automorphisms Tensor invariants, saturation problems, and Dynkin Advances in Mathematics

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Advances in Mathematics

www.elsevier.com/locate/aim

Tensor invariants, saturation problems, and Dynkin automorphisms

Jiuzu Hong^a,∗, Linhui Shen^b

aDepartmentofMathematics,YaleUniversity,NewHaven,CT 06511, United States

bDepartmentofMathematics,NorthwesternUniversity,Evanston,IL 60208, United States

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received30April2014

Receivedinrevisedform27June 2015

Accepted18August2015 Availableonline31August2015 CommunicatedbyRoman Bezrukavnikov

Keywords:

Tensormultiplicity Twiningformula Saturationproblems

GeometricSatakecorrespondence AffineGrassmannian

Satakebasis Tropical points

LetG beaconnected almostsimple algebraicgroupwitha Dynkinautomorphismσ.LetGσbetheconnectedalmostsim- plealgebraicgroupassociatedwithGandσ.Weprovethat thedimension of thetensor invariant spaceof Gσ isequal tothetraceofσonthecorrespondingtensorinvariantspace of G.Weprove that if G has thesaturationproperty then sodoesGσ.Asa consequence,weshowthat thespingroup Spin(2n+1) hassaturationfactor 2,whichstrengthensthere- sultsofBelkale–Kumar[1]andSam[28]inthecaseoftypeB_n.

© 2015ElsevierInc.All rights reserved.

Contents

1. Introduction . . . . 630

1.1. Mainresults . . . . 630

1.1.1. Twiningformula . . . . 630

1.1.2. Saturationproperty . . . . 631

* Correspondingauthor.

E-mailaddresses:[email protected](J. Hong),[email protected](L. Shen).

http://dx.doi.org/10.1016/j.aim.2015.08.015 0001-8708/© 2015ElsevierInc.All rights reserved.

1.2. Mainmethods . . . . 632

1.3. Strategies . . . . 633

1.4. Otherapplications . . . . 633

2. Basicsofreductivegroups . . . . 634

2.1. DynkinautomorphismsofG . . . . 634

2.2. TheassociatedgroupGσ. . . . 635

2.3. ThefixedpointgroupG^σ . . . . 636

3. Configurationspaceofdecoratedflagsanditstropicalization . . . . 637

3.1. Positivespacesandtheirtropicalpoints . . . . 637

3.2. Lusztig’spositiveatlasofU_∗ . . . . 639

3.3. Configurationspaceofdecoratedflags . . . . 642

3.4. TheautomorphismσofConf^×_n(A) . . . . 644

3.5. TheembeddingıfromConf^×_n(A^σ) toConf^×_n(A) . . . . 645

3.6. Summationoftropicalpoints . . . . 647

4. AffineGrassmannianandSatakebasis . . . . 649

4.1. Topcomponentsofcyclicconvolutionvariety . . . . 649

4.2. Parameterization oftopcomponents . . . . 649

4.3. GeometricSatakecorrespondenceandSatakebasis . . . . 651

5. Proofofmainresults . . . . 654

5.1. ProofofTheorem 1.1 . . . . 654

5.2. ProofofTheorem 1.2 . . . . 655

Acknowledgments . . . . 656

References . . . . 656

1. Introduction

LetGbe aconnectedalmostsimplealgebraicgroupwithaDynkinautomorphismσ.

Onecanassociatewithit anotheralmostsimplealgebraicgroupGσ(seeSection2.2).We investigatetherelationbetweenthetensorinvariantspacesofGandG_σ inthis paper.

In fact we can identify the dominant weights of G_σ and the σ-invariant dominant weights ofG.Letλ= (λ1,. . . ,λn) beasequenceofdominantweightsofGσ.Denote by Vλi (respectivelyWλi)the irreducible representation of G(respectivelyGσ) of highest weightλ_i.Weareinterestedinthepairoftensorinvariantspaces

V_λ^G:= (Vλ1⊗. . .⊗Vλn)^G, W_λ^Gσ := (Wλ1⊗. . .⊗Wλn)^Gσ. (1) 1.1. Mainresults

Wepresent twomainresultsrelatingV_λ^G andW_λ^Gσ. 1.1.1. Twiningformula

LetλbeadominantweightofGσ.TheDynkinautomorphismσuniquelydetermines anactionσontherepresentationV_λofGbykeepingthehighestweightvectorsinvariant.

Let μ be aweightofG_σ. Thetwining characterformula assertsthatthetraceof σ on theweightspaceVλ(μ) isequaltothedimensionofWλ(μ).

The twining character formula is originally due to Jantzen [11]. Since then, there hasbeen manydifferentproofs appearingintheliterature (e.g.[6,26,27,20,10]).Oneof theseapproachesusesnaturalbases oftherepresentationsthatarecompatiblewiththe actionσ. Itwas achieved viacanonical basisin[20], andviaMV cyclesin[10]. Dueto theworksofLusztig[22],Berenstein–Zelevinsky[2]andKamnitzer[13],canonicalbasis and MV cycles can be parameterized by many different but equivalent combinatorial objects, i.e., Lusztig’s data, BZ patterns,and MVpolytopes. These parameterizations arecruciallyused intheproofsof[20]and [10].

Thefirst resultof thepresent paper provides ananalogueof thetwining formulain thesettingoftensorinvariantspaces.NotethattheDynkinautomorphismσdetermines anactionσonV_λ^G.Ourfirstmaintheorem isasfollows.

Theorem1.1. Thetrace of σon thespace V_λ^G isequaltothedimensionof W_λ^Gσ: trace (σ:V_λ^G→V_λ^G) = dimW_λ^Gσ. (2) Theorem 1.1isprovedinSection5.1.WeremarkherethatTheorem 1.1impliessimilar twiningformulasformoregeneralmultiplicityspaces.

1.1.2. Saturation property

WesaythatareductivegroupGhassaturationfactork if

• for any dominant weights λ1,λ2,· · ·,λm such thatm

i=1λi is intheroot lattice of G, if(V_{N λ1}⊗V_{N λ2}⊗ · · · ⊗V_{N λm})^G = 0 forsome positiveintegerN, then(V_kλ1⊗ V_kλ2⊗ · · · ⊗V_kλn)^G= 0.

Kapovich–Millson[16]provedthateveryalmostsimplegrouphassaturationproperty but with a wild factor. There is a general saturation conjecture asserting that every simply-laced grouphassaturationfactor 1 [15]. When G= SLn,it wasfirst provedby Knutson–Tao[17]usinghoneycombs.AdifferentproofwasduetoDerksen–Weyman[3].

WhenG= Spin(8), itwasprovedbyKapovich–Kumar–Millson[14]. Itis stillopen for simply-lacedgroupsofothertypes.Foramorethoroughsurveyonsaturationproblems, see[19,Section 8].

Thesecond resultof thispaper shows thatthesaturationproperty of Gimplies the saturationpropertyofGσ.

Theorem1.2. IfG hassaturation factork,then Gσ has saturationfactorcσk,where

cσ =

⎧⎨

⎩

2 if Gis not of typeA2n andσ is of order2, 3 if σis of order3,

4 if σis of order2andGis of type A_2n.

(3)

Theorem 1.2isprovedinSection5.2.

Non-simply-lacedgroupsareexpectedtohavesaturationfactor2.Forsuchgroupsnot oftypeG2,ifweassumethesaturationconjectureofsimply-lacedgroups,thenitfollows fromTheorem 1.2.Inparticular,theworksofKnutson–TaoandDerksen–Weymanimply that

Corollary 1.3.The spingroup Spin(2n+ 1) has saturationfactor 2.

Proof. By Example 2.1 in Section 2.2, if G = SL_2n and σ is nontrivial, then G_σ = Spin(2n+ 1). Theorem 1.2impliesthatSpin(2n+ 1) hassaturationfactor 2. 2

Theideathatthesaturationpropertyofabiggroupimpliesthesaturationpropertyof theintimatelyrelatedsmallgroupwasalsoadoptedbyBelkale–Kumar[1],inwhichthey showed thatKnutson–Tao’s theorem impliesthatthe saturationfactors of SO(2n+ 1) and Sp(2n) are2.However,thetechniquesusedbythem areverydifferentfromours.

1.2. Mainmethods

The main methods of this paper are the geometric Satake correspondence [21,7,25]

and thework of Goncharov–Shen[8] on parameterizations ofbases of tensor invariant spaces.

Let G^∨ be the Langlands dual groupof G. Let K:= C((t)) and letO :=C[[t]]. We consider theaffineGrassmannian oftheLanglandsdualgroup

Gr_G∨ :=G^∨(K)/G^∨(O).

The geometric Satake correspondence provides a connection between the geometry of the affine Grassmannianof G^∨ and the representation theory of G. As aconsequence, the top componentsof certain cyclic convolution variety ofG^∨ provides abasis of the correspondingtensorinvariantspaceofG(Lemma 4.6).FollowingFontaine–Kamnitzer–

Kuperberg [5], wecallittheSatakebasisofG.

Another main tool iscertain tropical pointsintroduced byGoncharov–Shen[8]. The tropical points are obtained via the tropicalization of the configuration space of deco- ratedflagsofG.WecallthesetropicalpointsG-laminations,whosedefinitionisrecalled in Sections 3.2–3.5. One of the main results of Goncharov–Shen is that there exists a canonical bijection between G-laminations and the Satake basis of G^∨ (Theorem 4.2).

When G= PGL₂, theG-laminationsare exactlythe integrallaminationsonapolygon [4,Section 12]. WhenG= GL_n, theG-laminationsencapsulatethehives [8,Section 3].

Inthis sense theresultofGoncharov–Shengeneralizes theworkofKamnitzer[12] that hivesparameterize theSatake basesoftensorinvariantspacesofGLn.

WeemphasizethatTheorem 1.2isprovedbyessentiallyusingcombinatorialproperties of G-laminations. The geometric Satake correspondence is only used to establish that thecardinalityoftheset ofG^∨-laminationsequals thetensorproductmultiplicity.

1.3. Strategies

Letλbeatupleof dominantweightsof Gσ. DenotebyBλ,G theSatakebasisof V_λ^G DenotebyC_λ,G∨ thesetoftheG^∨-laminationsthatparameterizeBλ,G.DenotebyBλ,Gσ

andC_λ,(Gσ)∨ thecorresponding setsforthesmallgroupG_σ.

TheSatake basisBλ,G hasremarkable properties. One ofthem is that,there exists aDynkin automorphism σ of Gsuch that theinduced action on V_λ^G interchanges the elements intheSatake basisBλ,G (Proposition 4.8). Thusthetraceof σon V_λ^G equals thenumberofσ-invariantelementsinBλ,G.

TheDynkinautomorphismσofGgivesrisetoaDynkinautomorphismσ^∨ofG^∨.The latterinducesanautomorphismσ^∨onthesetofG^∨-laminations(seeSection3.4).The- orem 4.4andProposition 4.8 assertthattheσ^∨-actiononG^∨-laminationsiscompatible withtheσ-actionontheSatakebasisofG,i.e.,thefollowingdiagram commutes:

Cλ,G^∨

σ^∨

Bλ,G

σ

C_λ,G∨

Bλ,G

(4)

Thereforetheσ^∨-invariantG^∨-laminationsareinbijectionwiththeσ-invariantelements intheSatake basisofG.

Theorem 3.25 is one of the main technical results for proving Theorem 1.1. It as- serts that the σ^∨-invariant G^∨-laminations are in one-to-one correspondence with the (G_σ)^∨-laminations,i.e., thereexistsacanonicalbijection

(Cλ,G^∨)^σ∨ C_λ,(Gσ)∨. (5)

Combining(4)and(5),theσ-invariantelementsinBλ,Gareinbijectionwiththeelements inBλ,Gσ.Theorem 1.1followsasadirectconsequence.

Theorem 3.29providesasummationmap

Σ : Cλ,G^∨ −→(Ccσ·λ,G^∨)^σ∨ C_{cσ·λ,(Gσ)}∨, (6) where cσ is thenumberappearing inTheorem 1.2. Ifthe set Cλ,G^∨ is nonempty, then thesetC_{cσ·λ,(Gσ)}∨ isalsononempty.InthiswayweproveTheorem 1.2(seeSection5.2).

1.4. Other applications

AlongourproofsofTheorems 1.1,1.2,wegetseveral interestingnumericalresultsof representationtheoryrelatedto GandG_σ.

Proposition1.4. With thesame settingasin Theorem 1.1andTheorem 1.2,wehave

1. dimV_λ^G≥dimW_λ^Gσ.

2. If dimV_λ^G= 1,thendimW_λ^Gσ = 1.

3. If dimV_λ^G= 0,thendimW_c^Gσ

σ·λ= 0.

Proof. ThefirstandthesecondresultsfollowfromTheorem 3.25.Thethirdresultfollows from Theorem 3.29. 2

Recall aconjecture byW. Fultonassertingthatforatripleλ= (λ1,λ2,λ3) ofdomi- nantweightsofGL_n,ifdimV_λ^GLn= 1,thendimV_{N λ}^GLn= 1 forallN ∈N.Theconjecture was proved by Knutson–Tao–Woodward [18] using honeycomb models. Combining it with Proposition 1.4(2),wegetthefollowingresult.

Proposition 1.5.Letλ= (λ1,λ2,λ3)beatripleofσ-invariantdominantweights ofSLn. If dimV_λ^SLn= 1,thenforallN ∈Nwehave

dim(W_{N λ})^Spin(n+1)= 1 if nis even

dim(W_{N λ})^Sp(n−1)= 1 if nis odd , (7) where Sp(n−1)isthesymplectic group.

2. Basicsofreductive groups

Let G be aconnected almost simple groupwith a Dynkinautomorphism σ. Inthis section, weintroducetwodifferentgroupsGσ and G^σ related toG.

2.1. Dynkinautomorphismsof G

LetGbeaconnectedalmostsimplealgebraicgroupoverC.LetT beamaximaltorus inGand letB beaBorelsubgroup containingT.Denote byX^∨ and X thelatticesof cocharactersandcharactersofT.Weassociatearootdatum(X^∨,X,α^∨_i,αi,i∈I) with (G,B,T) together withaperfectpairing

, :X^∨×X→Z.

HereIistheindexsetofsimplecoroots{α^∨_i}andsimpleroots{α_i}.WehavetheCartan matrix(a_ij):=

α^∨_i,α_j .

Adiagramautomorphismσoftherootdatum(X^∨,X,α^∨_i,αi,i∈I) consistsofauto- morphismsofX^∨ andofX, andapermutationofI(withoutconfusion,allofthemare denotedbyσ)suchthat

1. σ(λ^∨),σ(μ) =λ^∨,μ forany λ^∨∈X^∨ andμ∈X.

2. σ(αi)=α_σ(i) andσ(α^∨_i)=α^∨_σ(i).

Letxi :C→Gand yi :C→Gbe root subgroups associatedwith the simple roots αi and −αi. Thedatum(T,B,xi,yi;i∈I) is called apinning of Gifit givesriseto a homomorphismγi: SL2→Gforeachi∈I suchthat

γi( 1 a 0 1

) =xi(a), γi( 1 0 a 1

) =yi(a), γi( a 0 0 a⁻¹

) =α_i^∨(a). (8) LetσbeanautomorphismofGthatpreservesB andT.Itinduces adiagramautomor- phismoftheroot datum(X^∨,X,αi,α^∨_i;i∈I),whichisstilldenotedbyσ.Wecall σa Dynkinautomorphism ofGifitpreservesapinning ofG,i.e.,

σ xi(a)

=x_σ(i)(a), σ yi(a)

=y_σ(i)(a), σ α^∨_i(a)

=α_σ(i)^∨ (a), ∀i∈I.

Bytheisomorphismtheorem ofthetheory ofreductivegroups(e.g. [29,Section 9]), everydiagram automorphismarisesfrom aDynkinautomorphism ofG.

2.2. The associatedgroupG_σ

Every diagram automorphism σ of aroot datum(X^∨,X,α^∨_i,αi,i ∈I) gives riseto thefollowingdatum:

1. LetX_σ bethelatticeofσ-fixedelements inX.LetX_σ^∨:= Hom_Z(X_σ,Z).

2. LetIσ bethesetof orbitsofσonI.Foreachelementη∈Iσ,weset α_η:= ^i∈ηαi ifaij = 0 for any two elementsi, j in η

2

i∈ηαi ifη ={i, j}and aij =−1.

Note thatitcoversallpossiblecasesofη.

3. TheembeddingofX_σintoX inducesanaturalmapθ:X^∨→X_σ^∨.Letα^∨_η :=θ(α^∨_i) withi inη.Clearlyα^∨_η doesnotdependonthechoiceofi.

By[11,p. 29], (X_σ^∨,Xσ,α^∨_η,αη,η∈Iσ) isarootdatum.Itdeterminesareductivegroup Gσ.IfGissimply-connected,thensoisGσ.HereisatableofGandGσfornontrivialσ [24,6.4]:

1. IfG=A_2n−1 andσisoforder 2,thenG_σ=B_n,n≥2.

2. IfG=A_2n and σisoforder2,thenG_σ=C_n,n≥1.

3. IfG=D_n andσisoforder2,thenG_σ=C_n−1,n≥4.

4. IfG=D₄ andσis oforder3,thenG_σ=G₂. 5. IfG=E6 andσisoforder 2,thenGσ=F4.

Example2.1.LetG= SL2n.Weconsidertheautomorphism σ:G−→G, g−→σ(g) :=w·(g^t)⁻¹·w⁻¹,

where g^tisthetranspositionofg andw= (wij) isamatrixwithentries wij :=

(−1)^j ifi+j= 2n+ 1

0 otherwise .

TheautomorphismσisaDynkinautomorphismonG.Inthiscase,Gσ= Spin(2n+ 1).

2.3. Thefixed pointgroupG^σ

Let usfix apinning(T,B,x_i,y_i;i∈I) of G. Letσ beaDynkin automorphismof G thatpreservesthepinning.LetG^σ betheidentitycomponentoftheσ-fixedpointsofG.

LetT^σ andB^σbetheidentitycomponentsoftheσ-fixedpointsofT andB respectively.

Recall theset I_σ oforbitsofσonI.Foreachorbitη∈I_σ,therearetwo cases:

1. Ifaij = 0 foranyi,j∈η, thenweset x_η(a) :=

i∈η

x_i(a), y_η(a) :=

i∈η

y_i(a), α^∨_η :=

i∈η

α^∨_i.

2. Ifη={i,j}anda_ij =−1,thenweset xη(a) :=xi(a)xj(2a)xi(a), yη(a) :=yi(a

2)yj(a)yi(a

2), α^∨_η := 2(α^∨_i +α^∨_j).

Note thatthedefinitionofxη and yη doesnotdependontheordering ofelementsinη.

Lemma 2.2.The datum(T^σ,B^σ,x_η,y_η;η∈I_σ)gives apinning ofG^σ.

Proof. Thefirstcaseisclear.Thesecond caseisduetoacomputationofSL3. 2 Remark 2.3. Let G^∨ be the Langlands dual group of G. By considering the diagram automorphismσonthedualrootdatumof(X^∨,X,α^∨_i,α_i;i∈I),wegeta Dynkinauto- morphismσ^∨ofG^∨.Let(G^∨)^σ∨ bethe identitycomponentoftheσ^∨-fixedpointsofG^∨. Note thatthe cocharactersof (G^∨)^σ∨ are identified with theσ^∨-invariantcocharacters of G^∨.So(G^∨)^σ∨ istheLanglandsdual groupofG_σ (see[20]).

Weyl groupsof GandG^σ

Let s_i (i ∈ I) be the simple reflections generating the Weyl group W of G. Set si :=yi(1)xi(−1)yi(1). Theelements si satisfythe braid relations.So wecanassociate with eachw∈W itsrepresentativewinsuchawaythatforanyreduceddecomposition w=s_i1· · ·s_ik onehasw=s_i1· · ·s_ik. Letw₀ bethelongestelement oftheWeylgroup.

SetsG :=w²₀. NotethatsG isacentralelementinG. Moreovers²_G= 1.

TheWeylgroupW^σ ofG^σ canbenaturallyembeddedintoW with generators s_η = ^i∈ηsi, ifaij= 0, ∀i, j∈η;

s_is_js_i, ifη={i, j}, a_ij =−1. (9) Thelongest element w0 of W coincides with the longestelement of W^σ. Westate the followingwell-knownfactforfutureuse.

Lemma 2.4. Each reduced decomposition sη1· · ·sηm of w0 in W^σ determines areduced decompositionofw0inW withsηi expressedby(9),oncewefixanorderingofelements ineach η.

Example2.5.Ifthepair(G,G^σ) isofCartan–Killingtype(A4,B2),then w0=sη1sη2sη1sη2 =s1s4·s2s3s2·s1s4·s2s3s2.

Weset ˆs_η :=y_η(1)x_η(−1)y_η(1) for η ∈I_σ. There is another representative s_η of s_η obtainedbyits decompositioninW. Adirectcalculation showsthatsˆ_η =h_ηs_η,where hη =α^∨_i(2)α^∨_j(2) ifη ={i,j},aij =−1,andhη = 1 otherwise.We associatewithw0 a representativewˆ0 viaareduceddecompositionofw0 inW^σ.Then

ˆ

w0=h·w0, whereh:=

1 ifG=A_2n n

k=1(α^∨_k(2)α^∨_2n+1−k(2))^k ifG=A_2n. (10) NotethatsG=sG^σ := ˆw²₀.

3. Configurationspace ofdecoratedflagsanditstropicalization

In this section, let us assume that G is defined over Q. Let us fix a pinning (T,B,xi,yi;i ∈ I) of G. Let σ be a Dynkin automorphism of G that preserves the pinning.

3.1. Positivespaces andtheir tropicalpoints

Below we briefly introduce the category of positive spaces and the tropicalization functor.

Positivespaces

ApositiverationalfunctiononasplitalgebraictorusT isanonzerorationalfunction onT whichinacoordinatesystem,given byaset ofcharacters ofT,canbepresented as aratio of two polynomials with positive integralcoefficients. Denote by Q+(T) the setofpositivefunctionsonT.

Apositivestructureonanirreduciblespace(i.e.,variety/stack)Yisabirationalmap γfromT toY.Arationalfunctionf onY iscalledpositiveiff◦γ∈Q+(T).Denoteby

Q+(Y) theset ofpositive functionsonY.TwopositivestructuresonY areequivalentif theydeterminethesamesetQ+(Y).Suchapair(Y,Q+(Y)) iscalledapositivespace.

Let(Y,Q+(Y)) and(Z,Q+(Z)) beapairofpositivespaces.Arationalmapφ:Y → Z is calledapositivemapiff◦φ∈Q+(Y) forallf ∈Q+(Z).

Tropicalization Let

Y,Q+(Y)

be apositive space. A tropical pointof Y is amap l : Q+(Y)→Z suchthat

∀f, g∈Q+(Y), l(f +g) = min{l(f), l(g)}, l(f g) =l(f) +l(g).

Denote byY(Z^t) theset oftropical pointsofY.Tautologically, eachf ∈Q+(Y) deter- minesaZ-valuedfunctionf^t ofY(Z^t) suchthatf^t(l):=l(f).

Thefollowing lemma isaneasyexercise.

Lemma 3.1. Letφ:Y → Z be apositivemap. There existsaunique mapφ^t:Y(Z^t)→ Z(Z^t),called thetropicalization ofφ,such that(f◦φ)^t=f^t◦φ^t forallf ∈Q+(Z).

The followinglemma isstandard. Itshows thatthetropicalizationis afunctorfrom thecategory ofpositivespacesto thecategoryofthesetsoftropicalpoints.

Lemma3.2.Letφ:X → Y andψ:Y → Z betwopositivemaps.Then(ψ◦φ)^t=ψ^t◦φ^t. Letf,g∈Q+(Y).Wesayf < gifg−f isstillapositivefunctiononY.

Lemma3.3.Letf,g∈Q+(Y).Ifthereexistsapositiveinteger N suchthatf < g < N f, then f^t=g^t.

Proof. Ifh:=g−f ∈Q+(Y),theng^t= min{h^t,f^t}≤f^t. Therefore(N f)^t≤g^t≤f^t. Note that(N f)^t=f^t.Thereforeg^t=f^t. 2

Example 3.4. Denote by X_∗(T) and X^∗(T) the lattices ofcocharacters and characters of asplitalgebraic torusT.There isaperfectpairing

, :X_∗(T)×X^∗(T)→Z. Each f ∈Q+(T) canbepresented as

f =

α∈X^∗(T)c_αX^α

α∈X^∗(T)dαX^α, c_α, d_α∈N={0,1,2,· · ·}.

Here X^α istheregularfunctiononT associatedwith αandc_α, d_α are zeroforallbut finitely manyα.Eachcocharacterl∈X_∗(T) determinesatropicalpointofT suchthat

l(f) := min

α|cα=0l, α − min

α|dα=0l, α .

ItiseasytoshowthatalltropicalpointsofT canbedefinedthisway.Thereforetheset T(Z^t) iscanonicallyidentifiedwithX_∗(T).Wetreatthemasthesamesetinthispaper.

Lemma3.5. If f ∈Q+(T)is aregularfunction¹ on T,thenf^t isconvex,i.e., f^t(l₁+l₂)≥f^t(l₁) +f^t(l₂), ∀l₁, l₂∈X_∗(T).

Proof. Thefunctionf isaLaurentpolynomialonT:

f =

α∈X^∗(T)

cαX^α, cα∈Z.

Itiseasytoshow thatf^t(l)= min_α|cα=0l,α .Theconvexityfollows. 2

Lemma 3.6. Let φ: T1 → T2 be apositive map betweentwo split algebraic tori. If φ is regular,then φ^t:X_∗(T1)→X_∗(T2)islinear.

Proof. Letuswrite themap φincoordinates:

φ:T1−→ T2, x:= (x1, . . . , xn)−→(φ1(x), . . . , φm(x)).

Ifφisaregularmap,theneveryφi(x) isinvertible.Thereforeφi(x) mustbe monomials ofx1,. . . ,xn withnontrivial coefficients.Soitstropicalizationislinear. 2

If thespace Y admits apositive structure defined by abirational map γ : T → Y, thenγ^tisabijectionfromT(Z^t) toY(Z^t).Forl∈ Y(Z^t),itspre-imageβ(l):= (γ^t)⁻¹(l) iscalled thecoordinateofl inT(Z^t).Note thatT(Z^t)=X_∗(T) isanabelian group,it inducesanextraoperation+γ onY(Z^t) suchthat

β(l+γl) =β(l) +β(l), l, l ∈ Y(Z^t). (11) 3.2. Lusztig’spositiveatlas of U_∗

LetU = [B,B] be the maximal unipotent subgroup insideB. Let B⁻ be the Borel subgroupsuchthatB∩B⁻ =T.LetU_∗ =U ∩B⁻w₀B⁻.

Letw₀=s_i1s_i2. . . s_iN beareduceddecompositioninW.Thesequencei= (i₁,. . . ,i_N) iscalledareducedwordforw0inW.There isanopenembedding

γi:G^Nm−→U_∗, (a1, . . . , aN)−→xi1(a1). . . xiN(aN). (12)

1 Forexample,f= ^1+X_1+X³ = 1−X+X²issuchafunctiononGm.

Thebirational mapγi definesapositivestructureofU_∗. Itisshownin[23] thatallthe reducedwordsforw0 giverisetotheequivalentpositivestructuresonU_∗,whichwecall Lusztig’spositiveatlas.

Note thattheDynkinautomorphismσ preservesB andB⁻.Soitpreserves U_∗. Lemma 3.7.The automorphismσ:U_∗−→U_∗ isapositivemap.

Proof. Leti= (i₁,. . . ,i_N) beareducedword forw₀. Thenσ(i)= (σ(i₁),. . . ,σ(i_N)) is also areducedwordforw₀. Foreachu=x_i1(a₁). . . x_iN(a_N)∈U_∗,wehave

σ(u) =x_σ(i1)(a1). . . x_σ(iN)(aN)∈U_∗.

Since thepositivestructuresgivenbyiandσ(i) areequivalent,thelemma follows. 2 Thetropicalization ofσisabijection

σ^t:U_∗(Z^t)−→^∼ U_∗(Z^t). (13) Denote by

U_∗(Z^t)σ

theset ofσ^t-fixed points.Below wegiveacharacterizationof the σ^t-fixedpoints.

Let j= (η₁,. . . ,η_n) beareducedword for w₀ inW^σ. Itdetermines areducedword i= (i₁,i₂,. . . ,i_N) forw₀ inW (Lemma 2.4).Thetropicalizationof(12)isabijection

γ_i^t:Z^N−→⁼ U_∗(Z^t).

Denote by(m₁,. . . ,m_N) thepre-imageofl∈U_∗(Z^t) inZ^N,whichiscalled thetropical coordinateofl providedbyγi.

Thefollowing lemmaisamanifestationofProposition 3.5in[10].

Lemma 3.8.A tropicalpointl isσ^t-invariantif andonly if

m1=m2=. . .=mrη1, mrη1+1=mrη1+2=. . .=mrη1+rη2, . . . , where r_η isthecardinality of theorbit η.

Proof. Firstwe provethe casewhen G isof type A2 and σ is of order 2.In this case, thesetI={1,2},andσ(1)= 2,σ(2)= 1.Soi= (1,2,1) isareducedwordofw₀ inW. Ifu=x₁(a)x₂(b)x₁(c),then

σ(u) =x₂(a)x₁(b)x₂(c) =x₁( bc

a+c)x₂(a+c)x₁( ab a+c).

Let(m1,m2,m3) bethecoordinateofl.Sothecoordinateofσ^t(l) is

(m2+m3−min{m1, m3},min{m1, m3}, m1+m2−min{m1, m3}). (14) Notethatl=σ^t(l) ifandonlyifm₁=m₂=m₃. Thelemma follows.

The generalcase canbe reduced to the abovecase and the casewhen Gis of type A1× · · · ×A1.Thelattercasefollowsbyasimilarbuteasierargument. 2

LetU^σ be theidentitycomponentoftheσ-fixedpointsof U. Thereducedwordj of w0inW^σ determinesapositivestructureof U_∗^σ:

γj:Gⁿm−→U_∗^σ, (a1, . . . , an)−→xη1(a1). . . xηn(an).

Lemma 3.9. The natural embedding ı:U_∗^σ → U_∗ isa positivemap. The tropicalization ofı identifiesthesetU_∗^σ(Z^t)with (U_∗(Z^t))^σ.

Proof. RecalltheconstructionofthepinningofG^σinSection2.3.Itprovidesanexplicit expressionofthemapıusingthecoordinatesprovidedbyγ_jandγ_i.Thenthepositivity ofıisclear.ThensecondpartfollowsdirectlyfromLemma 3.8. 2

The additiveWhittakercharacterχ

ThepinningofGdeterminesanadditivecharacterχofU suchthat

χ(xi(a)) =a, ∀i∈I; χ(u1u2) =χ(u1) +χ(u2), ∀u1, u2∈U. (15) Thefollowinglemma isclear.

Lemma3.10.TherestrictionofχonU_∗isapositivefunction.Thefunctionχisinvariant under theautomorphismσ,i.e., χ◦σ=χ.

Denote by χσ the additiveWhittaker character of U^σ such that χσ(xη(a))= a for η∈Iσ,andχσ(u1u2)=χσ(u1)+χσ(u2) foru1,u2∈U^σ.Therestrictionofχσ onU_∗^σ is apositivefunction.

Lemma3.11.Wehave χ^t_σ =χ^t◦ı^t.

Proof. Itiseasyto checkthatχ◦ı(xη(a))=κηχσ(xη(a)),where

κ_η=

⎧⎪

⎪⎨

⎪⎪

⎩

1 ifη={i}.

2 ifη={i, j}, andaij = 0.

3 ifη={i, j, k}, anda_ij =a_jk =a_ik= 0.

4 ifη={i, j}, anda_ij =−1.

Hence inany case, χσ ≤ χ◦ı ≤ 4χσ. By Lemma 3.3, χ^t_σ = (χ◦ı)^t. By Lemma 3.2, (χ◦ı)^t=χ^t◦ı^t.It concludestheproofofourlemma. 2

3.3. Configurationspaceof decoratedflags

LetA:=G/U.Theelements ofAarecalled decoratedflags.ThegroupGactsonA on the left. For each A∈ A, the stabilizer stabG(A) is amaximal unipotent subgroup ofG.LetπbethenaturalprojectionfromAtotheflag varietyBsuchthatπ(A) isthe Borel subgroup containing stab_G(A). It is easy to show thatfor each B ∈ B, its fiber π⁻¹(B) isaT-torsor.

Weconsider theconfigurationspace

Conf_n(A) :=G\Aⁿ. (16)

Wesayapair(B1,B2)∈ B² isgenericifB1∩B2 isamaximal torusofG.Weconsider thefollowing opensubspaceofConfn(A):

Conf^×_n(A) :={G\(A1, . . . , An)|(π(Ai), π(Ai+1)) is generic for eachi∈Z/n}. (17) Below we introduceapositivestructure onConf^×_n(A), following[4,Section 8].Wealso referthereadersto[8,Section 6]formoredetails.

Weconsider thespace

R:=G\{(B₁, A, B₂)|(π(A), B₁),(π(A), B₂) are generic} ⊂G\(A × B²). (18) Denote by R∗ theopen subspaceof Rwith requiringthe pair(B1,B2) is alsogeneric.

Abusing notation, denote by U the decorated flag corresponding to the coset of the identity inA.

Lemma 3.12. (See[4,Section 8].)Thereis an isomorphismed: Conf^×₂(A)−→^∼ T such that

(A₁, A₂) = (U,ed(A₁, A₂)w₀·U).

There isanisomorphism an:R−→^∼ U suchthat

(B1, A, B2) = (B⁻, U,an(B1, A, B2)·B⁻).

The restriction ofan onR∗ isan isomorphismfromR∗ toU_∗.

Lemma–Construction3.13.(See[4,Section 8].)There isanaturalopenembedding² p:Tⁿ⁻¹×U_∗ⁿ⁻²−→Conf^×_n(A), (h₂, . . . , h_n, u₂, . . . , u_n−1)−→(A₁, . . . , A_n) (19) such that

2 In fact, the images of p consist of configurations (A1,. . . ,An) ∈ Conf^×_n(A) such that the pairs (π(A1),π(Ai)) arealsogenericfori= 2,. . . ,n.

Fig. 1.The invariants assigned to Conf^×₅(A).

• hi=ed(Ai−1,Ai), i∈ {2,. . . ,n};

• uj =an(π(A1),Aj,π(Aj+1)), j∈ {2,. . . ,n−1}.

LetP_n beaconvexn-gon.Letusassign toeach vertexof P_n adecoratedflag A_i sothat A1,. . . ,An sit clockwisein thepolygon. Thenhi,uj are variablesassigned totheedges andanglesof Pn.SeeFig. 1.

The positive structure on U_∗ is defined via Lusztig’s atlas. Note that T is a split algebraicgroupandthereforeadmitsanaturalpositivestructure.SoTⁿ⁻¹×U_∗ⁿ⁻²admits apositivestructure.Fromnowon,wefixapositivestructureonConf^×_n(A)suchthatthe mappanditsinversep⁻¹ are both positivemaps.

Fockand Goncharov[4,Definition2.5]definedthetwistedcyclic shift map

r: Conf^×_n(A)−→Conf^×_n(A), (A₁, . . . , A_n)−→(s_G·A_n, A₁, . . . , A_n−1). (20) Theyshowedthat

Theorem 3.14. (See [4, Corollary 8.1].) The twisted cyclic shift map(20) is a positive map.

Corollary3.15.The followingmapisaregularpositivemap:

Ed: Conf^×_n(A)−→Tⁿ, (21)

(A1, . . . , An)−→

ed(sG·An, A1),ed(A1, A2), . . . ,ed(A_n−1, An) .

Proof. ThepositivityofthefirstfactorfollowsfromTheorem 3.14.Therestis clear. 2 RecalltheadditiveWhittakercharacterχofU inSection3.2.

Definition3.16.(See[8,Section2.1.4].)ThepotentialWisaregularfunctionofConf^×_n(A) suchthat

W(A1, . . . , An) :=

i∈Z/n

χ

an(π(A_i−1), Ai, π(Ai+1)) .

Corollary 3.17.The potentialW isapositive function.

Proof. Notethatan(π(A1),A2,π(A3)

isapartofthemap(19).ByLemma 3.10,χisa positivefunctiononU_∗.Sothesummandχ

an(π(A1),A2,π(A3)

isapositivefunction.

ThecentralelementsG is containedintheintersectionof Borelsubgroups. Therefore an(B1, sG·A, B2) =an(B1, A, B2).

Using Theorem 3.14,therestsummandsarepositivefunctions. 2 3.4. Theautomorphism σof Conf^×_n(A)

The automorphismσ ofG preservesU. Thusit descendsto anautomorphism ofA. Similarly, it descends to an automorphism of B. Recall the projectionπ from A to B. Clearly σcommuteswiththeprojectionπ(σ(A))=σ(π(A)) forA∈ A.

Abusingnotation,we considertheautomorphism

σ: Conf^×_n(A)−→^∼ Conf^×_n(A), (A1, . . . , An)−→(σ(A1), . . . , σ(An)). (22) Lemma 3.18.The mapσcommutes withtheinvariants inLemma 3.12:

ed(σ(A1), σ(A2)) =σ(ed(A1, A2)), (23) an(σ(B₁), σ(A), σ(B₂)) =σ(an(B₁, A, B₂)). (24) Proof. Let A1 = U and letA2 = ed(A1,A2)w0·U. Note thatσ preserves U and w0. Therefore σ(A1)=U and σ(A2)=σ(ed(A1,A2))w0·U.Thefirstidentity follows. The second identityfollows bythesameargument. 2

Lemma 3.19.The automorphism(22)isapositivemap.

Proof. Recallthebirationalmap pin(19).ByLemma 3.18,wehavetheisomorphism p⁻¹◦σ◦p: Tⁿ⁻¹×U_∗ⁿ⁻²−→^∼ Tⁿ⁻¹×U_∗ⁿ⁻², (25) (h2, . . . , hn, u2, . . . , u_n−1)−→(σ(h2), . . . , σ(hn), σ(u2), . . . , σ(u_n−1)).

ByLemma 3.7, itisapositivemap.Since pandp⁻¹ areboth positivemaps,theauto- morphism σispositive. 2