Zip Data

Zip Data

Master’s Thesis Paul Ziegler

Advisor Prof. Richard Pink

March 2010

Departement of Mathematics

ETH Z¨ urich

Contents

1 Introduction 1

2 Reductive Groups 3

3 Algebraic Zip Data 3

4 Coset Representatives in Coxeter Groups 8

5 The Orbits of an Algebraic Zip Datum 10

6 Closure 15

7 Stabilizers 20

8 Frobenius Zip Data 24

9 Abstract Zip Data 25

10F-Zips 28

1 Introduction

LetGbe a connected reductive algebraic group over an algebraically closed field, P andP⁰ two parabolic subgroups ofGwith Levi factorsLandL⁰ respectively.

Letϕ:L→L⁰ be an isogeny. Consider the algebraic group Z := (R_uP× R_uP⁰)oL,

where` ∈L acts onRuP by conjugation with` and on RuP⁰ by conjugation withϕ(`). The groupZ acts onGfrom the left by

(u, u⁰, `)·g=u<sup>0</sup>ϕ(`)g`⁻¹u⁻¹.

We call such a datum (G, P, P⁰, L, L⁰, ϕ) analgebraic zip datum. In this paper we study the orbit structure of such an action.

Let W be the Weyl group of G with respect to a maximal torus T of G contained in L and pick a Borel subgroup B of P containing T. This defines a set of simple reflectionsS. There exists a subset I of S such that the Weyl group W(L) ofL is generated by the elements of I. OnW there is a natural partial order, the Bruhat order, which we denote by ≤. In W there exists a natural set of representatives forW/W(L), namely

W^I ={w∈W |w≤ws for alls∈I}.

To eachw∈W^I we associate a locally closed subsetG^w ofGsuch that:

Theorem 1.1(see 5.19).

G= a

w∈W^I

G^w.

Theorem 1.2(see 6.12). The closure ofG^w is given by G^w=a

w⁰

G^w0,

where w⁰ ranges over the w⁰ ∈ W^I for which there exists v ∈W(L) such that ϕ(v)w⁰v⁻¹≤w.

If P and P⁰ are Borel subgroups of Gand L = L⁰ is a common maximal torus our decomposition is the Bruhat decomposition of Ginto double cosets P⁰wP forw∈W.

The orbits in each G^w correspond to the orbits of the action of a certain reductive group on itself by twisted conjugation (see Section 5). Of particular interest is the case that there are only finitely many orbits in G. We call algebraic zip data having this property Frobenius zip data. Using the Lang- Steinberg Theorem, we deduce a criterion for a zip datum to be Frobenius. It is satisfied in particular if the differential of ϕ vanishes, for example if ϕ is a Frobenius morphism (see Section 8).

Theorem 1.3. For Frobenius zip data, the pieces G^w are the orbits of Z. In particular there is a bijection between the set of orbits inGandW^I.

We also obtain a description of the stabilizers:

Theorem 1.4(see Theorem 8.8). The stabilizer of an element ofGunder the action of a Frobenius zip datum is the semidirect product of a finite group of Lie type and a connected unipotent algebraic group.

In Section 9 we determine which elements ofW lie in the same orbits under the action of a Frobenius zip datum. In order to do this, we introduce the notion of an abstract zip datum. This is a datum (W, X, X⁰, ψ), whereW is an abstract group with subgroups X and X⁰ and ψ: X → X⁰ is a homomorphism. Each algebraic zip datum gives rise to an abstract zip datum (W, W(L), W(L⁰), ψ), whereψis induced byϕ. For each abstract zip datum we define an equivalence relation on W, which is the equivalence relation defined by intersecting the orbits inGwithW in case the abstract zip datum comes from a Frobenius zip datum. We give an inductive and an explicit characterization of this relation.

The main tool we use to deduce our results is the following: For eachw∈W, the Bruhat cell P⁰wP is Z-invariant, and we show that the orbits in P⁰wP correspond to the orbits in L⁰ under the action defined by another algebraic zip datum. This allows us to prove statements inductively, starting from the caseL⁰ =L=G. In this case, we simply have the groupGacting on itself by conjugation twisted withϕ.

In order to study the closure of a pieceG^w, we show thatG^wis the minimal Z-invariant subset ofGcontaining the Bruhat cellBwB. Then we use the fact that the closure order between the Bruhat cells is the Bruhat order to deduce our result about the closure ofG^w.

An important application is the classification ofF-zips. Letk be a field of characteristicp >0. AnF-zip over kis a datum (M, C^•, D_•, ϕ_•) consisting of a finite-dimensional vector space M over k, a descending filtration C^• of M, an ascending filtrationD_•ofM and Frobenius-linear isomorphismsϕ_•between the graded pieces of these filtrations (see 10.2). This notion was introduced by Moonen and Wedhorn in [4]. There, they classify theF-zips over an algebraically closed field as follows:

The type τ of an F-zip is the function Z → Z≥0 sending i ∈ Z to the dimension of thei-th graded piece of C^•.

Theorem 1.5. Letkbe an algebraically closed field of characteristicp >0. Let τ:Z→Z≥0 be a function with finite supporti1> . . . > ir. Let nj =τ(ij) and n=n1+. . .+nr. Then there is a bijection

{isomorphism classes ofF-zips of typeτ overk} ←→(S_n1× · · · ×S_nr)\Sn

To prove this, they define a varietyXτwith an action ofG=GLnsuch that the orbits onXτ correspond to the isomorphism classes ofF-zips of typeτ and classify theG-orbits onXτ.

In [4], Moonen and Wedhorn use F-zips to define stratifications on certain moduli spaces. For this application, it is also important to know the closure order between the orbits inXτ. This order was determined by Wedhorn in [9].

In Section 10, we show that these results also follow from our theory of alge- braic zip data. We show that for a certain Frobenius zip datum (G, P, P⁰, L, L⁰, ϕ) there exists a morphismG→X_τ which induces a bijection between the orbits ofZinGand the orbits ofGinX_τ preserving the closure order. ThisLsatisfies W(L) =S_n1× · · · ×S_nr for n₁, . . . , n_r as in Theorem 1.5, hence Theorem 1.5 follows from our classification of the orbits in G. Furthermore, our result on the closure order of the orbits inGimplies the result of Wedhorn. Although we work in a different setting than Moonen and Wedhorn, our proof of the closure order uses methods similar to those of Wedhorn in [9].

Theorem 1.4 yields a similar statement about the automorphism group of anF-zip (see 10.16). We also define certain universal constructions forF-zips, namely direct sums and tensor products, and show how they can be realized as morphisms of the algebraic zip data which classify theF-zips of a certain type.

2 Reductive Groups

Except for the beginning of Section 10, we shall use the language of varieties over a fixed algebraically closed fieldk. By an algebraic group we shall always mean a linear algebraic group overk.

For any algebraic groupG, we denote byRuGits unipotent radical. For any w∈G, we shall denote the conjugation mapG→G, g7→wgw⁻¹ by int(w) or g7→^wg. LetGbe a connected reductive algebraic group.

Lemma 2.1 ([7], 8.4.6 (ii)). Let P andQ be parabolic subgroups of G. Then (P∩Q)RuP is a parabolic subgroup ofGwith unipotent radical(P∩RuQ)RuP. Lemma 2.2. Let H be a Levi factor of a parabolic subgroup of G and let T be a maximal torus ofH. If P is a parabolic subgroup of Galso containing T, then H∩P is a parabolic subgroup of H with unipotent radical H ∩ RuP. If P=LnRuP is a Levi decomposition of P withT ⊂L, then

H∩P= (H∩L)n(H∩ RuP) is a Levi decomposition ofH∩P.

Proof. This follows from [3], II.1.8.

3 Algebraic Zip Data

An isogeny between two connected algebraic groups is a surjective homomor- phism with finite kernel.

Definition 3.1. Analgebraic zip datum is a tuple (G, P, P⁰, L, L⁰, ϕ) consisting of a connected reductive algebraic groupG, two parabolic subgroupsP andP⁰ ofG, Levi components LandL⁰ ofP andP⁰ and an isogenyϕ:L→L⁰.

For each algebraic zip datumZ := (G, P, P⁰, L, L⁰, ϕ), we consider the alge- braic group

(RuP× RuP⁰)oL,

where ` ∈ L acts on R<sub>u</sub>P by conjugation and on R<sub>u</sub>P<sup>0</sup> by conjugation with ϕ(`). This group acts onGfrom the left by

(u⁰, u, `) :g7→u<sup>0</sup>ϕ(`)g`⁻¹u⁻¹. We call thisthe action of Z on G.

Definition 3.2. A morphism between two algebraic zip data (G, P, P⁰, L, L⁰, ϕ) and ( ˜G,P ,˜ P˜⁰,L,˜ L˜⁰,ϕ) is a homomorphism˜ f: G → G˜ such that f(P) ⊂ P˜, f(P⁰)⊂P˜⁰,f(L)⊂L,˜ f(L⁰)⊂L˜⁰ and the diagram

L ^f //

ϕ

L˜

˜

ϕ

L^{0 f} //L˜⁰ commutes.

The composition of two morphisms of algebraic zip data is the obvious one, and in this way we obtain the category of algebraic zip data.

A different choice of Levi componentLofP would differ from the given one only by conjugation by an element ofRuP, and, for anyu∈ RuP, the orbits of the action of the algebraic zip datum (G, P, P⁰,^uL, L⁰, ϕ◦int(u⁻¹)) are the same as the orbits of the action of (G, P, P⁰, L, L⁰, ϕ). The same is true for a different choice forL⁰, hence the orbit structure of the action of an algebraic zip datum only depends on the isogenyP/RuP →P⁰/RuP⁰ induced byϕ.

There exists a maximal torus ofGcontained inP∩P⁰and Levi components of P and P⁰ containing this torus. Since we are only interested in the orbit structure, we can take L and L⁰ to be these Levi components. So we will assume from now on thatL∩L⁰ contains a maximal torus ofG.

We pick a maximal torusTofGcontained inL∩L⁰. LetN be the normalizer ofT andW =N/T the Weyl group ofG with respect toT. We denote by Φ the root system ofGwith respect toT. For any α∈Φ, we denote byU_α the associated root subgroup ofG. For all w∈W, we fix a representative ˙win N.

The algebraic groupP⁰×P acts onGfrom the left by (p⁰, p) :x7→p⁰xp⁻¹.

Lemma 3.3. For anyn∈N, the Bruhat cellP⁰nP is a locally closed subvariety ofGthat is invariant under the action of Z.

Proof. Since P⁰nP is an orbit under the action of P⁰×P, it is locally closed (see [7], Lemma 2.3.3). The second part of the claim follows directly from the definition of the action ofZ.

We first show how to relate the orbits inP⁰nP to the orbits in the reductive groupL⁰ under the action of another algebraic zip datum.

Construction 3.4. For each n ∈ N, we construct a new zip datum Z_n as follows: Let

Q:=L∩ⁿ⁻¹P⁰, Q⁰:=L⁰∩ⁿP, M :=L∩ⁿ⁻¹L⁰, M⁰:=L⁰∩ⁿL.

By Lemma 2.2, Q and Q⁰ are parabolic subgroups of the reductive groups L and L⁰, and M and M⁰ are Levi factors of Q and Q⁰. The group ϕ(Q) is a parabolic subgroup of L⁰ andϕ(M) is a Levi factor of ϕ(Q). Hence, if we set

˜

ϕ:=ϕ◦int(n⁻¹) :M⁰→ϕ(M), we obtain an algebraic zip datum Z_n := (L⁰, Q⁰, ϕ(Q), M⁰, ϕ(M),ϕ).˜

IfH andH⁰ are two algebraic groups acting on varieties X andX⁰ respec- tively we say that a morphism f: X → X⁰ is equivariant with respect to a homomorphismg:H→H⁰ if for all x∈X andh∈H

f(h·x) =g(h)·f(x).

In this case,f induces a map from the orbits inX to the orbits inX⁰. By direct calculation, the morphism

in:L⁰→P⁰nP

`<sup>0</sup>7→`⁰n is equivariant with respect to the homomorphism

(RuQ⁰× Ruϕ(Q))oM⁰→(RuP× RuP⁰)oL (v, v⁰, m⁰)7→(ⁿ⁻¹v, v⁰,ⁿ

−1

m⁰).

We will show thatin induces a bijection between the orbits of the actionZnon L⁰ and the orbits of the action ofZ onP⁰nP.

The stabilizer ofnis

StabP⁰×P(n) =

(p⁰, p)∈P⁰×P |p⁰np⁻¹=n

=

(p⁰, n⁻¹p⁰n)|p⁰∈P⁰∩ⁿP . (1) This implies in particular

Lemma 3.5. The dimension ofP⁰nP isdimP+ dimP⁰−dim(P⁰∩ⁿP).

LetHnbe the image of StabP⁰×P(n) under the projectionP⁰×P →L⁰×L.

Lemma 3.6. Hn ={(v⁰m⁰, vn⁻¹m⁰n)|m⁰∈M⁰, v⁰∈ RuQ⁰, v∈ RuQ}.

Proof. By definition,

Hn={(`<sup>0</sup>, `)∈L⁰×L| ∃u⁰∈ RuP⁰, u∈ RuP:n⁻¹`<sup>0</sup>u<sup>0</sup>n=`u}.

Let (`<sup>0</sup>, `)∈H_n. Chooseu⁰∈ RuP⁰ andu∈ RuP such thatn⁻¹`<sup>0</sup>u<sup>0</sup>n=`u.

Let ˜P⁰ = (P⁰∩ⁿP)RuP⁰ and ˜P = (ⁿ

−1

P⁰∩P)RuP. By Lemma 2.1 these are parabolic subgroups ofG. From the Levi decomposition P =LnRuP it follows that

L∩P˜ =L∩ⁿ

−1

P⁰=Q and analogously we find

L⁰∩P˜⁰=Q⁰.

Since`<sup>0</sup>u<sup>0</sup> ∈P<sup>0</sup>∩<sup>n</sup>P this implies`⁰∈Q⁰, so we can write`⁰=v⁰m⁰ for uniquely determinedm⁰∈M⁰ andv⁰ ∈ RuQ⁰, and we have

`= (n⁻¹v⁰n)(n⁻¹m⁰n)(n⁻¹u⁰n)u⁻¹.

From u⁰ ∈ R_uP⁰∩(P⁰∩ⁿP) we getn⁻¹u⁰n ∈ ⁿ⁻¹R_uP⁰∩P. Lemma 2.2 impliesv⁰ ∈ R_uQ⁰=L⁰∩ R_u(ⁿP), so we getn⁻¹v⁰n∈ R_uP. Also,

n⁻¹m⁰n∈ⁿ⁻¹M⁰=M ⊂L.

This implies

`∈n⁻¹m⁰n(ⁿ

−1

R_uP⁰∩P)R_uP.

By Lemma 2.1, we have (ⁿ⁻¹RuP⁰∩P)RuP =RuP˜. Hence

`∈n⁻¹m⁰n(R_uP˜∩L).

Finally, again by Lemma 2.2, we haveRuP˜∩L=Ru( ˜P∩L) =RuQ, so we can write`=vn⁻¹m⁰nfor somev∈ RuQ.

On the other hand, let (`<sup>0</sup>, `) = (v⁰m⁰, vn⁻¹m⁰n) for some m⁰ ∈ M⁰, v⁰ ∈ R_uQ⁰ andv∈ R_uQ=L∩ⁿ⁻¹R_uP⁰. From

n⁻¹`<sup>0−1</sup>n`= (n⁻¹m⁰⁻¹n)(n⁻¹v⁰⁻¹n)v(n⁻¹m⁰n)∈(ⁿ

−1

L⁰∩ RuP)(L∩ⁿ

−1

RuP⁰)

⊂ⁿ⁻¹RuP⁰· RuP

it follows that there existu∈ RuP and u⁰ ∈ RuP⁰ such that n⁻¹`<sup>0</sup>u<sup>0</sup>n =`u.

This shows (`<sup>0</sup>, `)∈H_n.

Lemma 3.7. The morphism i_n induces a bijection between the orbits of the action ofZ_n on L⁰ and the orbits of the action of Z on P⁰nP.

Proof. Any g ∈ P⁰nP can be written as u⁰`<sup>0</sup>n`u for certain u ∈ RuP, u⁰ ∈ RuP⁰, `∈L and `⁰ ∈L⁰. Since such ag lies in the same orbit as ϕ(`)`⁰n, the map induced byin on the orbits is surjective.

Let`<sup>0</sup>, `⁰⁰∈L⁰ such that in(`<sup>0</sup>) and in(`⁰⁰) lie in the same orbit. Then there existu∈ RuP, u⁰∈ RuP⁰ and`∈Lsuch that

`<sup>0</sup>n=u<sup>0</sup>ϕ(`)`<sup>00</sup>n`⁻¹u⁻¹. This implies

(`<sup>0−1</sup>ϕ(`)`<sup>00</sup>, `)∈H_n.

By Lemma 3.6 the existm⁰∈M⁰, v∈ RuQand v⁰ ∈ RuQ⁰ such that

`<sup>0−1</sup>ϕ(`)`⁰⁰=v⁰m⁰

`=vn⁻¹m⁰n.

Together, this yields

`⁰ =ϕ(v)ϕ(ⁿ

−1

m⁰)`<sup>00</sup>m<sup>0−1</sup>v<sup>−1</sup>, so`⁰ and`⁰⁰lie in the same orbit under the action ofZn.

Since Lemma 3.7 relates the action of an algebraic zip datum on an algebraic group with the action of another zip datum on a group of smaller dimension, it will allow us to prove facts about such actions inductively. The base case of such an induction will always be the case where the above construction does not yield a smaller group, that is the case whereG=L⁰.

The following two lemmas will be needed later.

Lemma 3.8. The morphism

π: P⁰×P →P⁰nP (p⁰, p)7→p⁰np

is separable. In particular the differential ofπat any point is surjective.

Proof. The claim is equivalent to the fact that the multiplication mapP⁰×ⁿP → P⁰·ⁿP is separable. To prove this, it is sufficient to show that

Lie(P⁰∩ⁿP) = Lie(P⁰)∩Lie(ⁿP),

where Lie denotes the Lie algebra functor. SinceP⁰ and ⁿP both contain T, both sides of the equation are the direct sum of Lie(T) and the Lie(U_α) for all α∈Φ such thatU_α⊂P⁰∩ⁿP.

Sinceπis aP⁰×P-equivariant morphism of homogenousP⁰×P-spaces, the last point follows from [7], Theorem 4.3.7(ii).

For any varietyX and any pointxonX, we denote the tangent space ofX atxbyTxX. IfY andZ are non-singular subvarieties of a non-singular variety X, we say thatY andZ intersect transversally in X if for anyx∈Y ∩Z, the spaceTxX is spanned by the subspacesTxY andTxZ.

The varietiesL⁰n, P⁰nP and any orbit ofZ in Gare non-singular, because they are orbits of certain group actions (see [7], Theorem 4.3.7(i)).

Lemma 3.9. Any orbito ofZ inP⁰nP intersects L⁰n transversally inP⁰nP. Proof. Let x∈ o∩L⁰n. We need to show thatT_x(P⁰nP) is the sum of T_x(o) andT_x(L⁰n). Writex=`<sup>0</sup>nfor some`⁰ ∈L⁰. InP⁰×P we considerX =L⁰×1 and

Y ={(u⁰ϕ(`)`⁰, u`)|`∈L, u∈ RuP, u⁰∈ RuP⁰},

which map ontoL⁰n andorespectively under π:P⁰×P →P⁰nP. By Lemma 3.8, the differential of π at (`<sup>0</sup>,1) is surjective, so it is sufficient to show that T<sub>(`</sub>⁰_,1)X andT_{(`</sub><sup>0</sup><sub>,1)</sub>Y spanT<sub>(`}⁰_,1)(P⁰×P)∼=T`<sup>0</sup>P<sup>0</sup>×T1P. SinceX=L<sup>0</sup>`⁰×1 and (RuP⁰)`<sup>0</sup>×1⊂Y, the sum of the two vector spaces containsT`⁰P⁰×1. Since this sum also contains 1×T1(RuP), it suffices to show that the differential at (`<sup>0</sup>,1) of the projectionY →L,(u<sup>0</sup>ϕ(`)`<sup>0</sup>, u`)7→`is surjective. But this follows from the existence of the sectionL→Y, `7→(ϕ(`)`⁰, `).

4 Coset Representatives in Coxeter Groups

We collect some facts about Coxeter groups which we shall need in the sequel.

These can be found in [2], sections 2.3 and 2.7.

Let (W, S) be a finite Coxeter group, i.e. W is a group with a set of generators S={s1, . . . , s_n} such thatW has a presentation

W =< s₁. . . , s_n |(s_is_j)^mij = 1>

for certainm_ij ∈Nsuch thatm_ii = 1 for all 1≤i≤nand m_ij =m_ji for all 1≤i6=j≤n. Let`denote the length function onW, i.e. forw∈W the length

`(w) ofwis the smallest numberr such thatwis the product of relements of S. An expression ofwas the product of`(w) elements ofS is called reduced.

Let I be a subset of S. We denote by WI the subgroup of W generated by I and by W^I (respectively ^IW) the set of elements w of W which have minimal length in their cosetwWI (respectivelyWIw). Then everyw∈W can be written uniquely asw=w^I ·wI = ˜wI·^Iw withwI,w˜I ∈WI,w^I ∈W^I and

Iw∈^IW, and`(w) =`(wI) +`(w<sup>I</sup>) =`( ˜wI) +`(^Iw) (see [2], Proposition 2.3.3).

In particular,W^I and^IW are systems of representatives forW/WI andWI\W respectively. The fact that`(w) =`(w⁻¹) for allw∈W impliesW^I = (^IW)⁻¹. Furthermore, if J is a second subset ofS, let ^JW^I be set ofw∈W which have minimal length in the double cosetW_JwW_I. Then^JW^I =^JW ∩W^I and

JW^I is a system of representatives forW_J\W/WI (see [2], Proposition 2.7.3).

Theorem 4.1(Kilmoyer, [2], Theorem 2.7.4). If w∈^JW^I, then WJ∩^wWI =WK,

whereK=J∩^wI.

Proposition 4.2 (Howlett, [2], Proposition 2.7.5). Let ^Jw^I ∈^JW^I and K = J∩^wI ⊂S. Then every elementw of the double coset WJJw^IWI is uniquely expressible in the form w = w_J^Jw^Iw_I where w_J ∈ W_J ∩W^K and w_I ∈ W_I. Moreover, this decomposition satisfies

`(wJJ

w^IwI) =`(wJ) +`(^Jw^I) +`(wI).

Lemma 4.3. The setW^I is the set of elementswfor which in the decomposition of Proposition 4.2 the factorwI is1.

Proof. Let ^Jw^I ∈ ^JW^I and K = J ∩ ^JwII. If wJ ∈ WJ ∩W^K then for any wI ∈ WI the decomposition wJJw^IwI is the unique decomposition given by Proposition 4.2. So we get

`(wJJw<sup>I</sup>wI) =`(wJ) +`(<sup>J</sup>w<sup>I</sup>) +`(wI)≥`(wJ) +`(^Jw^I) =`(wJJw^I), which shows thatwJJw^I ∈W^I.

Now let w ∈ W^I and w = wJJw^IwI the decomposition from Proposition 4.2. Since by the preceding paragraphwJJw^I ∈W^I we must havewI = 1.

OnW there exists a natural partial order, the Bruhat order, which we shall denote by≤. It 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=si₁· · ·si_n as a product of simple reflections , one gets a reduced expression forx by removing certain si_j from this product. More information about the Bruhat order can be found in [1], Chapter 2.

Using this order, the set W^I can be described as W^I ={w∈W |w < wsfor alls∈I}

(see [1], Definition 2.4.2 and Corollary 2.4.5).

Assume that in addition 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⁻. ForI ⊂S, let ΦI

be the root system spanned by the basis elements corresponding toI, and let Φ^±_I := ΦI ∩Φ^±. Then

W^I ={w∈W |wΦ⁺_I ⊂Φ⁺} (2) (see [2], Proposition 2.3.3).

Also, forw∈W, the length ofwis the cardinality of the set

{α∈Φ⁺|wα∈Φ⁻} (3) (see [2], Proposition 2.2.7).

5 The Orbits of an Algebraic Zip Datum

Definition 5.1. A zip datum Z= (G, P, P⁰, L, L⁰, ϕ) isnice with respect to a maximal torusT ⊂L∩L⁰ and a Borel subgroupB ofG, ifϕ(T) =T,

T ⊂B⊂P∩P⁰ and

ϕ(L∩B) =L⁰∩B.

If there exist suchT andB, we shall also just say thatZ is nice.

Proposition 5.2. Let Z= (G, P, P⁰, L, L⁰, ϕ)be an algebraic zip datum. Then there exists a nice algebraic zip datum Z˜ = (G,P ,˜ P˜⁰,L,˜ L˜⁰,ϕ)˜ and an isomor- phism of varietes ψ:G→Gwhich maps each orbit of Z in Gbijectively onto an orbit ofZ˜ inG.

SuchZ˜ andψcan be obtained as follows: There exist a Borel subgroupB of G,w∈W andz∈L such that

B⊂P,

˙

wϕ(^z(L∩B)) =^w˙L⁰∩B and

˙

wϕ(^zT) =T.

For any suchB,wandz the algebraic zip datum

Z˜= (G, P,^w˙P⁰, L,^w˙L⁰,int( ˙w)◦ϕ◦int(z))

is nice with respect to T and B and the morphism ψ: G → G, g 7→ wgz˙ is equivariant with respect to the isomorphism

(RuP× RuP⁰)oL→(RuP× Ru(^w˙P⁰))oL (u, u⁰, `)7→(<sup>z1</sup>u,<sup>w˙</sup>u<sup>0</sup>,<sup>z−1</sup>`).

Proof. LetT be a maximal torus ofGcontained inL∩L⁰. There existsw∈W such thatP and^w˙P⁰ both contain a Borel subgroup B ofGwhich containsT. Since int( ˙w)◦ϕ: L→^w˙L⁰is an isogeny and^w˙L⁰∩Bis a Borel subgroup of^w˙L⁰, its preimageB⁰ := (int(w)◦ϕ)⁻¹(^w˙L⁰∩B) is a Borel subgroup ofL. Similarly, the subgroupT⁰:= (int(w)◦ϕ)⁻¹(T) is a maximal torus ofL.

Since the action ofLby inner automorphisms on pairs consisting of a Borel subgroup and a maximal torus contained in the Borel subgroup is transitive, there existsz∈Lsuch that^z(L∩B) =B⁰ and^zT =T⁰, that is such that

˙

wϕ(^z(L∩B)) =^w˙L⁰∩B

and

˙

wϕ(^zT) =T.

Hence the algebraic zip datum ˜Z := (G, P,^w˙P⁰, L,^w˙L⁰,int( ˙w)◦ϕ◦int(z)) is nice with respect toB andT.

Letψ:G→Gbe the isormorphism of varieties sendingg ∈Gto ˙wgz. For (u, u⁰, `)∈(RuP× RuP⁰)oLandg∈G

ψ(u⁰ϕ(`)g`⁻¹u⁻¹) =^w˙u⁰(int( ˙w)◦ϕ◦int(z))(^z−1`)ψ(g)(<sup>z−1</sup>`)(^z

−1

u⁻¹).

This shows thatψis equivariant with respect to the isomorphism (RuP× RuP⁰)oL→(RuP× Ru(^w˙P⁰))oL

(u, u⁰`)7→(<sup>z1</sup>u,<sup>w˙</sup>u<sup>0</sup>,<sup>z−1</sup>`),

which implies that ψ maps each orbit of Z in G bijectively to an orbit of ˜Z inG.

This allows us to reduce to the case of a nice algebraic zip datum for many questions, so we will only consider such data in the following.

So let Z = (G, P, P⁰, L, L⁰, ϕ) be nice with respect to T and B. The Borel subgroup B defines a set of simple reflections S of W. Let I and J be the subsets of S such that P and P⁰ are the standard parabolics of type I and J respectively. Then W(L) = W_I and W(L⁰) = W_J. Since ϕ(T) = T, the isogenyϕinduces an isomorphismW_I →W_J which we also denote byϕ. Since ϕ(L∩B) =ϕ(L⁰∩B), this isomorphism mapsIbijectively toJ, and so it is an isomorphism of Coxeter groups (W_I, I)→(W_J, J).

We denote by Φ be the root system of G with respect to T, and for any closed subgroupH ofG which is normalized byT, we denote by ΦH the root system of H. Also, we denote by Φ⁺ the set of positive roots with respect to B, by Φ⁻ the set of negative roots, and we let Φ^±_H:= ΦH∩Φ^±.

Definition 5.3. Forw∈W, letK^w:=J∩^wI.

Lemma 5.4. Let ^Jw^I ∈ ^JW^I. Let ZJw˙^I = (L⁰, Q⁰, ϕ(Q), M⁰, M,ϕ)˜ be the algebraic zip datum from 3.4. Then:

(a) The algebraic zip datumZJw˙^I is nice with respect toT andL⁰∩B.

(b) The type ofQ⁰ isK^JwI.

(c) The elements w∈WJJw^IWI ∩W^I are exactly the elements of the form w=wJJw^I for somewJ ∈WJ having minimal length inwJW(M⁰). For suchw, wJ and ^Jw^I we have`(w) =`(wJ) +`(^Jw^I).

Proof. First we show:

Claim. (i) L∩^(Jw˙I)−1B=L∩B.

(ii) L⁰∩^Jw˙IB=L⁰∩B.

Proof. By assumption on ^Jw^I we have ^Jw^IΦ⁺_L ⊂Φ⁺. This implies ^Jw^IΦ⁺_L =

Jw^IΦL∩Φ⁺or equivalently Φ⁺_L = ΦL∩(^Jw^I)⁻¹Φ⁺. Since Φ⁺_L is the root system ofL∩B and ΦL∩(^Jw^I)⁻¹Φ⁺ is the root system of L∩^(Jw˙I)−1B, we get (i).

The second part can be shown similarly.

Now (i) showsL⁰∩B =ϕ(L∩B) =ϕ(L∩^(Jw˙I)−1B)⊂ϕ(Q) and from (ii) we getL⁰∩B=L⁰∩ ^Jw˙IB ⊂Q⁰.

Using (i) again we get

˜

ϕ(M⁰∩B) =ϕ(^(Jw˙I)−1B∩M)⊂ϕ(M)∩B.

Because both ˜ϕ(M⁰∩B) andϕ(M)∩Bare Borel subgroups ofϕ(M), they must be equal. Since ˜ϕ(T) =T this shows (a).

Theorem 4.1 impliesW^KJwI =W(M⁰), which shows (b). Then (c) is just a restatement of Lemma 4.3.

Definition 5.5. Let w ∈ W. For any collection of subsets of I which are mapped into themselves under the homomorphism int(w⁻¹)◦ϕ:WI →W, the union of these sets is also mapped into itself under int(w⁻¹)◦ϕ. Hence there exists a unique maximal subset ofI having this property, which we denote by I_w.

Lemma 5.6. Forw∈W, the map int(w⁻¹)◦ϕ:I_w→I_w is a bijection.

Proof. Since int(w⁻¹) :W →W andϕ: W_I →W_J are bijective, the composite int(w⁻¹)◦ϕis injective. SinceI_w is finite, this implies the claim.

We give an inductive description ofIw forw∈W^I:

Lemma 5.7. Let w∈W^I andw=wJJw^I the decomposition from Lemma 5.4 and ZJw˙^I = (L⁰, Q⁰, ϕ(Q), M⁰, M,ϕ)˜ be the algebraic zip datum from 3.4. Let K_w^JwI

j be the largest subset ofK^JwI invariant underint(w⁻¹_J )◦ϕ. Then˜ K_w^JwI

j = ^JwII_w. Proof. The definition of K_w^JwI

J implies

(^Jw^I)⁻¹

[(int(wJ)⁻¹◦ϕ◦int(^Jw^I)⁻¹)(K_w^Jw_j^I)]⊂⁽

Jw^I)⁻¹

[K_w^Jw_j^I].

This shows that the subset⁽

Jw^I)⁻¹

K_w^Jw_j^I ofIis invariant under int(w⁻¹)◦ϕ, so that

(^Jw^I)⁻¹

K_w^Jw_j^I ⊂Iw.

Lemma 5.6 implies

Jw^II_w=^w−1J ϕ(I_w) =^w

−1

J ϕ(˜ ^JwII_w),

hence ^JwIIwis contained inK^JwI =J∩^JwII and invariant under int(w⁻¹_J )◦ϕ.˜ This shows ^JwIIw⊂K_w^Jw_j^I.

For any set of simple reflectionsR, there exists a unique parabolic subgroup of typeRofGcontainingB, which is called thestandard parabolic subgroup of typeR. This pararabolic subgroup has a unique Levi factor containingT, which is called thestandard Levi subgroup of typeR.

Definition 5.8. For w ∈ W, let L_w be the standard Levi subgroup of G of typeI_w.

Lemma 5.9. Forw ∈W, the morphism int( ˙w⁻¹)◦ϕ:L→ Gmaps Lw into itself.

Proof. Since the group Lw is generated by T and the Uα for α ∈ ΦI_w, it is sufficient to show that these subgroups are mapped intoLw by int( ˙w⁻¹)◦ϕ.

ForT this is clear and for theUα it follows from the definition ofIw. We give an inductive description ofLwforw∈W^I:

Lemma 5.10. Let w ∈W^I and w =wJJw^I the decomposition from Lemma 5.4 and ZJw˙^I = (L⁰, Q⁰, ϕ(Q), M⁰, M,ϕ)˜ be the algebraic zip datum from 3.4.

LetM_w⁰_J be the standard Levi subgroup ofQ⁰ of typeK_w^Jw_J^I. Then M_w⁰_J = ^Jw˙ILw.

Proof. This follows directly from Lemma 5.7.

Remark 5.11. Using the inductive description ofL_win the preceding Lemma, one can show that L_w is the unique maximal subgroup ofL which is mapped into itself by int( ˙w⁻¹)◦ϕ:L→G. But we shall not need this.

For anyX ⊂G, we denote the union of the orbits of all elements of X by o(X).

Lemma 5.12. If X is a constructible subset of G, theno(X) is constructible.

Proof. This follows from the fact that o(X) is the image of the constructible subset ((RuP× RuP⁰)oL)×X of ((RuP× RuP⁰)oL)×Gunder the mult- plication morphism toG.

Definition 5.13. Forw∈W^I, letG^w˙ := o( ˙wLw).

Lemma 5.14. For w ∈ W^I, the Z-stable piece G^w˙ does not depend on the choice of representativew.˙

Proof. By definition the group LwcontainsT. Hence ˙wLwdoes not depend on the choice of ˙w, which implies the claim.

This justifies the following definition:

Definition 5.15. Forw∈W^I, letG^w=G^w˙.

Remark 5.16. A priori theG^wforw∈W^I are only constructible subsets ofG.

However we shall see later (Corollary 6.15) that they are in fact locally closed inG.

Definition 5.17. Forw∈W^I letjw˙ be the continuous map Lw→G^w

`7→w`˙

Lemma 5.18. Let w ∈W^I and w =wJJw^I the decomposition from Lemma 5.4. Let ZJw˙^I = (L⁰, Q⁰, ϕ(Q), M⁰, M,ϕ)˜ be the algebraic zip datum from 3.4.

Assumew˙ = ˙wJJ

˙ w^I. (i) We have M_w⁰_˙

J = ^Jw˙IL_w˙ and the diagram L_w ^jw˙ //P⁰wP˙

M_w⁰_˙J

int(^Jw˙^I)⁻¹

OO

j_wJ˙

//L⁰

iJwI˙

OO

commutes.

(ii) The morphismiJw˙^I maps(L⁰)^wJ intoG^wand induces a bijection between the ZJw˙^I-orbits in(L⁰)^wJ and theZ-orbits inG^w. In particular

(L⁰)^wJJw˙^I =G^w∩L^0Jw˙^I.

Proof. The first part of (i) follows from Lemma 5.10, and the second statement in (i) can be directly verified. Then (ii) follows from (i), the definition of (L⁰)^wJ andG^w and Lemma 3.7.

Theorem 5.19. (i) The G^w for w ∈ W^I form a disjoint decomposition of G.

(ii) For all w ∈ W^I, the continuous map j_w˙:L_w → G^w induces a bijection between the orbits on L_w of the action ofL_w on itself by

(`, g)7→(^w˙

−1

ϕ(`))g`⁻¹ and the orbits ofZ inG^w.

Proof. We prove everything by induction on dimG. IfL⁰=G, we haveW^I = {1}

andL=L1=L⁰. Since in this case the action of Z is just the action ofLon itself given in (ii), both claims are true.

Assume dimL⁰ <dimG. For ^Jw^I ∈ ^JW^I it follows from Lemmas 3.7, 5.4 and 5.18 and the induction hypothesis applied to the nice algebraic zip datum ZJw˙^I thatP^0Jw˙^IP is the disjoint union of theG^w for thew∈W^I of the form w=w_J^Jw^I withw_J∈W_J. SinceG=`

Jw˙^I∈^JW^IP^0Jw˙^IP this proves (i).

The induction step for (ii) follows from Lemma 5.18.

6 Closure

In this section, we will show that forw∈W^I, the closure ofG^w is the union of otherZ-stable piecesG^w0 for certainw⁰∈W^I.

We shall need the following lemma (see [8, Lemma 7.3]):

Lemma 6.1. Let G be a connected linear algebraic group and ϕ a surjective endomorphism ofGwhich leaves a Borel subgroupB of Ginvariant. Then the morphismG×B →G,(x, b)7→ϕ(x)bx⁻¹ is surjective.

Proposition 6.2. Forw∈W^I, we have G^w= o(BwB).˙

Proof. We proceed by induction on dimG. In the base case we haveG =L⁰, and the claim follows from Lemma 6.1.

Assume dimL⁰ <dimGand let b, b⁰ ∈ B. We can decompose b and b⁰ as b =`u and b<sup>0</sup> = u<sup>0</sup>`⁰ with ` ∈B∩L, `⁰ ∈ B∩L⁰, u ∈ RuP and u⁰ ∈ RuP⁰. Since o(b⁰wb) = o(ϕ(`)`˙ ⁰w) with˙ ϕ(`)`⁰∈L⁰∩B we get

o(BwB) = o((B˙ ∩L⁰) ˙w).

Hence it will be sufficient to prove thatG^w= o((B∩L⁰) ˙w).

Let w = wJJw^I ∈ ^JW^I be the decomposition given by Lemma 5.4. The morphismiJw˙^I:L⁰→P⁰wP˙ maps (L⁰∩B) ˙w_J onto (L⁰∩B) ˙w. Since by Lemma 5.4 the zip datumZJw˙^I is nice with respect to T and B∩L⁰, the claim now follows from the induction hypothesis applied toZJw^I and Lemma 3.7.

Corollary 6.3. IfP is a Borel subgroup ofG, then G^w=BwBfor allw∈W. The group B×B acts onGby (b⁰, b)·g=b⁰gb⁻¹. The set{w˙ |w∈W}is a set of representatives for this action, and forw, w⁰ ∈W

w≤w⁰ if and only ifBwB˙ ⊂Bw˙⁰B.

Proposition 6.2 will allow us to use the closure order of this action to determine the closure order of the action ofZ. For this, we shall need the following lemma (see [9, 5.3]):

Lemma 6.4. LetH be any algebraic group acting on a varietyZ and letP ⊂H be an algebraic subgroup such that H/P is proper. Then for any P-invariant subvarietyY ⊂Z we have

H·Y =H·Y .

Lemma 6.5. Forw∈W^I, we have G^w= [

x∈W x≤w

o(Bx).˙

Proof. We letLact onGby`∗g=ϕ(`)g`⁻¹. SinceBwB˙ =R_uP⁰(BwB)R˙ _uP, it follows from Proposition 6.2 that

G^w=L∗BwB.˙

This together with Lemma 6.4, applied to H = L, P = L∩B, Z = G and Y =BwB˙ yields

G^w=L∗BwB˙ = [

x≤w

L∗BxB.˙

Because again for each suchxwe have RuP⁰(BxB)R˙ uP =BxB, this implies˙ the lemma.

Lemma 6.6. For all x, z ∈W and b∈B there exists v ∈W such thatv ≤z and

˙

zbx˙ ∈Bv˙xB.˙

Proof. We prove the statement by induction on `(z). If z = 1, we may take v = 1. For the induction step write z =sz<sup>0</sup> for some simple reflection ssuch that`(z⁰) =`(z)−1. By the induction hypothesis there existsv⁰ ≤z⁰ such that

˙

z⁰bx˙ ∈Bv˙⁰xB. Hence ˙˙ zbw˙ ∈sB˙ v˙⁰xB˙ ⊂Bs˙v˙⁰xB˙ ∪Bv˙⁰xB, so either˙ v=sv⁰ or v=v⁰ will have the required property.

Lemma 6.7. Let w ∈ W^I, b, b⁰ ∈ B and x ∈ W such that o(bw) = o(b˙ ⁰x).˙ Then there existsy∈WI such that ϕ(y)wy⁻¹≤x.

Proof. We proceed by induction on dimG. In the base case we haveG=L, so wmust be 1 and we may takey= 1.

So assume that dimL⁰ < dimG. We also may and do assume that b, b⁰ ∈ L∩B. Letx=xJJx^IxI be the decomposition ofxgiven by Proposition 4.2, so we havexJ ∈WJ, xI ∈ WI and ^Jx^I ∈^JW^I. It follows from Lemma 4.3 that x^I :=xJJx^I ∈W^I.

By Lemma 6.6, there existsv ∈ W such that v ≤ϕ(xI) and ϕ( ˙xI)b⁰w˙^I ∈ Bv˙x˙^IB. Hence there exists b⁰⁰ ∈L⁰∩B such that ϕ( ˙x_I)b⁰w˙^I lies in the same orbit asb⁰⁰v˙x˙^I. Altogether we get

o(bw) = o(b˙ ⁰x˙^Ix˙_I) = o(ϕ( ˙x_I)b⁰x˙^I) = o(b⁰⁰v˙x˙^I) = o(b⁰⁰v˙x˙_J^Jx˙^I).

Let ZJx˙^I = (L⁰, Q⁰, ϕ(Q), M⁰, ϕ(M),ϕ).˜ By Lemma 5.4 we can decompose w = wJJw^I with wJ ∈ WJ minimal in wJW(M⁰) and ^Jw^I ∈ ^JW^I. From o(bw) = o(b˙ ⁰x) we get˙

PJJw˙^IPI =PJbwP˙ I =PJb⁰xP˙ I =PJJx˙^IPI,

which implies ^Jx^I = ^Jw^I.

Now Lemma 5.4 implies that bw˙J and b⁰⁰v˙x˙J lie in the same orbit under the action of ZJx˙^I on L⁰. Hence, by the induction hypothesis, there exists a y⁰∈W(M⁰) such that

˜

ϕ(y⁰)wJy⁰⁻¹≤vxJ.

Since both sides lie inWJ and since ^Jx^I ∈^JW, this implies z:= ˜ϕ(y⁰)w_Jy^0−1Jx^I ≤vx_J^Jx^I =vx^I.

Since ˜ϕ=ϕ◦int((^Jw˙^I)⁻¹), if we let ˜y= (^Jw^I)⁻¹y^0Jw^I ∈W_I we can writezas ϕ(˜y)w˜y⁻¹.

Because of the Bruhat relation z ≤ vx^I we can write z = v⁰x⁰ for certain v⁰, x⁰∈W withv⁰≤vandx⁰≤x^I. Sinceϕ(I) =J, we getϕ⁻¹(v⁰)≤ϕ⁻¹(v)≤ xI, where ϕ⁻¹ is the inverse of the isomorphism ϕ: WI → WJ induced by ϕ.

Sincex^I ∈W^I, we get

v⁰⁻¹zϕ⁻¹(v⁰) =x⁰ϕ⁻¹(v⁰)≤x^Iϕ⁻¹(v)≤x^IxI =x.

Soy:=ϕ⁻¹(v⁰)⁻¹y˜has the required property.

For w ∈ W, the set Tw˙ is independent of the choice of representative ˙w.

This justifies the following definition:

Definition 6.8. Forw∈W^I let ˜G^w:= o(Tw)˙ ⊂G^w.

We shall show that ˜G^w is dense in G^w. The crucial case is the following lemma. The proof we give is a slight modification of the proof of Lemma 6.1 given in [8].

Lemma 6.9. LetGbe a reductive algebraic group andϕa surjective endomor- phism ofG leaving invariant a Borel subgroup B of Gand a maximal torus T ofB. Then the morphismα:G×T →G,(g, t)7→ϕ(g)tg⁻¹ has dense image.

Proof. Equivalently we may show that for some t0∈T, the image of the mor- phism ˜α:G×T →G,(g, t)7→t0ϕ(g)t⁻¹₀ t⁻¹g⁻¹ is dense. It will be enough to show that the differential of ˜α at 1 is surjective. This differential is the linear map

L(G)×L(T)→L(G)

(X, Y)7→T+L(int(t0)◦ϕ)(X)−X.

This linear map has image

Lie(T) + (L(int(t₀)◦ϕ)−1) Lie(G).

Letϕt₀ = int(t0)◦ϕ. LetB⁻ be the Borel subgroup opposite toBwith respect to T. Since ϕ(B) = B, the differential of ϕt₀ at 1 preserves L(RuB) and L(RuB⁻). If we find at0such thatL(ϕt₀) has no fixed points onL(RuB) and L(RuB⁻) we will be done.

For each α∈Φ, let xα be a basis vector of L(Uα). The isogeny ϕinduces a bijection ˜ϕ: Φ → Φ such that ϕ(Uα) = U_ϕ(α). For each α ∈ Φ there ex- ists a c(α) ∈ k such that L(ϕ)(xα) = c(α)xϕ(α)˜ . This implies L(ϕt₀)(xα) = α(t0)c(α)xϕ(α)˜ . Sinceϕt0 fixes RuB and RuB⁻, its differential permutes Φ⁺ and Φ⁻. HenceL(ϕt0) can only have a fixed point inL(RuB) orL(RuB⁻) if there exists a cycle (α₁,· · · , α_n) of the permutation ˜ϕin Φ⁺ or Φ⁻ such that

n

Y

i=1

α_i(t₀)c(α_i) = 1.

This shows that for t₀ in some non-empty open subset of T, the differential L(ϕ_t0) has no fixed points onL(R_uB) andL(R_uB⁻).

Lemma 6.10. For each w∈W^I, the setG˜^w is dense in G^w.

Proof. By Theorem 5.19, the continuous map jw˙:Lw˙ → G^w gives a bijection between the orbits inLw˙ under the action ofL⁰_w˙ on itself by twisted conjugation and the orbits ofZ in G^w. By Lemma 6.9, the orbit ofT is dense inLw˙. Since jw˙ is continous we get

L⁰_w˙w˙ =jw˙(o(T))⊂jw˙(o(T))⊂G˜^w. Since ˜G^w isZ-invariant, this implies ˜G^w=G^w.

Definition 6.11. Forwandw⁰in W^I we let w4w⁰ if and only if there exists y∈WI such thatϕ(y)wy⁻¹≤w⁰.

Theorem 6.12. Forw∈W^I

G^w= a

w⁰∈W^I w⁰4w

G^w0.

Proof. First, considerw⁰ ∈W^I such thatG^w0 intersectsG^w. Then by Proposi- tion 6.2 and Lemma 6.5 there exist b, b⁰ ∈B and x∈W such thatx≤wand o(bw⁰) = o(b⁰x). Hence Lemma 6.7 impliesw⁰4wand this shows ”⊂”.

For ”⊃” letw⁰ ∈W^I withw⁰4w. By definition there existsy∈W^I such thatϕ(y)w⁰y⁻¹≤w. Since by Lemma 6.10 the orbit ofTw˙⁰ is dense inG^w˙0, in order to showG^w0 ⊂G^w it is sufficient to showTw˙⁰⊂G^w. Fort∈T

o(tw˙⁰) = o(ϕ( ˙y)tw˙⁰y˙⁻¹) = o((ϕ( ˙y)tϕ( ˙y⁻¹))(ϕ( ˙y) ˙w⁰y˙⁻¹)).

Hence, since ϕ( ˙y)tϕ( ˙y⁻¹) ∈ T and ϕ(y)w⁰y⁻¹ ≤ w, Lemma 6.5 shows tw˙⁰ ∈ G^w.

Remark 6.13. This theorem was motivated by a similar result of Wedhorn in a different setting in [9]. Also, the proof we give here was inspired by Wedhorn’s proof in [9].