On a class of double cosets in reductive algebraic groups

0.37 MB·English

by  Jiang-Hua Lu

ON A CLASS OF DOUBLE COSETS IN REDUCTIVE ALGEBRAIC
GROUPS
5
0 JIANG-HUALUANDMILENYAKIMOV
0
2
Abstract. We study a class of double coset spaces RA\G1×G2/RC, where G1 and
n G2 areconnected reductive algebraicgroups, and RA andRC arecertainspherical sub-
a groupsofG1×G2obtainedby“identifying”LevifactorsofparabolicsubgroupsinG1and
J G2. Such double cosets naturally appear in the symplectic leaf decompositions of Pois-

0 sonhomogeneous spaces ofcomplexreductivegroupswiththe Belavin–DrinfeldPoisson
2 structures. TheyalsoappearinorbitdecompositionsoftheDeConcini–Procesicompact-
ificationsofsemi-simplegroupsofadjointtype. Wefindexplicitparametrizationsofthe

] doublecoset spaces and describethedouble cosets as homogeneous spaces of RA×RC.
T We further show that all such double cosets give rise to set-theoretical solutions to the
R quantum Yang–Baxter equationonunipotent algebraicgroups.
.
h
t
a
m 1. The setup
[
1.1. The setup. Let G1 and G2 be two connected reductive algebraic groups over an alge-
2 braically closed base field k. For i=1,2, we will fix a maximal torus H in G and a choice
i i
v ∆+ of positive roots in the set ∆ of all roots for G with respect to H . For each α ∈ ∆ ,
6 wei willuseUα to denotethe one-piarameterunipotenitsubgroupofG deiterminedbyα. Leit
0 i i
Γ be the setofsimple rootsin∆+. Fora subsetA ofΓ ,let P be the standardparabolic
0 i i i i Ai
0 subgroupof Gi containingthe BorelsubgroupBi =HiUi, where Ui = α∈∆+i Uiα. LetMAi
1 and U be respectively the Levi factor of P containing H and the unipotent radical of
4 P . TAhien P =M U and M and U Ainitersect trivialliy. DenoteQby M′ the derived
0 Ai Ai Ai Ai Ai Ai Ai
subgroups of the Levi factors M . We will also use ∆ to denote the set of roots in ∆
/ Ai Ai i
h that are in the linear span of A . Set ∆+ =∆ ∩∆+.
t i Ai Ai
a
Definition 1.1. Identify Γ and Γ with the Dynkin diagrams of G and G respectively.
m 1 2 1 2
By a partial isometry from Γ to Γ we mean an isometry from a subdiagram of Γ to a
1 2 1
:
v subdiagram of Γ2. We will denote by P(Γ1,Γ2) the set of all partial isometries from Γ1 to
i Γ . If a ∈ P(Γ ,Γ ), and if A and A are the domain and the range of a respectively,
X 2 1 2 1 2
we define a generalized a-graph to be an abstract (not necessarily algebraic) subgroup K of
r
a M ×M such that K∩(Uα×Ua(α))⊂Uα×Ua(α) is the graphof a group isomorphism
A1 A2 1 2 1 2
from Uα to Ua(α) for every α∈∆ .
1 2 A1
By an admissible pair for G ×G we mean a pair (a,K), where a∈P(Γ ,Γ ) and K is
1 2 1 2
a generalized a-graph. If A = (a,K) is an admissible pair, we define the subgroup R of
A
P ×P by
A1 A2
(1.1) R =K(U ×U ).
A A1 A2
1.2. An Example. Assume that G and G are connected and simply-connected. Fix a
1 2
partial isometry a ∈ P(Γ ,Γ ) with domain A and range A . Then the derived subgroups
1 2 1 2
M′ of the Levi factors M are connected and simply-connected [24, Corollary 5.4], and a
Ai Ai
can be lifted to a group isomorphism θ : M′ →M′ . Let
a A1 A2
Graph(θ )={(m ,θ (m ))|m ∈M′ }⊂M′ ×M′
a 1 a 1 1 A1 A1 A2
2000 Mathematics Subject Classification. Primary20G15; Secondary14M17,53D17.
1
2 JIANG-HUALUANDMILENYAKIMOV
bethegraphofθ ,andletZ bethecenterofM fori=1,2. Then,anyabstractsubgroup
a Ai Ai
X of Z ×Z gives rise to the group K = XGraph(θ ) which can be easily seen to be a
A1 A2 a
generalized a-graph for G ×G , and for the corresponding admissible pair A =(a,K), we
1 2
have
R =XGraph(θ )(U ×U ).
A a A1 A2
As a direct consequence of Lemma 3.2 in §3, one sees that, in the case when G and G
1 2
are connected and simply-connected, all admissible pairs A and groups R are of the above
A
type.
1.3. Main problem. Fix two admissible pairs A = (a,K) and C = (c,L). In this paper,
we obtain the solutions to the following problems:
1.) Find an explicit parametrization of all (R ,R ) double cosets in G ×G .
A C 1 2
2.) Describe explicitly each (R ,R ) double coset as a homogeneous space of R ×R .
A C A C
3.) Find a closed formula for the dimension of each (R ,R ) double coset.
A C
Remark 1.2. For an admissible pair A=(a,K), we will set
(1.2) R− =K(U ×U− ),
A A1 A2
where U− is the subgroup of G generated by the one-parameter subgroups Uα for α ∈
A2 2 2
−∆+ . It is easy to see that solutions to the problems in §1.3 lead to solutions to the
A2
corresponding problems for the (R−,R−) double cosets in G ×G .
A C 1 2
1.4. Motivation. LetGbeacomplexsemi-simpleLiegroupofadjointtype. Thewonderful
compactificationGofG,constructedbyDeConciniandProcesiin[4],isasmoothprojective
(G×G)-varietywith finitely manyG×Gorbits. It wasprovedin[4]thateveryG×Gorbit
onGisisomorphicto(G×G)/R− foranadmissiblepairC =(c,L)forG×G,wherecisthe
C
restriction of the identity automorphism of the Dynkin diagram of G to some subdiagram.
Let g be the Lie algebra of G. De Concini and Procesi further defined an embedding of the
wonderful compactification of G in the projective variety of all Lie subalgebras of g⊕g. Its
image is the closure of the G×G orbit of the diagonalsubalgebraof g⊕g which clearly lies
in the variety L ofall Lagrangiansubalgebrasof g⊕g. Here,a subalgebral of g⊕g is said
g
to be Lagrangianif diml=dimg and if l is isotropic with respect to the symmetric bilinear
form on g⊕g given by
hx +y ,x +y i=≪x ,x ≫−≪y ,y ≫, x ,y ,x ,y ∈g,
1 1 2 2 1 2 1 2 1 1 2 2
where ≪, ≫ denotes the Killing form of g. The variety L was recently studied in [12] in
g
connection with Drinfeld’s classification [7] of Poisson homogeneous spaces of Poisson Lie
groups. It was further shown in [12] that all G×G orbits in L (under the adjoint action)
g
are again of the type (G×G)/R− for some admissible pairs C = (c,L) for G×G, but this
C
time the partial isometries c can be arbitrary. Thus, solutions to the double coset problems
in §1.3 can be used to classify R− orbits in L for any admissible pair A.
A g
OurmotivationforstudyingR− orbitsinL comesfromthetheoryofPoissonLiegroups.
A g
In [1], Belavin and Drinfeld classified all quasi-triangular Poisson structures on G. Denote
such a Poisson structure on G by π . The Poisson Lie groups (G,π ) have received a
BD BD
lot of attention since the appearance of [1]. They were explicitly quantized by Etingof–
Schedler–Schiffmannin[10]andtheirsymplecticleaveswerestudiedbythesecondauthorin
[25]. Oneimportantproblemregardingthe PoissonLie groups(G,π )isthe studyoftheir
BD
Poisson homogeneous spaces [7]. The variety L of Lagrangian subalgebras of g⊕g serves
g
as a “master” variety of Poisson homogeneous spaces of (G,π ). Indeed, identify G with
BD
the diagonal subgroup G of G×G. It is shown in [11] that L carries a natural Poisson
∆ g
structure Π , such that every G orbit in L , when equipped with Π , is a Poisson
BD ∆ g BD
homogeneous space of (G,π ). To identify these homogeneous spaces of G, one needs to
BD
classify G orbits in L , and to study their symplectic leaf decompositions with respect to
∆ g
DOUBLE COSETS IN REDUCTIVE ALGEBRAIC GROUPS 3
Π ,oneneedstoclassifyN(G∗ )orbitsinL ,whereG∗ isthedualPoissonLiegroupof
BD BD g BD
(G,π )insideG×G,andN(G∗ )isthenormalizersubgroupofG∗ inG×G. BothG∗
BD BD BD BD
and N(G∗ ) are of the type R−, where the partialisometry a in A=(a,K) is the Belavin-
BD A
Drinfeld triple defined in [1]. In the forthcoming paper [19], we will show that intersections
of G and N(G∗ ) orbits in L are all regular Poissonsubvarieties of (L ,Π ). Solutions
∆ BD g g BD
totheproblemsin§1.3willbeusedin[19]toclassifysuchorbitsandtocomputetheranksof
the PoissonstructuresΠ . Basedon the results of this paper,in [19] we will also treatthe
BD
moregeneralclassofPoissonstructuresonL ,obtainedfromarbitrarylagrangiansplittings
g
of g⊕g. The latter were classified by Delorme in [5].
For the so-called standard Poisson structure on G, we have N(G∗ ) = B ×B−, where
BD
(B,B−) is a pair of opposite Borel subgroups of G. In [23], Springer classified the B×B−
orbits in the wonderful compactification G of G and studied the intersection cohomologyof
the B×B− orbit closures. A classification of G orbits in G was given by Lusztig in [21].
∆
It was used by He [15] to study the closure of the unipotent variety in G. Our Theorem 2.2
generalizesbothSpringer’sandLusztig’sresults. We alsopointoutthatintersectionsofG
∆
andB×B− orbits in the closedG×G orbits in L are closelyrelatedto the double Bruhat
g
cells in G and intersections of dual Schubert cells on the flag variety of G (see [2, 12, 13]).
Itwouldbe veryinterestingto understandintersectionsofarbitraryG andN(G∗ )orbits
∆ BD
in L in the framework of the theory of cluster algebras of Fomin and Zelevinsky [14], and
g
to find toric charts on the symplectic leaves of (L ,Π ) using the methods of Kogan and
g BD
Zelevinsky [18].
Our approachto the double cosetspaceR \G ×G /R reliesonaninductive argument
A 1 2 C
that relates an (R ,R ) double coset in G ×G to cosets of similar type in the product
A C 1 2
M ×M of Levi subgroups M ⊂ G and M ⊂ G . Such an argument first appeared in
1 2 1 1 2 2
the work of the second author [25] on the symplectic leaf decomposition of the Poisson Lie
group (G,π ). It dealt with the case G = G = G and A = C. Recently the first author
BD 1 2
and Evens [12] extended the methods of [25] to the case G = G = G and R = G . The
1 2 A ∆
advantage of the approach in this paper to the one in [25] is that in this paper we find an
explicitrelationbetweenthedoublecosetsinG ×G andthedoublecosetsinM ×M ,while
1 2 1 2
[25] deals with various complicatedZariskiopen subsets of the double cosets in G ×G . In
1 2
particular,theiterationoftheinductiveprocedurein[25]isonlyhelpfulforstudyingZariski
open subsets of the symplectic leaves of π , while we expect that the results of this paper
BD
will be useful in understanding globally all the symplectic leaves of (L ,Π ).
g BD
Therootsofthemethodsusedinthispapercanbetracedbacktotheinductiveprocedure
ofDuflo[8]andMoeglin–Rentschler[22]fordescribingtheprimitivespectrumoftheuniversal
enveloping algebra of a Lie algebra which is in general neither solvable nor semi-simple.
The reason for this relation is that the Poisson structures Π on L vanish at many
BD g
points and the corresponding linearizations give rise to Lie algebras that are rarely semi-
simple or solvable. Due to the Kirillov–Kostant orbit method, one expects a close relation
betweentheprimitivespectrumofthequantizedalgebrasofcoordinateringsofvariousaffine
subvarietiesofL (whichareinfactdeformationsofthe universalenvelopingalgebrasofthe
g
linearizations)and the symplectic leaves of such affine Poissonsubvarieties of L . From this
g
point of view, our method can be considered as a nonlinear quasiclassical analog of [8, 22].
1.5. Organization of paper. Thestatementsofthemainresultsinthispaperaregivenin
§2. Thetwosections,§3and§4,serveaspreparationsfortheproofsofthemainresultswhich
are givenin §5 and§6. In §7, we describe some solutions to the set-theoreticalYang–Baxter
equation that arise in our solution to 2) of the main problems in §1.3. In §8, we classify
(R ,P) double cosets in G ×G for any admissible pair A and any standard parabolic
A 1 2
subgroup P of G × G , and we show in particular that R is a spherical subgroup of
1 2 A
G ×G .
1 2
4 JIANG-HUALUANDMILENYAKIMOV
1.6. Notation. Throughout this paper, the letter e always denotes the identity element of
any group. We will denote by Ad the conjugation action of a group G on itself, given by
Ad (h)=ghg−1 for g,h∈G. IfM and M aretwo sets andifK ⊂M ×M , we canthink
g 1 2 1 2
of K as a correspondence between M and M . If M is another set and if L ⊂ M ×M ,
1 2 3 2 3
then one defines the composition of L and K to be
(1.3) L◦K ={(m ,m )∈M ×M |∃m ∈M such that (m ,m )∈K, (m ,m )∈L}.
1 3 1 3 2 2 1 2 2 3
1.7. Acknowledgements. ThisworkwasstartedwhiletheauthorswerevisitingtheErwin
Schr¨odinger institute for Mathematical Physics, Vienna in August 2003. We would like
to thank the organizers A. Alekseev and T. Ratiu of the program on Moment Maps for
the invitation to participate. During the course of the work, the first author enjoyed the
hospitality of IHES and UCSB and the second author of the University of Hong Kong. We
are grateful to these institutions for the warm hospitality. We would also like to thank S.
Evens, K. Goodearl, G. Karaali and A. Moy for helpful discussions. We are indebted to
the referee for carefully reading the paper, pointing out many typos, and suggesting several
improvements in the exposition. The research of the first author was partially supported
by (USA)NSF grant DMS-0105195 and HKRGC grant 701603. The research of the second
author was partially supported by NSF grant DMS-0406057 and a UCSB junior faculty
researchincentive grant.
2. Statements of the main results
Themainparametersthatcomeintoourclassificationof(R ,R )doublecosetsinG ×G
A C 1 2
arecertainelementsintheWeylgroupsofG andG andcertain“twistedconjugacyclasses”
1 2
in Levi subgroups of G .
1
2.1. Minimal length representatives. For i=1,2, let W be the Weyl group of Γ . For
i i
a subset D of Γ , we will use W to denote the subgroup of W generated by D . Let
i i Di i i
WDi ⊂ W and DiW ⊂ W be respectively the set of minimal length representatives of the
i i i i
cosets from W /W and W \W . It is well-known that
i Di Di i
w ∈WDi if and only if w (D )⊂∆+
i i i i i
and
w ∈DiW if and only if w−1(D )⊂∆+.
i i i i i
2.2. Twisted conjugacy classes.
Definition 2.1. Let D be a subset of Γ , and let d be a partial isometry from Γ to Γ
1 1 1 1
with D as both its domain and its range. If J ⊂ M ×M is a generalized d-graph, we
1 D1 D1
let J act on M from the left by
D1
J ×M →M : ((l,n),m)7→lmn−1, (l,n)∈J, m∈M .
D1 D1 D1
By a d-twisted conjugacy class in M we mean a J orbit in M for any generalized d-
D1 D1
graph J. If J is the graph of a group automorphism j: M → M , the action of J on
D1 D1
M becomes the usual j-twisted conjugation action of M on itself:
D1 D1
M ×M →M : (l,m)7→lmj(l)−1, l,m∈M
D1 D1 D1 D1
whose orbits will be referred to as j-twisted conjugacy classes.
DOUBLE COSETS IN REDUCTIVE ALGEBRAIC GROUPS 5
2.3. Classification of double cosets. Let A = (a,K) and C = (c,L) be two admissible
pairs, and let A and A be the domain and the range of a and C and C the domain and
1 2 1 2
the range of c respectively. We now state our classification of (R ,R ) double cosets in
A C
G ×G . In doing so, we will need to refer to several lemmas that will be proved in §3-§6.
1 2
For v1 ∈W1C1 and v2 ∈A2W2, consider the sets
(2.1) A (v ,v )={α∈A |(v c−1v−1a)nαis defined
1 1 2 1 1 2
and is inA forn=0,1,...}
1
(2.2) C (v ,v )={β ∈C |(v−1av c−1)nβ is defined
2 1 2 2 2 1
and is inC forn=0,1,...}
2
ThenA (v ,v )isthe largestsubsetofA thatisinvariantunderv c−1v−1a,andC (v ,v )
1 1 2 1 1 2 2 1 2
is the largest subset of C that is invariant under v−1av c−1. Note that
2 2 1
(2.3) v−1a and cv−1: A (v ,v )→C (v ,v )
2 1 1 1 2 2 1 2
are both partialisometries from Γ to Γ . Fix a representative v˙ in the normalizer of H in
1 2 i i
G , and define
i
(2.4) K(v ,v )= M ×M ∩Ad−1 K
1 2 A1(v1,v2) C2(v1,v2) (e,v˙2)
(2.5) L(v1,v2)=(cid:0)MA1(v1,v2)×MC2(v1,v2)(cid:1)∩Ad(v˙1,e)L.
It is easyto see that K(v ,v )is (cid:0)a generalized(v−1a)-gra(cid:1)ph,whileL(v ,v ) is a generalized
1 2 2 1 2
(cv−1)-graph for the partial isometries in (2.3). Define
1
(2.6) J(v ,v )={(m,m′)∈M ×M |∃n∈M
1 2 A1(v1,v2) A1(v1,v2) C2(v1,v2)
such that(m,n)∈K(v ,v )and(m′,n)∈L(v ,v )}.
1 2 1 2
Then J(v ,v ) is a generalized (v c−1v−1a)-graph for the partial isometry
1 2 1 2
v c−1v−1a: A (v ,v )→A (v ,v ).
1 2 1 1 2 1 1 2
This follows from Lemma 3.5 in §3 and the facts that if σ is the map
σ: M ×M →M ×M : (m,n)7→(n,m),
A1(v1,v2) C2(v1,v2) C2(v1,v2) A1(v1,v2)
then σ(L(v ,v )) is a generalized (v c−1)-graph, and J(v ,v ) is the composition
1 2 1 1 2
(2.7) J(v ,v )=σ(L(v ,v ))◦K(v ,v )
1 2 1 2 1 2
(of group correspondences, see §1.6). Let J(v ,v ) act on M from the left by
1 2 A1(v1,v2)
(2.8) (l ,n )·m :=l m n−1, (l ,n )∈J(v ,v ), m ∈M .
1 1 1 1 1 1 1 1 1 2 1 A1(v1,v2)
By Definition 2.1, J(v ,v ) orbits in M are (v c−1v−1a)-twisted conjugacy classes
1 2 A1(v1,v2) 1 2
in M . Let Z be the center of M , let η : G × G → G be the
A1(v1,v2) C2(v1,v2) C2(v1,v2) 2 1 2 2
projection, and let
Z(v ,v )=Z / Z ∩v−1(η (K)) Z ∩η (L) .
1 2 C2(v1,v2) C2(v1,v2) 2 2 C2(v1,v2) 2
For each s ∈ Z(v1,v2), fix a repre(cid:0)sentative s˙ of s in ZC2(v(cid:1)1,(cid:0)v2). For (g1,g2) ∈(cid:1)G1×G2, we
will use [g ,g ] to denote the double coset R (g ,g )R in G ×G . We can now state our
1 2 A 1 2 C 1 2
main theorem on the classification of (R ,R ) double cosets in G ×G .
A C 1 2
Theorem 2.2. Let A = (a,K) and C = (c,L) be any two admissible pairs for G ×G .
1 2
Then every (R ,R ) double coset of G ×G is of the form
A C 1 2
[m v˙ ,v˙ s˙]for somev ∈WC1,v ∈A2W , m ∈M ,s∈Z(v ,v ).
1 1 2 1 1 2 2 1 A1(v1,v2) 1 2
Twosuchdoublecosets[m v˙ ,v˙ s˙]and[m′v˙′,v˙′s˙′]coincideifandonlyifv′ =v fori=1,2,
1 1 2 1 1 2 i i
s=s′, and m and m′ are in the same J(v ,v ) orbit in M .
1 1 1 2 A1(v1,v2)
6 JIANG-HUALUANDMILENYAKIMOV
Example 2.3. If η (K) = M or η (L) = M , then Z(v ,v ) = {e} for all choices of
2 A2 2 C2 1 2
(v ,v ). In particular, every (R ,R ) double coset of G ×G is of the form
1 2 A C 1 2
[m v˙ ,v˙ ]for somev ∈WC1,v ∈A2W , m ∈M .
1 1 2 1 1 2 2 1 A1(v1,v2)
Two cosets of this type [m v˙ ,v˙ ] and [m′v˙′,v˙′] coincide if and only if v′ = v for i = 1,2,
1 1 2 1 1 2 i i
and m and m′ are in the same J(v ,v ) orbit in M .
1 1 1 2 A1(v1,v2)
Example 2.4. If K and L are respectively the graphs of group isomorphisms
θ : M →M and θ : M →M ,
a A1 A2 c C1 C2
the group J(v ,v ) is the graph of the group automorphism
1 2
j(v ,v ):=Ad θ−1Ad−1θ : M →M .
1 2 v˙1 c v˙2 a A1(v1,v2) A1(v1,v2)
Then every (R ,R ) orbit in G ×G is of the form [m v˙ ,v˙ ] for a unique pair (v ,v ) ∈
A C 1 2 1 1 2 1 2
W1C1 ×A2W2 and some m1 ∈MA1(v1,v2). Two such cosets [m1v˙1,v˙2] and [m′1v˙1,v˙2] coincide
if and only if m and m′ are in the same j(v ,v )-twisted conjugacy class (see §2.2).
1 1 1 2
2.4. Structure of double cosets. Let A= (a,K) and C = (c,L) be two admissible pairs
for G ×G . For q =(g ,g )∈G ×G , set
1 2 1 2 1 2
Stab(q)=Ad−1 (R )∩Ad (R ).
(e,g2) A (g1,e) C
Thenthedouble cosetR qR inG ×G ,consideredasanR ×R -spaceunderthe action
A C 1 2 A C
(r ,r )·q′ =r q′r−1, is given by
A C A C
R qR =(R ×R )/{(Ad (r),Ad−1 (r))|r ∈Stab(q)}.
A C A C (e,g2) (g1,e)
For q = (m1v˙1,v˙2s2) as in Theorem 2.2, where m1 ∈ MA1(v1,v2),v1 ∈ W1C1,v2 ∈ A2W2, and
s ∈ Z , we will now give an explicit description of Stab(q). Let π : P → M
2 C2(v1,v2) A1 A1 A1
and π : P → M be, respectively, the projections with respect to the decompositions
C2 C2 C2
P =M N and P =M N . By 2)of Lemma 3.4 in §3 (by taking D =∅ therein),
A1 A1 A1 C2 C2 C2 1
thegroupK givesrisetoagroupisomorphismθ : U ∩M →U ∩M suchthatθ (Uα)=
a 1 A1 2 A2 a 1
Ua(α)foreveryα∈∆ . Similarly,wehavethegroupisomorphismθ : U ∩M →U ∩M
2 A1 c 1 C1 2 C2
induced from the group L. We will prove that
(2.9) Stab(q)⊂ P ∩v (P ) × P ∩v−1(P ) ,
A1(v1,v2) 1 C1 C2(v1,v2) 2 A2
and we will use the maps π(cid:0)A1,πC2,θa, and θc to(cid:1)de(cid:0)scribe Stab(q). (cid:1)
Note that v1 ∈A1(v1,v2)W1C1 and v2 ∈A2W2C2(v1,v2) because
(2.10) v−1(A (v ,v ))=c−1(C (v ,v ))⊂C and v (C (v ,v ))=a(A (v ,v ))⊂A .
1 1 1 2 2 1 2 1 2 2 1 2 1 1 2 2
By Lemma 4.2 in §4, we have
(2.11) P ∩v (P )=M U ∩v (U ) ,
A1(v1,v2) 1 C1 A1(v1,v2) A1(v1,v2) 1 1
(2.12) PC2(v1,v2)∩v2−1(PA2)=MC2(v1,v2)(cid:0)UC2(v1,v2)∩v2−1(U2(cid:1)) .
Consider the group homomorphisms (cid:0) (cid:1)
(2.13) φ =Ad θ−1π : U →Ad (U ),
1 m1v˙1 c C2 2 m1 1
(2.14) φ =Ad−1 θ π : U →U .
2 v˙2s2 a A1 1 2
In §6 we will show that
(2.15) σ =φ φ ∈End(U ∩v (U )),
1 1 2 A1(v1,v2) 1 1
(2.16) σ =φ φ ∈End(U ∩v−1(U ))
2 2 1 C2(v1,v2) 2 2
DOUBLE COSETS IN REDUCTIVE ALGEBRAIC GROUPS 7
are well-defined and that σk+1 ≡e and σk+1 ≡e for some integer k ≥1, where e stands for
1 2
the trivial endomorphism. Consider the subgroups
(2.17) U ∩v (U )=U ∩v (U )⊂U ∩v (U ),
1 1 C1 A1(v1,v2) 1 C1 A1(v1,v2) 1 1
(2.18) U ∩v−1(U )=U ∩v−1(U )⊂U ∩v−1(U ),
2 2 A2 C2(v1,v2) 2 A2 C2(v1,v2) 2 2
and define
ψ : (U ∩v (U ))× U ∩v−1(U ) →U ∩v (U ),
1 1 1 C1 2 2 A2 A1(v1,v2) 1 1
ψ2: (U1∩v1(UC1))×(cid:0)U2∩v2−1(UA2)(cid:1)→UC2(v1,v2)∩v2−1(U2),
respectively by (cid:0) (cid:1)
(2.19) ψ (n ,n )=φ (n )σ (n φ (n ))σ2(n φ (n ))···σk(n φ (n )),
1 1 2 1 2 1 1 1 2 1 1 1 2 1 1 1 2
(2.20) ψ (n ,n )=φ (n )σ (n φ (n ))σ2(n φ (n ))···σk(n φ (n )).
2 1 2 2 1 2 2 2 1 2 2 2 1 2 2 2 1
Theorem 2.5. Let the notation be as in Theorem 2.2. For q = (m v˙ ,v˙ s ), where m ∈
1 1 2 2 1
MA1(v1,v2),v1 ∈ W1C1,v2 ∈ A2W2, and s2 ∈ ZC2(v1,v2), we have the semi-direct product
decomposition
Stab(q)=Stab (q)Stab (q),
M U
where
Stab (q)=K(v ,v )∩Ad L(v ,v )
M 1 2 (m1,e) 1 2
= M ×M ∩Ad−1 K∩Ad L,
A1(v1,v2) C2(v1,v2) (e,v˙2) (m1v˙1,e)
StabU(q)={(cid:0)(n1ψ1(n1,n2), n2ψ2(n1(cid:1),n2))|(n1,n2)∈(U1∩v1(UC1))× U2∩v2−1(UA2) }
⊂ UA1(v1,v2)∩v1(U1) × UC2(v1,v2)∩v2−1(U2) . (cid:0) (cid:1)
In particular (cid:0)Stab(q) is the semi-(cid:1)dire(cid:0)ct product of (cid:1)
Stab (q)=Stab(q)∩ M ×M and
M A1(v1,v2) C2(v1,v2)
StabU(q)=Stab(q)∩(cid:0)(UA1(v1,v2)∩v1(U1))×(cid:1)(UC2(v1,v2)∩v2−1(U2)) ,
cf. (2.9) and (2.11)–(2.12). (cid:0) (cid:1)
Remark 2.6. Form ∈M ,letJ (v ,v )be the stabilizersubgroupofJ(v ,v )at
1 A1(v1,v2) m1 1 2 1 2
m for the action of J(v ,v ) on M given by (2.8), cf. Definition 2.1. We will show
1 1 2 A1(v1,v2)
in Lemma 6.2 in §6 that Stab (q) is a central extension of J (v ,v ) by
M m1 1 2
(kerη | )∩(Id×v−1)(kerη | )⊂{e}×Z ,
1 L 2 1 K C2(v1,v2)
where, for i=1,2, η : G ×G →G is the projection.
i 1 2 i
2.5. A dimension formula.
Theorem 2.7. In the notation of Theorem 2.5, if K and L are algebraic subgroups of
G ×G , or if the base field is C and K and L are Lie subgroups of G ×G , then the
1 2 1 2
double coset [m v˙ ,v˙ s˙] (which is clearly smooth subvariety or submanifold, respectively) has
1 1 2
dimension equal to
l(v )+l(v )+dimP −dimM +dimP −dimZ
1 2 A1 A1(v1,v2) C2 C2(v1,v2)
+dimη KL∩(v−1Z ×v Z ) +dim(J(v ,v )·m ).
2 1 A1(v1,v2) 2 C2(v1,v2) 1 2 1
Here l(·) denotes the(cid:0)length function on the Weyl groups(cid:1)W and W , and J(v ,v )·m is
1 2 1 2 1
the J(v ,v ) orbit through m for the J(v ,v ) action on M given in (2.8).
1 2 1 1 2 A1(v1,v2)
8 JIANG-HUALUANDMILENYAKIMOV
Let us note that in the two cases in Theorem2.7, the double coset[m v˙ ,v˙ s˙] is a locally
1 1 2
closed algebraic subset of G ×G (as an orbit of an algebraic group) or a submanifold of
1 2
G ×G . It will be shown in §6 that in these two cases J(v ,v ) is respectively an algebraic
1 2 1 2
grouporaLiegroup. Notethatη KL∩(v−1Z ×v Z ) isalsoanalgebraic
2 1 A1(v1,v2) 2 C2(v1,v2)
group or a Lie group because it is a quotient of KL∩(v−1Z ×v Z ).
(cid:0) 1 A1(v1,v2) (cid:1)2 C2(v1,v2)
3. Properties of the groups R
A
Let a∈P(Γ ,Γ ) be a partialisometry fromΓ to Γ , and let A andA be respectively
1 2 1 2 1 2
the domain and the range of a. In this section, we first give a characterization of those
subgroupsKofM ×M thataregeneralizeda-graphs. Wewillthenprovesomeproperties
A1 A2
of the groups R associatedto an admissible pairs A=(a,K). These properties are crucial
A
in the proofs of the main results in this paper.
Recall that for i = 1,2, η : G ×G → G denote the projections to the i’th factor. For
i 1 2 i
a subset D of Γ , M′ denotes the derived subgroup of M . We will denote by Z the
i i Di Di Di
center of M .
Di
Definition 3.1. By an a-quintuple we mean a quintuple (K ,X ,K ,X ,θ), where, for
1 1 2 2
i = 1,2, K is an abstract subgroup of M containing M′ , X is an abstract subgroup of
i Ai Ai i
K ∩Z , and θ: K /X →K /X is a group isomorphismthat maps Uα to Ua(α) for each
i Ai 1 1 2 2 1 2
α∈∆ , where Uα is identified with its image in K /X and similarly for Ua(α).
A1 1 1 1 2
Lemma 3.2. Let K ⊂M ×M be a generalized a-graph. Define
A1 A2
K =η (K)⊂M , X =η (ker(η | ))={x ∈M |(x ,e)∈K}⊂K
1 1 A1 1 1 2 K 1 A1 1 1
K =η (K)⊂M , X =η (ker(η | ))={x ∈M |(e,x )∈K}⊂K ,
2 2 A2 2 2 1 K 2 A2 2 2
and let
(3.1) θ: K /X →K /X where θ(k X )=k X if (k ,k )∈K.
1 1 2 2 1 1 2 2 1 2
Then θ is well-defined, and Θ(K):=(K ,X ,K ,X ,θ) is an a-quintuple. Moreover, K can
1 1 2 2
be expressed in terms of Θ(K) as
(3.2) K ={(k ,k )∈K ×K | θ(k X )=k X }.
1 2 1 2 1 1 2 2
The assignment K 7→ Θ(K) is a one to one correspondence between the set of generalized
a-graphs and the set of a-quintuples.
Proof. Let i = 1,2. Since K contains all one-parameter unipotent subgroups Uα of M
i i Ai
and the latter generate M′ as an abstract group, we have that K ⊃M′ . Moreover X is
Ai i Ai i
an abstract normal subgroup of K which does not intersect any one-parameter unipotent
i
subgroup Uα of M because of the main condition for an a-graph, cf. Definition 1.1.
i Ai
First we show that X ⊂Z . We will make use of the following fact which can be found
i Ai
e.g. in [16, Corollary 29.5].
(∗) Any simple algebraic group of adjoint type (over an algebraically closed field) is simple
as an abstract group.
Assume that X is not a subgroup of Z . Because of K ⊃ M′ , we have that M =
i Ai i Ai Ai
K Z . Since X is normal in K , X Z /Z is a nontrivial abstract normal subgroup of
i Ai i i i Ai Ai
M /Z .ThegroupM /Z issemi-simpleandhastrivialcenter,soitisadirectproduct
Ai Ai Ai Ai
of simple algebraic groups of adjoint types. We can now apply (∗) to get that X Z /Z
i Ai Ai
contains a simple factor of M /Z . Therefore (X Z )′ contains M′ for some nontrivial
Ai Ai i Ai Di
subset D of A . (Here (.)′ refers to the derived subgroup of an abstract group.) As a
i i
consequence X ⊃ X′ = (X Z )′ ⊃ M′ which contradicts with the fact that X does not
i i i Ai Di i
intersect any one-parameter unipotent subgroup of M . This shows that X ⊂Z .
Ai i Ai
It is easy to see that θ is a well-defined group isomorphism and that (3.2) holds. The
definition of K implies that θ induces a group isomorphism from Uα to Ua(α) for each
1 2
DOUBLE COSETS IN REDUCTIVE ALGEBRAIC GROUPS 9
α ∈ ∆ . Thus Θ(K) is an a-quintuple. It is straightforward to check that the map
A1
K 7→Θ(K) is bijective. (cid:3)
Notation 3.3. For i=1,2 and for a subset D ⊂A , let
i i
PAi =P ∩M , UAi =U ∩M .
Di Di Ai Di Di Ai
For subsets S ⊂M , we will set
i Ai
(3.3) K(S ) ={m ∈M |∃m ∈S such that (m ,m )∈K}
1 2 A2 1 1 1 2
(3.4) K(S ) ={m ∈M |∃m ∈S such that (m ,m )∈K}.
2 1 A1 2 2 1 2
Lemma 3.4. For any subset D of A ,
1 1
1) K(M ) ⊂ M , K(M ) ⊂ M , and the intersection (M ×M )∩K is
D1 a(D1) a(D1) D1 D1 a(D1)
generalized a| -graph;
D1
2) the intersection (UA1 ×UA2 )∩K is the graph of a group isomorphism θ: UA1 →
D1 a(D1) D1
UA2 ;
a(D1)
3) we have the decompositions
(3.5) (P ×M )∩K = (M ×P )∩K =(P ×P )∩K
D1 A2 A1 a(D1) D1 a(D1)
(3.6) = (M ×M )∩K (UA1 ×UA2 )∩K
D1 a(D1) D1 a(D1)
(3.7) (PD1 ×PA2)∩RA = (cid:0)(PA1 ×Pa(D1))∩RA =(cid:1)(cid:16)(PD1 ×Pa(D1))∩RA(cid:17)
(3.8) = (M ×M )∩R (U ×U )∩R
D1 a(D1) A D1 a(D1) A
(3.9) = (cid:0)(PD1 ×Pa(D1))∩K (U(cid:1)A(cid:0)1 ×UA2). (cid:1)
Proof. We first prove that K(H1)(cid:0)⊂ H2 (correspondin(cid:1)g to the special case when D1 = ∅).
Assumethat(h ,m )∈(H ×M )∩K. Forβ ∈∆ ,letα=a−1(β),andletθ : Uα →Uβ
1 2 1 A2 A2 α 1 2
bethegroupisomorphismwhosegraphis(Uα×Uβ)∩K. Forevery(u ,u )∈(Uα×Uβ)∩K,
1 2 1 2 1 2
we have
(h u h−1, m u m−1)∈K, and (h u h−1,θ (h u h−1))∈K.
1 1 1 2 2 2 1 1 1 α 1 1 1
Thus m u m−1θ (h u h−1)−1 ∈ X ⊂ Z , so m u m−1 ∈ H Uβ. Thus m normalizes
2 2 2 α 1 1 1 2 A2 2 2 2 2 2 2
both Borel subgroups of M defined by ∆+ and by −∆+ . This implies m ∈ H which
A2 A2 A2 2 2
shows that K(H )⊂H . Similarly, one shows that K(H )⊂H .
1 2 2 1
Let now D be any subset of A . Assume that (m ,m ) ∈ (M ×M )∩K. Write
1 1 1 2 D1 A2
m = h m′, where h ∈ H and m′ ∈ M′ . Since M′ is generated by the Uα’s for
1 1 1 1 1 1 D1 D1 1
α ∈ ∆ , there exists m′ ∈ M′ such that (m′,m′) ∈ K. Thus (h ,m (m′)−1) ∈ K.
D1 2 a(D1) 1 2 1 2 2
It follows from K(H ) ⊂ H that m (m′)−1 ∈ H . Thus m ∈ H M′ = M . This
1 2 2 2 2 2 2 a(D1) a(D1)
shows that K(M ) ⊂ M ). One proves similarly that K(M ) ⊂ M . It is clear
D1 a(D1 a(D1) D1
from the definition that (M ×M )∩K is a generalized a| -graph. This proves 1).
D1 a(D1) D1
Letθ: K /X →K /X begivenasin(3.1). SinceX ∩UA1 ={e}andX ∩UA2 ={e},
1 1 2 2 1 D1 2 a(D1)
wecanembedUA1 intoK /X andUA2 intoK /X . Thenθinducesagroupisomorphism
D1 1 1 a(D1) 2 2
from UA1 to UA2 whose graph is the intersection (UA1 ×UA2 )∩K. This proves 2).
D1 a(D1) D1 a(D1)
To prove (3.5) and (3.6), assume that (p ,m ) ∈ (P ×M )∩K. Write p = m u ,
1 2 D1 A2 1 1 1
where m ∈ M and u ∈ U . Note that u ∈ UA1 because p ,m ∈ M . By 2), there
1 D1 1 D1 1 D1 1 1 A1
existsu ∈UA2 suchthat(u ,u )∈K. Thus(m ,m u−1)∈K. By1),m u−1 ∈M .
2 a(D1) 1 2 1 2 2 2 2 a(D1)
Thus m ∈M UA2 ∈P , and
2 a(D1) a(D1) a(D1)
(p ,m )=(m ,m u−1)(u ,u )∈ (M ×M )∩K (UA1 ×UA2 )∩K .
1 2 1 2 2 1 2 D1 a(D1) D1 a(D1)
This shows that (cid:0) (cid:1)(cid:16) (cid:17)
(P ×M )∩K =(P ×P )∩K = (M ×M )∩K (UA1 ×UA2 )∩K .
D1 A2 D1 a(D1) D1 a(D1) D1 a(D1)
(cid:0) (cid:1)(cid:16) (cid:17)
10 JIANG-HUALUANDMILENYAKIMOV
Similarly, one proves that (M ×P )∩K is also equal to any one of the above three
A1 a(D1)
groups. This proves (3.5) and (3.6).
The identities in (3.7) follow directly from those in (3.5). To prove (3.8) and (3.9),
assume now that (p ,p ) ∈ (P ×P )∩R . Write p = m u and p = m u , where
1 2 D1 a(D1) A 1 1 1 2 2 2
m ∈M ,u ∈U ,m ∈M and u ∈U . Further write
1 D1 1 D1 2 a(D1) 2 a(D1)
u =uA1u , u =uA2 u ,
1 D1 A1 2 a(D1) A2
where
uA1 ∈UA1, u ∈U , uA2 ∈uA2 , u ∈U .
D1 D1 A1 A1 a(D1) a(D1) A2 A2
Since (u ,u )∈R , we have
A1 A2 A
m ,uA1, m uA2 ∈ PA1 ×PA2 ∩R = PA1 ×PA2 ∩K.
1 D1 2 a(D1) D1 a(D1) A D1 a(D1)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
By (3.6), we see that (m ,m )∈R . This shows (3.8). Note that (M ×M )∩R =
1 2 A D1 a(D1) A
(M ×M )∩K. It is also easy to show that
D1 a(D1)
(U ×U )∩R = (UA1 ×UA2 )∩K (U ×U ).
D1 a(D1) A D1 a(D1) A1 A2
Now (3.9) follows from (3.6). (cid:16) (cid:17) (cid:3)
Finally we use Lemma 3.2 to treat compositions of generalized graphs.
Lemma 3.5. Let G , G and G be connected reductive algebraic groups (with fixed choices
1 2 3
of maximal tori and Borel subgroups). Assume that (a,K) and (c,L) are two admissible
pairs for G ×G and G ×G , respectively, such that the domain of c coincides with the
1 2 2 3
range of a. Then the composition (ca,L◦K) is an admissible pair for G ×G .
1 3
Proof. Lemma 3.2 implies that for any root α of G in the span of the domain of a, there
1
exist group isomorphisms θ: Uα →Ua(α) and ϕ: Ua(α) →Uca(α) such that
1 2 2 3
K∩(Uα×G )=({e}×X )(Id×θ)(Uα) and
1 2 2 1
L∩(G ×Uca(α))=(Y ×{e})(Id×ϕ)(Ua(α))
2 3 2 3
for some subgroups X and Y of the fixed maximal torus of G . Then
2 2 2
(L◦K)∩(Uα×Uca(α))=(Id×ϕθ)(Uα).
1 3 1
ThereforeL◦K isageneralizedca-graphand(ca,L◦K)isanadmissiblepairforG ×G .Let
1 3
us note that it is not hard to determine the ca-quintuple corresponding to the composition
L◦K, cf. Lemma 3.2, but it will not be needed and will omit it. (cid:3)
4. Some facts on Weyl groups and intersections of parabolic subgroups
4.1. The Bruhat Lemma. Fix a connected reductive algebraic group G over k with a
maximaltorusH anda set∆+ of positive rootsfor (G,H). LetΓ be the set ofsimple roots
in ∆+. We will denote by W the Weyl group of (G,H) and by l(·) the standard length
function on W. For w ∈W, w˙ will denote a representative of w in the normalizer of H in G.
Given A ⊂ Γ, we will denote by W the subgroup of the Weyl group W generated by
A
elements in A. For A⊂Γ, let P be the corresponding parabolic subgroup of G containing
A
theBorelsubgroupofGdeterminedby∆+,andletM andU berespectivelyitsLevifactor
A A
containing H and its unipotent radical. For A,C ⊂ Γ, the standard Bruhat decomposition
for (P ,P ) double cosets of G states that
A C
(4.1) G= P w˙P .
A C
[w]∈WaA\W/WC