} → W_{J}^{J_{x}} denote the bijection obtained by applying Proposition 9.13 to A_{x}. Then Zẋ has the frame (M ∩ B, T, 1) by Proposition 4.12, and so the induction hypothesis implies that

M^{w_{J}} = M^{σ_{x}(w_{J})} = o_{Zẋ}((M ∩ B)σ̇_{x}(w_{J})(M ∩ B)).

By Lemma 5.13 this is equal to o_{Zẋ}((M ∩ B)σ̇_{x}(w_{J})), and so by Lemmas 5.9 and 4.9 we have

G^{w} = o_{Z}(gẋM^{w_{J}}) = o_{Z}(gẋ o_{Zẋ}((M∩B)σ̇_{x}(w_{J}))) = o_{Z}(gẋ(M∩B)σ̇_{x}(w_{J})).

Recall from Lemma 9.15 that σ(w) = w_{I}x with w_{I} := ψ^{−1}(σ_{x}(w_{J})) ∈ W_{I}. It follows that σ_{x}(w_{J}) = ψ(w_{I}) and therefore σ̇_{x}(w_{J}) ∈ T · φ(^{g}ẇ_{I}) and σ̇(w) ∈ T · ẇ_{I} ẋ. Since T ⊂ M ∩ B, using the action (3.3) of E_{Z} we deduce that

G^{w} = o_{Z}(gẋ(M ∩ B)φ(^{g}ẇ_{I})) = o_{Z}(gẇ_{I} ẋ(M ∩ B)) = o_{Z}(gσ̇(w)(M ∩ B)).

Using (3.3) and (3.9) for the action of V, respectively Lemma 5.13, we conclude that

G^{w} = o_{Z}(gσ̇(w)B) = o_{Z}(gBσ̇(w)B) = G^{σ(w)},

as desired.

Theorem 11.3. The G^{w} for all w ∈ W^{J} form a disjoint decomposition of G by nonsingular subvarieties of dimension dim P + ℓ(w).

Proof. Combine Theorems 5.10, 5.11, 11.2 and Proposition 9.14.

Next, in analogy to Definition 6.1 we define:

Definition 11.4. For w, w′ ∈ W^{J} we write w′ ≼ w if and only if there exists y ∈ W_{I} such that yw′ψ(y)^{−1} ≤ w.

Theorem 11.5. For any w ∈ W^{J} we have

\overline{G^{w}} = ∐_{\begin{matrix}w′∈W^{J} \\ w′≼w \end{matrix}} G^{w′}.

Proof. By combining Theorems 11.2 and 6.2 we already know that \overline{G^{w}} is the disjoint union of G^{w′} for certain w′ ∈ W^{J}; it only remains to determine which. First consider w′ ∈ W^{J} with G^{w′} ⊂ \overline{G^{w}}. Then gẇ′ ∈ \overline{G^{w}}, and so by Lemma 6.6 there exist b ∈ B and w′′ ∈ W such that w′′ ≤ w and o_{Z}(gẇ′) = o_{Z}(gẇ′′b). Set w̃′ := σ^{−1}(w′) ∈ ^{I}W and take y ∈ W_{I} satisfying w′ = yw̃′ψ(y)^{−1}. Then ẇ′ = ẏ ˙w̃′tφ(^{g}ẏ)^{−1} for some t ∈ T, and thus o_{Z}(gw′) = o_{Z}(gw̃′t). Therefore o_{Z}(gw̃′t) = o_{Z}(gw′′b), and so Lemma 6.9 implies that y′w̃′ψ(y′)^{−1} ≤ w′′ for some y′ ∈ W_{I}. Together it follows that

(y′y^{−1})w′ψ(y′y^{−1})^{−1} = y′w̃′ψ(y′)^{−1} ≤ w′′ ≤ w

and hence w′ ≼ w, proving “⊂”. Conversely consider w′ ∈ W^{J} with w′ ≼ w, and take y ∈ W_{I} such that w′′ := yw′ψ(y)^{−1}≤ w. Lemma 6.6 then shows that o_{Z}(gẇ′′T) ⊂ \overline{G^{w}}. Therefore

o_{Z}(gẇ′T) = o_{Z}(gẏẇ′T φ(gẏg^{−1})^{−1}) =

= o_{Z}(gẏẇ′φ(gẏg^{−1})^{−1}T ) = o_{Z}(gẇ′′T) ⊂ \overline{G^{w}}.

Since also o_{Z}(gẇ′T) ⊂ G^{w′}, this with the preliminary remark on \overline{G^{w}} shows that G^{w′} ⊂ \overline{G^{w}}, proving “⊃”.

Remark 11.6. In Definitions 5.7 and 11.1 we have introduced the subsets G^{w} := o_{Z}(gBẇB) only for w ∈ ^{I}W ∪ W^{J}, not for arbitrary w ∈ W. Our results do not say anything directly about the latter. Note that in case φ is an isomorphism their closures have been determined in [LY1] Theorem 5.2 and [He3] Proposition 5.8.

11.2 The non-connected case

Now we return to the notations from Section 10. We begin with an analogue of Proposition 9.13:

Proposition 11.7. There exists a unique bijection σ̂: ^{I}W Ω → ΩW^{J} with the property that for any ŵ ∈ ^{I}W Ω there exists y ∈ W_{I} such that σ̂(ŵ) = yŵψ(y)^{−1}.

Proof. The equation requires that σ̂(ŵ) ∈ Ŵ lie in the same W -coset as ŵ. Thus for any fixed ω ∈ Ω, we need a unique bijection ^{I}W ω → ωW^{J} sending wω to an element of the form ywωψ(y)^{−1} for some y ∈ W_{I}. Multiplying both elements on the right by ω^{−1} this amounts to a unique bijection IW → ωW^{J}ω^{−1} = W ^{^{ω}J} sending w to an element of the form ywωψ(y)^{−1}ω^{−1} for some y ∈ W_{I}. But int(ω) ∘ ψ: W_{I} → ωW_{J}ω^{−1} = W^{ω}_{J} is precisely the isomorphism associated to the conjugate connected algebraic zip datum ω̇Z from Lemma 10.2. Thus a unique bijection with that property exists by Proposition 9.13 applied to ω̇Z.

For any ŵ ∈ ΩW^{J} we now define

Ĝŵ := oẐ(gB ˙ŵB).  (11.8)

Again this does not depend on the representative ˙ ŵ of ŵ and conforms to Definition (10.5).

Theorem 11.9. For any ŵ ∈ ^{I}W Ω we have Ĝŵ = Ĝ^{σ̂(ŵ)}.

Proof. Write ŵ = wω with w ∈ ^{I}W and ω ∈ Ω. In the proof of Proposition 11.7 we have seen that σ̂(ŵ) = w′ω, where w′ ∈ W ^{^{ω}J} is the image of w under the isomorphism given by Proposition 9.13 applied to ω̇Z. Thus by Theorem 11.2 we have o ω̇_{Z}(gBẇB) = o ω̇_{Z}(gBẇ′B) inside G. On multiplying on the right by ω̇ and applying Lemma 10.2 to both sides we deduce that

o_{Z}(gB ˙ŵB) = o_{Z}(gBẇBω̇) = o_{Z}(gBẇ′Bω̇) = o_{Z}(gB ˙σ̂(ŵ)B).

The desired equality follows from this by applying oẐ.

Lemma 11.10. (a) The map (υ, ŵ) ↦ υŵψ̂(υ)^{−1} defines a left action of Ω_{I} on ΩW^{J}.

(b) The bijection σ̂: ^{I}W Ω → ΩW^{J} from Proposition 11.7 is Ω_{I}-equivariant.

Proof. Take υ ∈ Ω_{I} and ŵ ∈ ΩW^{J}. To prove (a) observe that conjugation by ψ̂(υ) ∈ Ω_{J} preserves the set of simple reflections J and thus the subset W^{J} ⊂ W. We therefore have υŵψ̂(υ)^{−1} = υψ̂(υ)^{−1} ·^{ψ̂(υ)}ŵ ∈ Ω · ΩW^{J} = ΩW^{J}, as desired. In (b) write σ̂(ŵ) = yŵψ(y)^{−1} with y ∈ W_{I}. Then

υσ̂(ŵ)ψ̂(υ)^{−1} = (υyυ^{−1})(υŵψ̂(υ)^{−1})ψ̂(υyυ^{−1})^{−1} = σ̂(υŵψ̂(υ)^{−1}),

because the left hand side is in ΩW^{I} and υyυ^{−1} ∈ W_{I}. This proves (b).

Theorem 11.11. The subsets Ĝŵ for all ŵ ∈ ΩW^{J} modulo the action of Ω_{I} from 11.10 (a) form a disjoint decomposition of Ĝ.

Proof. Combine Theorems 10.6 and 11.9 with Lemma 11.10.

Definition 11.12. For ŵ, ŵ′ ∈ ΩW^{J} we write ŵ′ ≼ ŵ if and only if there exists ŷ ∈ Ŵ_{I} such that ŷŵ′ψ̂(ŷ)^{−1} ≤ ŵ.

Theorem 11.13. For any ŵ ∈ ΩW^{J} we have

\overline{Ĝŵ} = ⋃_{\begin{matrix}ŵ′∈ΩW^{J} \\ ŵ′≼ŵ \end{matrix}} Ĝ^{ŵ′}.

Proof. Write ŵ = wω with ω ∈ Ω and w ∈ W ^{^{ω}J}. Applying Theorem 11.5 to the conjugate zip datum ω̇Z shows that \overline{o ω̇_{Z}(gBẇB)} is the union of the subsets o ω̇_{Z}(gBẇ′B) for all w′ ∈ W ^{^{ω}J} such that yw′ωψ(y)^{−1}ω^{−1} ≤ w for some y ∈ W_{I}. On multiplying on the right by ω̇ and applying Lemma 10.2 to everything we deduce that \overline{o_{Z}(gB ˙ŵB)} = \overline{o_{Z}(gBẇBω̇)} is the union of the subsets o_{Z}(gBẇ′Bω̇) = o_{Z}(gBẇ′ω̇B) for the same elements w′. Writing ŵ′ = w′ω this is equal to the union of the subsets o_{Z}(gB ˙ŵ′B) for all ŵ′ ∈ ΩW^{J} such that yŵ′ψ(y)^{−1} ≤ w for some y ∈ W_{I}. The theorem follows from this by applying oẐ.

12 Generalization of certain varieties of Lusztig

In this section we consider a certain type of algebraic variety with an action of a reductive group G whose orbit structure is closely related to the structure of the E_{Z}-orbits in G for an algebraic zip datum Z. Special cases of such varieties have been defined by Lusztig ([Lus2]) and by Moonen and the second author in [MW].

12.1 The coset variety of an algebraic zip datum

Remark 12.1. To keep notations simple, we restrict ourselves to connected zip data, although everything in this section directly extends to non-connected ones by putting ˆ in the appropriate places.

In this section we use only the definition of algebraic zip data and the action of the associated zip group from Section 3, but none of the other theory or notations from the preceding sections, not even the concept of a frame. Fix a connected algebraic zip datum Z = (G, P, Q, φ). Recall that E_{Z} is a subgroup of P × Q and hence of G × G. We also consider the image of G under the diagonal embedding Δ: G ↪ G × G, g ↦ (g, g). We are interested in the left quotient Δ(G)\(G × G) and the right quotient (G × G)/E_{Z}. The first is isomorphic to G via the projection morphism

λ: G × G → G, (g, h) ↦ g^{−1}h.  (12.2)

Turn the right action of E_{Z} on G×G into a left action by letting (p, q) ∈ E_{Z} act by right translation with (p, q)^{−1}. Then with E_{Z} acting on G as in the definition of algebraic zip data, a direct calculation shows that λ is E_{Z}-equivariant. To describe the second quotient recall that φ is a homomorphism P/U → Q/V, where U and V denote the unipotent radicals of P and Q. Consider a left P - coset X ⊂ G and a left Q-coset Y ⊂ G. Then X/U is a right torsor over P/U, and Y/V is a right torsor over Q/V. By a P/U-equivariant morphism Φ: X/U → Y/V we mean a morphism satisfying Φ(x̄p̄) = Φ(x̄)φ(p̄) for all x̄ ∈ X/U and p̄ ∈ P/U.

Definition 12.3. The coset space of Z is the set C_{Z} of all triples (X, Y, Φ) consisting of a left P -coset X ⊂ G, a left Q-coset Y ⊂ G, and a P/U-equivariant morphism Φ: X/U → Y/V.

For any X, Y as above and any (g, h) ∈ G × G, left multiplication by g induces an isomorphism ℓ_{g}: X/U →^{∼} gX/U, and left multiplication by h induces an

isomorphism ℓ_{h}: Y/V →^{∼} hY/V. Therefore (X, Y, Φ) ↦ (gX, hY, ℓ_{h} ∘ Φ ∘ ℓ_{g}^{−1})

defines a left action of G × G on C_{Z}. By applying this action to the canonical base point (P, Q, φ) ∈ C_{Z} we obtain a morphism

ρ: G × G → C_{Z}, (g, h) ↦ (gP, hQ, ℓ_{h} ∘ φ ∘ ℓ_{g}^{−1}).  (12.4)

Clearly this morphism is equivariant under the left action of G × G and hence under the subgroup Δ(G).

Lemma 12.5. There is a unique structure of algebraic variety on C_{Z} such that ρ identifies C_{Z} with the quotient variety (G × G)/E_{Z}.

Proof. The action of G×G is obviously transitive on the set of all pairs (X, Y). Moreover, any P/U-equivariant morphism of right torsors P/U → Q/V has the form p̄ ↦ π_{Q}(q)φ(p̄) = ℓ_{q} ∘ φ(p̄) for some q ∈ Q. Thus the subgroup 1 × Q acts transitively on the set of all triples of the form (P, Q, Φ). Together it follows that the action of G × G on C_{Z} is transitive. On the other hand (g, h) lies in the stabilizer of (P, Q, φ) if and only if g ∈ P and h ∈ Q and ℓ_{h} ∘ φ ∘ ℓ_{g}^{−1} = φ. But under the first two of these conditions, we have for all p̄ ∈ P/U

ℓ_{h} ∘ φ ∘ ℓ_{g}^{−1}(p̄) = π_{Q}(h)φ(π_{P} (g)^{−1}p̄) = π_{Q}(h)φ(π_{P} (g))^{−1}φ(p̄),

and so the third condition is equivalent to φ(π_{P} (g)) = π_{Q}(h). Together this means precisely that (g, h) ∈ E_{Z}, which is therefore the stabilizer of (P, Q, φ). It follows that ρ induces a bijection (G × G)/E_{Z} →^{∼} C_{Z}. Since the quotient variety exists by [Ser], Section 3.2, this yields the unique structure of algebraic variety on C_{Z}.

Following Lemma 12.5 we call C_{Z} also the coset variety of Z. Recall from [Ser] Prop. 2.5.3 that the quotient of an algebraic group by an algebraic subgroup is always a torsor. To summarize we have therefore constructed morphisms with the following properties:

E_{Z}-equivariant Δ(G)-torsor

{ \xymatrix{ & G × G \\ λ \ar[d] & & ρ \ar[d] \\ G & & C_{Z}} }

Δ(G)-equivariant

E_{Z}-torsor (12.6)

Recall that the actions of Δ(G) and E_{Z} on G × G commute and thus combine to an action of Δ(G) × E_{Z}. Therefore (12.6) directly implies:

Theorem 12.7. There are natural isomorphisms of algebraic stacks

\xymatrix{[E_{Z}\G] & [(Δ(G) × E_{Z})\(G × G)] \ar[l]_{[λ]}^{∼} \ar[r]^{[ρ]}_{∼} & [Δ(G)\C_{Z}].}

Even without stacks, we can deduce:

Theorem 12.8. (a) There is a closure-preserving bijection between E_{Z}- invariant subsets A ⊂ G and Δ(G)-invariant subsets B ⊂ C_{Z}, defined by A = λ(ρ^{−1}(B)) and B = ρ(λ^{−1}(A)).

(b) The subset A in (a) is a subvariety, resp. a nonsingular subvariety, if and only if B is one. In that case we also have dim A = dim B.

(c) In particular (a) induces a bijection between E_{Z}-orbits in G and Δ(G)- orbits in C_{Z}.

(d) For any g ∈ G and (X, Y, Φ) ∈ C_{Z} whose orbits correspond, there is an

isomorphism

Stab_{E_{Z}} (g) ≅ Stab_{Δ(G)}((X, Y, Φ)).

Proof. By (12.6) any Δ(G) × E_{Z}-invariant subset of G × G must be simultaneously of the form λ^{−1}(A) for an E_{Z}-invariant subset A ⊂ G and of the form ρ^{−1}(B) for a Δ(G)-invariant subset B ⊂ C_{Z}. Then A = λ(ρ^{−1}(B)) and B = ρ(λ^{−1}(A)), giving the bijection in (a). The bijection preserves closures because λ and ρ are smooth. This proves (a), the first sentence in (b), and the special case (c). In (b) it also proves that dim A+dim G = dim B+dim E_{Z}. But dim G = 2 dim U + dim L = 2 dim V + dim M and dim L = dim M imply that dim U = dim V, and thus using (3.9) that dim E_{Z} = dim U + dim L + dim V = dim G. Therefore dim A = dim B, proving the rest of (b). In (c) by assumption there exists a point \underline{x} ∈ G × G such that λ(\underline{x}) lies in the E_{Z}-orbit of g and ρ(\underline{x}) lies in the Δ(G)-orbit of (X, Y, Φ). Thus after replacing \underline{x} by a suitable translate under Δ(G) × E_{Z} we may assume that λ(\underline{x}) = g and ρ(\underline{x}) = (X, Y, Φ). Then the fact that λ and ρ are torsors implies that the two projection morphisms

\xymatrix{Stab_{E_{Z}} (g) & Stab_{Δ(G)×E_{Z}} (\underline{x}) \ar[l] \ar[r] & Stab_{Δ(G)}((X, Y, Φ))}

are isomorphisms, proving (c). (The isomorphism may depend on the choice of \underline{x}.)

With Theorem 12.8 we can translate many results about the E_{Z}-action on G from the preceding sections to the Δ(G)-action on C_{Z}, in particular Theorems 5.10, 5.11, 6.2, 7.5, 8.1, and their counterparts from Sections 10 and 11.

12.2 Algebraic zip data associated to an isogeny of G

In this subsection we consider algebraic zip data whose isogeny extends to an isogeny on all of G. (Not every connected algebraic zip datum has that property, for instance, if L and M have root system A_{1} associated to long and short roots, respectively, and the square of the ratio of the root lengths is different from the characteristic of k.) Fix a connected reductive algebraic group G over k and an isogeny φ: G → G. Choose a Borel subgroup B ⊂ G and a maximal torus T ⊂ B, and let W be the corresponding Weyl group of G and S its set of simple reflections. Choose an element γ ∈ G such that φ(^{γ}B) = B and φ(^{γ}T) = T. Then φ ∘ int(γ): Norm_{G}(T) → Norm_{G}(T) induces an isomorphism of Coxeter systems

φ̄: (W, S) →^{∼} (W, S).

For any subset I ⊂ S recall from Subsection 2.3 that P_{I} denotes the standard parabolic of type I. Thus the choices imply that φ(^{γ}P_{I}) = P_{φ̄(I)}. We denote the unipotent radicals of arbitrary parabolics P, Q, P′, Q′ by U, V, U′, V′, respectively. Let Ĝ be a linear algebraic group over k having identity component G, and let G^{1} be an arbitrary connected component of Ĝ. Choose an element g_{1} ∈ Norm_{G^{1}}(B)∩Norm_{G^{1}}(T). Then int(g_{1}) induces an automorphism of G that we use to twist φ. Let δ: (W, S) → (W, S) be the isomorphism of Coxeter systems induced by int(g_{1}). Then for any subset I ⊂ S we have ^{g_{1}}P_{I} = P_{δ(I)}. Fix subsets I, J ⊂ S and an element x ∈ ^{J}W^{δφ̄(I)} with J = ^{x}δφ̄(I). Set y := (δφ̄)^{−1}(x) ∈ W.

Lemma 12.9. (a) xΦ_{δφ̄(I)} = Φ_{J}.

(b) xΦ_{δφ̄(I)}^{+} = Φ_{J}^{+}.

Proof. Part (a) follows from J = ^{x}δφ̄(I). By (2.11) the fact that x ∈ ^{J}W^{δφ̄(I)} ⊂ W^{δφ̄(I)} implies xΦ_{δφ̄(I)}^{+} ⊂ Φ^{+}. Together with (a) this implies (b).

Construction 12.10. Set Q := P_{J} and P := ^{γẏ}P_{I} and let L be the Levi component of P containing ^{γẏ}T. Then ^{g_{1}}φ(P) = ẋ(^{g_{1}}φ(^{γ}P_{I})) = ẋP_{δφ̄(I)} and Q = P_{J} have relative position x. Set M := ^{g_{1}}φ(L); this is a Levi component of ^{g_{1}}φ(P) containing ^{g_{1}}φ(^{γẏ}T) = ^{g_{1}}φ(^{γ}T) = ^{g_{1}}T = T. Since the root system of M is xΦ_{δφ̄(I)}, Lemma 12.9 shows that it is also the Levi component of Q containing T. Let ^{g_{1}} φ̃: P/U → Q/V denote the isogeny corresponding to int(g_{1}) ∘ φ | L: L → M. Then we obtain a connected algebraic zip datum Z := (G, P, Q,^{g_{1}} φ̃).

Lemma 12.11. The triple (B, T, γẏ) is a frame of Z, and the Levi components determined by it are M ⊂ Q and L ⊂ P.

Proof. The statements about M and L follow from the inclusions T ⊂ M and γẏT ⊂ L. They also imply that the isogeny L → M corresponding to ^{g_{1}} φ̃ is simply the restriction of int(g_{1}) ∘ φ. Conditions (a) and (b) in Definition 3.6 assert that B ⊂ Q and ^{γẏ}B ⊂ P, which hold by the construction of Q and P. Condition (d) translates to ^{g_{1}}φ(^{γẏ}T ) = T, which was already shown in 12.10. To prove (c) note first that by Lemma 12.9 we have xΦ_{γφ̄(I)}^{+} = Φ_{J}^{+} and therefore ẋB ∩ M = B ∩ M. The definition of y implies that ^{g_{1}}φ(γẏγ^{−1}) ∈ ẋT and hence

g_{1}φ(^{γẏ}B) = ^{^{g_{1}} φ(γẏγ^{−1})·g_{1}}φ(^{γ}B) = ^{^{g_{1}} φ(γẏγ^{−1})}B = ẋB.

From this we can deduce that

g_{1}φ(^{γẏ}B ∩ L) = ^{g_{1}}φ(^{γẏ}B) ∩^{g_{1}}φ(L) = ẋB ∩ M = B ∩ M,

proving the remaining condition (c).

The automorphism ψ defined in (3.11) for the algebraic zip datum Z is given by

ψ := δ ∘ φ̄ ∘ int(y) = int(x) ∘ δ ∘ φ̄: (W_{I}, I) →^{∼} (W_{J}, J)  (12.7)

Definition 12.12. Let X_{I,φ,x} be the set of all triples (P′, Q′, [g′]) consisting of parabolic subgroups P′, Q′ of G of type I, J and a double coset [g′] := V′g′φ(U′) ⊂ G^{1} of an element g′ ∈ G^{1} such that

relpos(Q′,^{g′}φ(P′)) = x.

One readily verifies that the condition on the relative position depends only on [g′], and that ((g, h), (P′, Q′, [g′])) ↦ (^{g}P′,^{h}Q′, [hg′φ(g)^{−1}]) defines a left action of G × G on X_{I,φ,x}. We also have a standard base point (P, Q, [g_{1}]) ∈ X_{I,φ,x}. One can use the definition of X_{I,φ,x} to endow it with the structure of an algebraic variety over k, but in the interest of brevity we define that structure using the following isomorphism:

Proposition 12.13. There is a natural G × G-equivariant isomorphism

C_{Z} ⟶^{∼} X_{I,φ,x}, (gP, hQ, ℓ_{h} ∘^{g_{1}} φ̃ ∘ ℓ_{g}^{−1}) ↦ (^{g}P,^{h}Q, [hg_{1}φ(g)^{−1}]).

Proof. In view of Lemma 12.5 the assertion is equivalent to saying that the action of G × G on X_{I,φ,x} is transitive and the stabilizer of (P, Q, [g_{1}]) is E_{Z}. The transitivity follows directly from the definition of the action. For the stabilizer note that (^{g}P,^{h}Q, [hg_{1}φ(g)^{−1}]) = (P, Q, [g_{1}]) if and only if g ∈ P and h ∈ Q and V hg_{1}φ(g)^{−1}φ(U) = V g_{1}φ(U). Write g = uℓ for u ∈ U, ℓ ∈ L and h = vm for v ∈ V, m ∈ M. Then the last condition is equivalent to V mg_{1}φ(ℓ)^{−1}φ(U) = V g_{1}φ(U), or again to m ·^{g_{1}}φ(ℓ)^{−1} ∈ V ·^{g_{1}}φ(U) ∩ M. But for any element v′ ·^{g_{1}}φ(u′) = m′ ∈ V ·^{g_{1}}φ(U) ∩ M we have ^{g_{1}}φ(u′) = v^{′−1}m′ ∈ g_{1}φ(U) ∩ V M, and since M is also a Levi component of ^{g_{1}}φ(P), it follows that g_{1}φ(U) ∩ V M = ^{g_{1}}φ(U) ∩ V and hence m′ = 1. The last condition is therefore equivalent to m = ^{g_{1}}φ(ℓ). Together this shows that the stabilizer is E_{Z}, as desired.

Lemma 12.14. For any w ∈ ^{I}W ∪ W^{J} the subset G^{w} ⊂ G corresponds via Theorem 12.8 (a) and Proposition 12.13 to the subset

X_{I,φ,x}^{w} := {(^{g}P_{I},^{gẇ}P_{J}, [gẇẋg_{1}bφ(γg^{−1})]) | g ∈ G, b ∈ B} ⊂ X_{I,φ,x},

which is a nonsingular variety of dimension dim P + ℓ(w).

Proof. Since (B, T, γẏ) is a frame of Z by Lemma 12.11, Theorem 5.14 for w ∈ ^{I}W, respectively (11.1) and Lemma 5.13 for w ∈ W^{J}, show that G^{w} = o_{Z}(γẏBẇ). In other words G^{w} is the union of the E_{Z}-orbits of γẏbẇ for all b ∈ B. But by (12.2) and (12.4) we have

λ((γẏb)^{−1}, ẇ) = γẏbẇ,  and
ρ((γẏb)^{−1}, ẇ) = ((γẏb)^{−1}P, ẇQ, ℓẇ ∘ φ ∘ ℓ_{(γẏb)^{−1}}^{−1}),

and so the E_{Z}-orbit of the former corresponds to the Δ(G)-orbit of the latter under the correspondence from Theorem 12.8. Moreover, under the isomorphism from Proposition 12.13 the latter corresponds to the triple

(^{(γẏb)^{−1}}P , ẇQ, [ẇg_{1}φ((γẏb)^{−1})^{−1}]).

The definitions of P and Q show that ^{(γẏb)^{−1}}P = ^{b^{−1}}P_{I} = P_{I} and ẇQ = ẇP_{J}. The definition of y means that ^{g_{1}}φ(γẏγ^{−1}) = ẋt for some t ∈ T; hence

ẇg_{1}φ((γẏb)^{−1})^{−1} = ẇg_{1}φ(γẏb) =

= ẇ · ẋt · g_{1} · φ(γbγ^{−1}) · φ(γ) = ẇ · ẋ · g_{1} ·^{g_{1}^{−1}}t φ(γbγ^{−1}) · φ(γ).

Since φ(^{γ}B) = B, the factor b′ := ^{g_{1}^{−1}}t φ(γbγ^{−1}) runs through B while b runs through B. Thus altogether it follows that G^{w} corresponds to the union of the Δ(G)-orbits of the triples

(P_{I}, ẇP_{J}, [ẇẋg_{1}b′φ(γ)])

for all b′ ∈ B. This union is just the set X_{I,φ,x}^{w} in the lemma. The rest follows from Theorems 5.11, 11.3, and 12.8.

Combining this with Theorems 5.10 and 6.2 and 12.8 we conclude:

Theorem 12.15. (a) The X_{I,φ,x}^{w} for all w ∈ ^{I}W form a disjoint decomposition of X_{I,φ,x} by nonsingular subvarieties of dimension dim P + ℓ(w).

(b) For any w ∈ ^{I}W we have

\overline{X_{I,φ,x}^{w}} = ∐_{\begin{matrix}w′∈^{I}W \\ w′≼w \end{matrix}} X_{I,φ,x}^{w′}.

Analogously, using Theorems 11.3 and 11.5 and 12.8 we obtain:

Theorem 12.16. (a) The X_{I,φ,x}^{w} for all W^{J} form a disjoint decomposition of X_{I,φ,x} by nonsingular subvarieties of dimension dim P + ℓ(w).

(b) For any w ∈ W^{J} we have

\overline{X_{I,φ,x}^{w}} = ∐_{\begin{matrix}w′∈W^{J} \\ w′≼w \end{matrix}} X_{I,φ,x}^{w′}.

12.3 Frobenius

Keeping the notations of the preceding subsection, we now assume that k has positive characteristic and that φ: G → G is the Frobenius isogeny coming from a model G_{0} of G over a finite subfield 𝔽_{q} ⊂ k of cardinality q. Then G_{0} is quasi-split; hence we may, and do, assume that B and T come from subgroups of G_{0} defined over 𝔽_{q} and therefore satisfy φ(B) = B and φ(T) = T. We can thus take γ := 1. In this case, our varieties X_{I,φ,x} coincide with the varieties Z_{I} used in [MW] to study F -zips with additional structures. The isogeny ^{g_{1}} φ̃ in the connected algebraic zip datum Z then has vanishing differential; hence Z is orbitally finite by Proposition 7.3. Thus by Theorem 7.5 each G^{w} is a single E_{Z}-orbit, and so by Theorem 12.8 and Theorem 12.15 we deduce:

Theorem 12.17. (a) If φ is the Frobenius isogeny associated to a model of G over a finite field, each X_{I,φ,x}^{w} in Theorem 12.15 is a single Δ(G)-orbit. In particular the set

{(P_{I}, ẇP_{J}, [ẇẋg_{1}]) | w ∈ ^{I}W }

is a system of representatives for the action of Δ(G) on X_{I,φ,x}.

(b) For any w ∈ ^{I}W, the closure of the orbit of (P_{I}, ẇP_{J}, [ẇẋg_{1}]) is the union of the orbits of (P_{I},^{˙w′}P_{J}, [ẇ′ẋg_{1}]) for those w′ ∈ ^{I}W satisfying w′ ≼ w.

Theorem 12.17 (a) was proved in [MW], Theorem 3 and (b) answers the question of the closure relation that was left open there.

12.4 Lusztig’s varieties

Now we apply the results of Subsection 12.2 to the special case φ = id. In this case we can choose γ := 1 and obtain φ̄ = id. Then our varieties X_{I,φ,x} coincide with the varieties Z_{I,x,δ} defined and studied by Lusztig in [Lus2]. There he defines a decomposition of X_{I,φ,x} into a certain family of Δ(G)- invariant subvarieties. In [He2], He shows how to parametrize this family by the set W^{δ(I)}. We will denote the piece corresponding to w ∈ W^{δ(I)} in this parametrization by X̃_{I,φ,x}^{w}. (In [He2], He denotes X_{I,φ,x} by Z̃_{I,x,δ} and X̃_{I,φ,x}^{w} by Z̃_{I,x,δ}^{w}.) We will show that this decomposition is the same as ours from Theorem 12.16 up to a different parametrization.

Lemma 12.18. The map w ↦ wx induces a bijection W^{J} ⟶^{∼} W^{δ(I)}.

Proof. Take any w ∈ W^{J}. Using Lemma 12.9 and (2.11) we get wxΦ_{δ(I)}^{+} = wΦ_{J}^{+} ⊂ Φ^{+}. By (2.11) this shows that wx ∈ W^{δ(I)}. A similar argument shows that wx^{−1} ∈ W^{J} for any w ∈ W^{δ(I)}, which finishes the proof.

Theorem 12.19. For any w ∈ W^{J} we have X_{I,φ,x}^{w} = X̃_{I,φ,x}^{wx}.

Proof. The statement makes sense by Lemma 12.18. Let w ∈ W^{J} and w′ := wx ∈ W^{δ(I)}. In [He2], Proposition 1.7, He shows that

X̃_{I,φ,x}^{w′} = Δ(G) · {(P_{I},^{b ˙w′ẋ^{−1}}P_{J}, [bẇ′g_{1}b′]) | b, b′ ∈ B}.

(In [He2], it is assumed that G is semi-simple and adjoint. But this assumption is not needed for the proof of Proposition 1.7 in [loc. cit.].) By acting on such a point (P_{I},^{b ˙w′ẋ^{−1}}P_{J}, [bẇ′g_{1}b′]) with Δ(b^{−1}) and using w = w′x^{−1} we get

X̃_{I,φ,x}^{w′} = Δ(G) · {(P_{I}, ẇP_{J}, [ẇẋg_{1}b′]) | b′ ∈ B}.

Since γ = 1, comparison with Lemma 12.14 proves the claim.

From Theorem 12.16 we can now deduce the closure relation between the X̃_{I,φ,x}^{w}:

Theorem 12.20. For any w ∈ W^{δ(I)} we have

\overline{X̃_{I,φ,x}^{w}} = ∐_{\begin{matrix}w′∈W^{δ(I)} \\ w′x^{−1}≼wx^{−1} \end{matrix}} X̃_{I,φ,x}^{w′}.

In the special case x = 1 this result is due to He (see [He2], Proposition 4.6).

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Richard Pink Dept. of Mathematics ETH Zürich CH-8092 Zürich Switzerland pink@math.ethz.ch Torsten Wedhorn Dept. of Mathematics University of Paderborn D-33098 Paderborn Germany wedhorn@math.unipaderborn.de

Paul Ziegler Dept. of Mathematics ETH Zürich CH-8092 Zürich Switzerland paul.ziegler@math.ethz.ch