Simplifying and Unifying Bruhat Order for BGB, PGB, KGB, and KGP
aa r X i v : . [ m a t h . R T ] D ec SIMPLIFYING AND UNIFYING BRUHAT ORDER FOR B \ G/B , P \ G/B , K \ G/B , AND K \ G/P
WAI LING YEE
This paper is dedicated to the memory of Angela Sodan.
Abstract.
This paper provides a unifying and simplifying approach to Bruhat order inwhich the usual Bruhat order, parabolic Bruhat order, and Bruhat order for symmetricpairs are shown to have combinatorially analogous and relatively simple descriptions. Suchanalogies are valuable as they permit the study of P \ G/B and K \ G/B by reducing to B \ G/B rather than by introducing additional machinery. A concise definition for reducedexpressions and a simple proof of the exchange condition for P \ G/B are provided as appli-cations of this philosophy. A geometric argument for spherical subgroups, which includesall of the cases considered, shows that Bruhat order has property Z and therefore satisfiesthe subexpression property. Thus, Bruhat order can be described using only simple rela-tions, and it is the simple relations which we simplify combinatorially. A parametrization of K \ G/P is a simple consequence of understanding the Bruhat order of K \ G/B restricted toa P -orbit. In P \ G/B , if x, y ∈ W G are maximal length W L \ W G coset representatives, then W L x ≤ W L y ⇐⇒ x ≤ y . Similarly, viewing KuP and
KvP as unions of B orbits for which KuB and
KvB have maximal dimension, then
KuP (cid:22)
KvP ⇐⇒ KuB (cid:22)
KvB . K \ G/P is in order preserving bijection with the P -maximal elements of K \ G/B in the same waythat P \ G/B is in order preserving bijection with the P -maximal elements of B \ G/B . Introduction
Bruhat order is an important tool in many branches of representation theory, in partbecause of the importance of studying orbits on the flag variety. Category O and the categoryof Harish-Chandra modules (see [Kna02] p. 375) are two categories for which representationsare related to orbits on the flag variety. Let G be a complex reductive linear algebraicgroup with Lie algebra g , θ a Cartan involution of G (specifying a real form), K = G θ , B a θ -stable Borel subgroup, and P a standard parabolic subgroup containing B . UsingBeilinson-Bernstein’s geometric construction, irreducible representations in Category O oftrivial infinitesimal character are known to be in correspondence with B -orbits on the flagvariety while irreducible Harish-Chandra modules of trivial infinitesimal character are inbijection with K -equivariant local systems on K -orbits on the flag variety. The moduleconstructions may be modified suitably to produce modules for other infinitesimal characters.Multiplicities of irreducible composition factors in standard modules for each of thesecategories can be computed using Kazhdan-Lusztig-Vogan polynomials. Finding efficientmeans of computing such polynomials is a heavily studied problem. Since the recursionformulas for computing Kazhdan-Lusztig-Vogan polynomials are expressed in terms of theBruhat order on orbits and on local systems, we hope that the simplifications to Bruhat order Mathematics Subject Classification.
Primary 22E50, Secondary 05E99.The author is grateful for the support from a Discovery Grant and UFA from NSERC, NSF grants DMS-0554278 and DMS-0968275, and the hospitality of the American Institute of Mathematics. or P \ G/B and for K \ G/B contained in this paper may lead to a deeper understanding of theKazdhan-Lusztig-Vogan polynomials in these categories and the relationships among them.(Recall that local systems are parametrized by certain orbits in pairs of flag varieties.)Beyond parametrizing representations of various categories, orbits on the flag variety andBruhat order appear in geometry (symmetric spaces, spherical homogeneous spaces) and innumber theory problems in which one studies the fixed points of an involution. Bruhat orderis ubiquitous in mathematics and is of fundamental importance.We begin this paper by showing that Bruhat order can be described using only simplerelations by:(1) a simple geometric argument for spherical subgroups, which includes all of the casesconsidered, showing that Bruhat order satisfies property Z (see [Deo77] Theorem 1.1and [RS90] Property 5.12(d) or definition 3.10)(2) using property Z to show that Bruhat order has the subexpression property.Simple relations for Bruhat order are examined from the following perspectives for eachof B \ G/B , P \ G/B , and K \ G/B : • topological (closure order) • cross actions and Cayley transforms • roots and the Weyl group • roots and pullbacks.Strong analogies are drawn between the different cases B \ G/B , P \ G/B , and K \ G/B . Thispermits definitions of standard objects and proofs of properties to be simplified for P \ G/B and for K \ G/B : it is more efficient to exploit similarities with B \ G/B than it is to introducenew machinery to accommodate the differences. The simplest combinatorial descriptions ofthe simple Bruhat relations are Theorems 4.6, 5.8, and 8.23.This paper is structured as follows. Section 2 contains notation and the setup which willremain fixed for the duration of the paper. In section 3 we show that Bruhat order forspherical subgroups can be described using only simple relations. Sections 4 and 5 discussBruhat order for B \ G/B and for P \ G/B , respectively.We discuss reduced expressions and the exchange property in sections 6 and 7 for each of B \ G/B and P \ G/B .In section 8, Bruhat order for K \ G/B is simplified and shown to be analogous to Bruhatorders for B \ G/B and for P \ G/B .In section 9, we give a simple combinatorial parametrization of K \ G/P (Theorem 9.14).In the same way that P \ G/B is in bijection with maximal length coset representatives, K \ G/P is in bijection with “ P -maximal” elements of K \ G/B . We show that if
KuB and
KvB are “ P -maximal”, then KuP (cid:22)
KvP ⇐⇒ KuB (cid:22)
KvB . We discuss how themonoidal action descends (or fails to descend) from K \ G/B to K \ G/P .The final section contains a discussion of future work.1.1.
Acknowledgements.
I would like to thank Annegret Paul and Siddhartha Sahi fromwhom I have learned a tremendous amount. I would also like to thank Athony Henderson andJohn Stembridge for helpful feedback and David Vogan for his comments and his suggestionto look for the simplest possible geometric explanation that Bruhat order depends only onsimple relations. . Setup and Notation
The following notation will be fixed for the duration of this paper. • G : complex reductive linear algebraic group • g : the Lie algebra of G . Analogous notation will be used for Lie algebras of othergroups. • θ ∈ Aut( G ): an algebraic automorphism of order two • K = G θ • B = T U : the Levi decomposition of a Borel subgroup. We may assume that weselected both B and its Levi decomposition B = T U to be θ -stable (exists by Stein-berg, see [Spr85] 2.3). Therefore T is maximally compact, i.e. dim t ∩ g θ is maximalso all of the roots with respect to t have non-trivial restriction to t θ (see Proposition6.70 of [Kna02]). • ∆( g , t ): the roots of g with respect to t • W = W G : the Weyl group N G ( T ) /T of G • Π: the set of simple roots corresponding to B • S : the set of simple reflections corresponding to Π • P J : the standard parabolic subgroup corresponding to J ⊂ Π • P = LN : the T -stable Levi decomposition of P • I ⊂ Π: the subset corresponding to P • W L : the Weyl Group N L ( T ) /T of L • x α : R → G : for the simple root α , the one-parameter subgroup of G associated to α . This satisfies tx α ( τ ) t − = x α ( tτ t − ) for all t, τ ∈ T . Then x − α : R → G is chosento be the unique one-parameter subgroup such that x α (1) x − α ( − x α (1) ∈ N ( T ). • φ α : SL → G : for the simple root α , the group homomorphism satisfying x α ( m ) = φ α (cid:18) m (cid:19) , x − α ( m ) = φ α (cid:18) m (cid:19) , α ∨ ( t ) = φ α (cid:18) t t − (cid:19) . • ˙ s α = x α ( − x − α (1) x α ( −
1) = φ α (cid:18) −
11 0 (cid:19) ∈ N ( T ) for the simple root α ( ˙ s α = n α in the notation of [RS90]) • ˙ w : Given w = s i s i · · · s i k ∈ W a reduced expression, define ˙ w = ˙ s i ˙ s i · · · ˙ s i k ∈ N ( T ). It is known that ˙ w is independent of the reduced expression chosen. • H g := gHg − for every g ∈ G , H ≤ G • B := the variety of Borel subgroups of G . This is in bijection with G/B where B g ↔ gB .3. Spherical Subgroups: Reducing Bruhat Order to its Simple Relations
In this section, we prove that Bruhat order on B \ G/B , P \ G/B , and K \ G/B may bedescribed using only simple relations; more generally, if H is a spherical subgroup of G , thenBruhat order on H \ G/B can be described using only simple relations. For readers far moreaccustomed to Bruhat order on the Weyl group than Bruhat order on K \ G/B , it might beeasier to read this section last.Often, one first encounters Bruhat order as a partial order on the Weyl group and onquotients of the Weyl group. That definition of Bruhat order does not immediately lend itselfto generalization. Since our goal is to unify Bruhat order for B \ G/B , P \ G/B , K \ G/B , nd for K \ G/P , before we obtain combinatorially analogous definitions of Bruhat order,we focus on orbits on the flag variety
G/B and use the equivalent topological definition ofBruhat order which is common to all of our settings. Our subgroups B , P , and K share thecommon property that they are all spherical subgroups of G . Definition 3.1.
A closed subgroup H of G is spherical if it has an open orbit in G/B .It is well-known that if H is spherical, then H \ G/B is finite. This will also follow fromthe subexpression property.
Definition 3.2.
Let H be a spherical subgroup of G . Then the closure order on H \ G/B isdefined by: O (cid:22) H O if O ⊂ O . Bruhat order on H \ G/B is defined to be closure order.It is well-known that under the bijections B \ G/B ↔ W G and P \ G/B ↔ W L \ W G , thetopological definition of Bruhat order agrees with the usual definition.We will see that this topological definition can easily be used to find a common proof forall of our cases that Bruhat order satisfies property Z and hence satisfies the subexpressionproperty. Thus, Bruhat order for B \ G/B , P \ G/B , and for K \ G/B may be described usingonly what we will call simple relations.
Notation 3.3.
Given α a simple root, we define P α = P { α } (the standard parabolic of type α containing B ). Then we have the canonical projection: π α : G/B → G/P α . which may be viewed as a projection from B to P α , the variety of parabolics of type α .The set of Borel subgroups contained in P α is in bijection with P . Therefore we see that: Lemma 3.4. ( [Vog83] ) π α is a P -fibration: π − α π α ( x ) ∼ = P for all x ∈ B . P → G/B ↓ π α G/P α . Because of the P -fibration, Corollary 3.5.
Let H be a subgroup of G and let O be an H -orbit on G/B . Then dim π − α π α ( O ) = dim O or dim O + 1 . Lemma 3.6. [Kno95]
For spherical subgroups H of G and O an H -orbit on G/B , there isa unique dense orbit O ′ ⊂ π − α π α ( O ) . π − α π α ( O ) is a union of 1, 2, or 3 orbits.Proof. This is a consequence of the geometry of each P -fibre. Here, we outline Knop’s workfor its relevance to finiteness of H \ G/B .Given a B -variety, the complexity of the variety is defined to be the minimal codimensionof a B -orbit in the variety ([Kno95], p. 287). Thus, since H is spherical, the complexity of G/H is 0. By Theorem 2.2 of [Kno95], any B -stable subvariety of G/H also has complexity0. Therefore π − α π α ( O ) has a codimension 0 H -orbit, O ′ .Let x ∈ G be such that B x ∈ O ′ . Using the notation of this paper, there is a correspon-dence ([Kno95] p. 290) between H ∩ ( P α ) x -orbits on ( P α ) x /B x ∼ = P and the H -orbits in − α π α ( O ′ ) = π − α π α ( O ) = H \ xP α /B via H ∩ ( P α ) x xpx − xB HxpB for p ∈ P α . This mapis well-defined and surjective. To show injectivity, suppose that Hxp B = Hxp B for some p i ∈ P α . Then we may assume that xp = hxp b for some h ∈ H and b ∈ B . Rearranging,we see that h ∈ H ∩ ( P α ) x , whence H ∩ ( P α ) x xp B = H ∩ ( P α ) x xp B .Focusing on the action of H ∩ ( P α ) x on ( P α ) x /B x ∼ = P , we may instead study the actionof ¯ H which denotes the quotient of H ∩ ( P α ) x by the kernel of the action on P . Then ¯ H may be viewed as a subgroup of Aut( P ) ∼ = P SL ( C ). According to [Kno95] Lemma 3.1,since the complexity of G/H is 0, therefore ¯ H has positive dimension. It is easy to classifythe orbits of positive dimension subgroups of P SL ( C ) on P , and thus π − α π α ( O ) is a unionof 1, 2, or 3 orbits with O ′ the unique dense orbit (see [Kno95] p. 291). (cid:3) Definition 3.7.
If in the previous corollary dim O ′ = dim O + 1, then write O α
7→ O ′ or O (cid:22) Hα O ′ . This is a simple relation for Bruhat order.
Remark . (1) For B \ G/B , we will see that for some w ∈ W G , O = BwB and O ′ = Bws α B with ℓ ( ws α ) = ℓ ( w ) + 1.(2) Since O is an arbitrary orbit in π − α π α ( O ′ ) different from O ′ , therefore O α
7→ O ′ forany orbit O different from O ′ in π − α π α ( O ′ ). Lemma 3.9.
Let P ≤ P ≤ G be parabolic subgroups. Since parabolic subgroups are closed,each G/P i is equipped with the quotient topology. Then, letting p i : G → G/P i and π : G/P → G/P be the natural projections, we have the commutative diagram: G p > G/P G/P .p ∨ π > Then given X i ⊂ G/P i : (1) π ( X ) = π ( X )(2) π − ( X ) = π − ( X )(3) π − π ( X ) = π − π ( X ) .Proof. Each
G/P i is a projective variety, and hence complete. Then each π ( X ) must beclosed (see [Spr09], 6.1.2), whence π ( X ) ⊃ π ( X ). (Note that this containment could alsofollow from Lemma 2, p. 68, of [Ste74], just as Lemma 4.5 of [RS90] did.) or the opposite containment, since the topology on G/P i is the quotient topology, for X ⊂ G/P : π ( X ) = p ( p − ( X )) ⊃ \ V ⊃ p − ( X ) closed p ( V ) by definition of quotient topology= \ V ⊃ p − ( X ) closed π ◦ p ( V ) ⊃ \ X ⊃ X closed π ( X )= π ( X ) . Given X ⊂ G/H , repeating the above formulas with p in place of p , p in place of p ,and π − in place of π yields π − ( X ) ⊃ π − ( X ). We have the opposite containment since π is continuous.The final statement follows from the first two statements. (cid:3) Definition 3.10. ([Deo77], [RS90]) Let H be a spherical subgroup of G and consider Bruhatorder on H \ G/B . Let α be a simple root and let u , u , v , v ∈ H \ G/B with u α u and v α v . We say that Bruhat order satisfies property Z ( α, u , v ) if the following areequivalent:(1) u (cid:22) H v or there exists x such that x α u and x (cid:22) H v (2) u (cid:22) H v (3) u (cid:22) H v .Bruhat order is said to satisfy property Z if it satisfies every property Z ( α, u , v ). Theorem 3.11. If H is a spherical subgroup of G , then Bruhat order on H \ G/B satisfiesproperty Z .Proof. We use the notation of the previous definition, replacing u with O u , for example,to remind us that we are studying H -orbits.(1) ⇒ (2): Let x (cid:22) H v be such that x α u , possibly setting x = u . Applying Lemma 3.9with H = B , H = P α , π − α π α ( O x ) ⊂ π − α π α ( O v ) = π − α π α ( O v ) ∪ kO u O v since O v is dense in π − α π α ( O v ).(2) ⇒ (3): It suffices to note that O u ⊂ π − α π α ( O u ) = O u since O u is dense in π − α π α ( O u ).(3) ⇒ (1): If O u ⊂ O v = π − α π α ( O v ) = π − α π α ( O v ), then O u ⊂ π − α π α ( ∪ u (cid:22) H v O u ).Therefore O u ⊂ π − α π α ( O x ) for some x (cid:22) H v . If x = u or x = u , then u (cid:22) H v .Otherwise, since O u is the unique dense orbit in π − α π α ( O x ), x satisfies x α u in additionto x (cid:22) H v , whence (1) is satisfied. (cid:3) Definition 3.12. ([RS90], 5.7) Let H be a spherical subgroup of G . A pair of sequences(( v , v , . . . , v k ) , ( α , . . . , α k )), where the v i are H -orbits and the α i are simple roots, is a educed decomposition of v ∈ H \ G/B if v is a closed orbit, v k = v , and v i − α i v i for i = 1 , . . . , k . The length of the reduced decomposition is k . Note that dim O v = dim O v + k . Definition 3.13.
Given (( v , . . . , v k ) , ( α , . . . , α k )) a reduced decomposition of v ∈ H \ G/B ,a subexpression of that reduced decomposition is a sequence ( u , . . . , u k ) such that u = v and for i = 1 , . . . , k :(1) u i = u i − , or(2) ∃ u such that both u i − α i u and u i α i u , or(3) u i − α i u i . Remark . We see by the deletion condition that this generalizes the usual definition forsubexpression for elements of the Weyl group.
Theorem 3.15.
Let H be a spherical subgroup of G . Then Bruhat order for H \ G/B hasthe subexpression property. That is, u (cid:22) H v if and only if there is a reduced expression (( v , . . . , v k ) , ( α , . . . , α k )) of v and a subexpression ( u , . . . , u k ) of the reduced expressionsuch that u k = u .Proof. Deodhar’s proof that property Z implies the subexpression property from [Deo77]works in this general setting. ⇐ : We prove by induction on j that for each 0 ≤ j ≤ k , u j (cid:22) H v j . The statement holdsfor j = 0. Assume that the statement holds for j = i −
1. In case (1) where u i = u i − , thestatement clearly holds for j = i . In case (3), property Z implies that since u i − α i u i and v i − α i v i , then u i − (cid:22) H v i − ⇒ u i (cid:22) H v i . In case (2), since u i − α i u and v i − α i v i , thus u i − (cid:22) H v i − ⇒ u (cid:22) H v i . Then by definition of u i , u i (cid:22) H u (cid:22) H v i . ⇒ : We prove the converse by induction on the length k of the reduced expression. If k = 0, v = v and the only subexpressions are those which terminate in u = v . Assume that theconverse holds when the reduced expressions have lengths less than k . (Note that due tothe relationship between reduced expression length and dimensions, u (cid:22) H v implies that thelength of a reduced expression for u is less than or equal to the length for v .) Now, v k − α k v .If u (cid:22) H v k − , then by induction, there is a subexpression for u of a reduced expression for v k − , whence there is a subexpression for u of a reduced expression for v . Otherwise, there is u ′ such that u ′ α k u and, by property Z , u ′ (cid:22) H v k − . By induction, there is a subexpressionfor u ′ of a reduced expression for v k − . By appending u and v = v k , we see that there is asubexpression for u of a reduced expression for v . (cid:3) Corollary 3.16. If H is a spherical subgroup of G , then H \ G/B is finite.Proof.
This was pointed out by Vogan. Since the dimension of an H -orbit is finite, thereforethere are finitely many reduced expressions for the unique open orbit in G/B . There arefinitely many subexpressions of each reduced expression, and thus by the subexpressionproperty, H \ G/B is finite. (cid:3)
Corollary 3.17.
Bruhat order for B \ G/B , P \ G/B , and K \ G/B satisfy the subexpressionproperty, whence Bruhat order can be defined using only simple relations.
It is the focus of the subsequent chapters to find elementary means of describing thesesimple relations and showing how their descriptions are analogous in all of our settings. . Bruhat Order for B \ G/B
Equivalence of Closure Order and Bruhat Order on W . As mentioned, it iswell-known that B \ G/B is in bijection with the Weyl group:
Proposition 4.1.
Bruhat Decomposition: G = ∐ w ∈ W B ˙ wB = ∐ w ∈ W BwB.
Therefore B \ G/B ↔ W . We review the definition of Bruhat order on the Weyl group W G arising from viewing itas a reflection group. Definition 4.2.
For u, v ∈ W , t a (not necessarily simple) reflection, we write u t −→ v if v = ut and ℓ ( u ) < ℓ ( v ). Bruhat order on W is defined by u ≤ v if there exists a sequence ofelements w , w , . . . , w k ∈ W such that u = w → w → · · · → w k = v . Theorem 4.3.
Let O w = BwB for w ∈ W and let α be a simple root. Then O w α
7→ O if and only if O = O ws α and w s α −→ ws α . Proof.
This is well-known, but we provide a discussion to illuminate the source of thesimilarities between the various double cosets for which we consider Bruhat order. First, π − α π α ( BwB ) =
BwP α /B = BwB ∪ Bws α B by the parabolic Bruhat decomposition (Propo-sition 5.1) and by Lemma 8.3.7 of [Spr09].Recall that ℓ ( w ) = { α ∈ ∆ + : wα < } . Under the correspondence G/B ↔ B where gB gBg − = B g , we may view B ˙ w as a point in the orbit O w and B ∩ B ˙ w as the stabilizerof that point. If ℓ ( ws α ) = ℓ ( w )+1, then dim B ∩ B ˙ w = dim B ∩ B ˙ w ˙ s α +1. Since the dimensionof a B -orbit is the dimension of B minus the dimension of the stabilizer of a point in theorbit, dim BwB + 1 = dim
Bws α B . (cid:3) Remark . The relationship between the dimension of an orbit and the dimension of thestabilizer of a point in the orbit is the source of the similarities between descriptions of simpleBruhat relations for our various settings in terms of Weyl group elements and positive roots.Observing now that the notion of reduced decomposition arising from viewing W as areflection group corresponds with definition 3.12, we conclude: Corollary 4.5.
Under the correspondence arising from the Bruhat decomposition (Proposi-tion 4.1), Bruhat order for B \ G/B agrees with Bruhat order for W . We now proceed to reformulate simple relations for Bruhat order. We find that ourreformulations apply not only to simple relations.4.2.
Weyl Group and Roots.
Another means of describing (not necessarily simple) Bruhatrelations is:
Theorem 4.6.
Let α be a positive root and w ∈ W . Then: w s α −→ ws α if wα ∈ ∆ + = ∆( u , t ) w s α ←− ws α if wα ∈ ∆ − = ∆( u − , t ) . roof. This is known if α is a simple root ([Hum90], Lemma 1.6). In general, suppose ℓ ( w ) <ℓ ( ws α ). Let ws α = s s · · · s r be a reduced expression for ws α . By the strong exchange con-dition, w = s s · · · ˆ s i · · · s r . Then s α = w − s s · · · s r = s r s r − · · · s i +1 s i s i +1 · · · s r − s r . Weconclude that α = s r s r − · · · s i +1 α i . (Note that since s s · · · s r is a reduced expression, α = s r s r − · · · s i +1 α i is a positive root, [Hum90] p. 14.) Then wα = s · · · ˆ s i · · · s r s r · · · s i +1 α i = s · · · s i − α i > s s · · · s r is a reduced expression.Similarly, if ℓ ( w ) > ℓ ( ws α ), then wα < (cid:3) Roots and Pullbacks.
Let ∆ = ∆( g , t ) and ∆ + = ∆( b , t ). Recall the notation ofsection 2: B = T U int( g ) B g = T g U g . Considering Lie algebras, b = t ⊕ u Ad( g ) b g = t g ⊕ u g .We have the map between Cartan subalgebras Ad( g − ) : t g → t , while pullback allows us tomap between duals of Cartan subalgebras: Ad( g − ) ∗ : t ∗ → t ∗ g :(Ad( g − ) ∗ α )( t ) = α (Ad( g − ) t ) for all t ∈ t g . Letting α g = Ad( g − ) ∗ α , it is easy to see thatAd( g ) : u = M α ∈ ∆ + g α → u g = M α ∈ ∆ + g α g with Ad( g ) g α = g α g . It is straightforward to prove (use int rather than Ad):
Lemma 4.7.
For w ∈ W and n ∈ N ( T ) any representative of w , wα = α n . In particular, wα = α ˙ w . Using pullbacks, Bruhat order may be reformulated as follows:
Proposition 4.8.
Let α be a positive root and w ∈ W . Then: w s α −→ ws α if α ˙ w ∈ ∆( u , t ) w s α ←− ws α if α ˙ w ∈ ∆( u − , t ) . This may also be written
BwB α Bws α B if α ˙ w ∈ ∆( u , t ) BwB α ← [ Bws α B if α ˙ w ∈ ∆( u − , t ) . Remark . Pullbacks turn out to be particularly useful in studying Bruhat order in moregeneral situations: for example, if one of the subgroups with respect to which you take doublecosets is twisted by conjugation or if that subgroup is a more general spherical subgroup.See [PSY] for details.4.4.
Cross Actions.Definition 4.10.
The cross action of W on B \ G/B is the action generated by s α × B ˙ wB := B ˙ w ˙ s − α B where α is a simple root.Under the correspondence between B \ G/B and W , cross action corresponds to the naturalleft action of W on itself by right multiplication by the inverse.It follows immediately: heorem 4.11. Let α be a positive root and w ∈ W . Then: BwB α s α × BwB = Bws − α B = Bws α B if wα ∈ ∆( u , t ) BwB α ← [ s α × BwB = Bws − α B = Bws α B if wα ∈ ∆( u − , t ) . Bruhat Order on P \ G/B
Equivalence of Closure Order with Bruhat Order for W L \ W G . It is well-knownthat:
Proposition 5.1. ( [DM91] Proposition 1.6, [Hum75]
Lemma 8.3.7) Bruhat Decomposition: P = ∐ w ∈ W L B ˙ wB = ∐ w ∈ W L BwB and thus P \ G/B ↔ W L \ W G . Furthermore, P wB = ∐ x ∈ W L BxwB.
The commonly used definition for Bruhat order on W L \ W G viewed as a reflection groupquotient is this: Definition 5.2.
Bruhat order on W L \ W G is the partial order induced from Bruhat orderon W G . That is, W L u ≤ W L v if there are coset representatives u and v , respectively, suchthat u ≤ v .That is, if u, v ∈ W G are such that u ≤ v , then W L u ≤ W L v . The converse holds forminimal length coset representatives: Proposition 5.3.
Let u, v ∈ W G be minimal length coset representatives for W L u and for W L v . Then W L u ≤ W L v ⇐⇒ u ≤ v. Proof.
This is a special case of [Deo77], Lemma 3.5. For a purely topological proof usingmaximal length coset representatives, adapt the proof of Proposition 9.16. (cid:3)
Because closure order for B \ G/B corresponds to Bruhat order for W G , by the Bruhatdecomposition applied to P wB , closure order for P \ G/B corresponds to Bruhat order for W L \ W G : Theorem 5.4.
Closure order on P \ G/B and Bruhat order on W L \ W G correspond. For u, v ∈ W G minimal length coset representatives of W L u and W L v : O u (cid:22) P O v ⇐⇒ W L u ≤ W L v. Let w ∈ W G and let α ∈ ∆ + ( g , t ) . Using the nomenclature of Casian-Collingwood from [CC87] for cases which may be shown to correspond, P wB = P ws α B if wα ∈ ∆( l , t ) i.e. α is Levi type P wB (cid:22) Pα P ws α B if wα ∈ ∆( n , t ) i.e. α is complex upward P wB (cid:23) Pα P ws α B if wα ∈ ∆( n − , t ) i.e. α is complex downward. roof. First, u ≤ v implies that B ˙ uB (cid:22) B B ˙ vB which implies that P ˙ uB (cid:22) P P ˙ vB . Con-versely, P ˙ uB ⊂ P ˙ vB ⇐⇒ ∪ w ∈ W L B ˙ w ˙ uB ⊂ ∪ w ∈ W L B ˙ w ˙ vB = ∪ w ∈ W L B ˙ w ˙ vB ⇐⇒ for every w ∈ W L , B ˙ w ˙ uB ⊂ B ˙ x ˙ vB for some x ∈ W L ⇒ u ≤ xv for some x ∈ W L ⇒ u ≤ v by Lemma 3.5 of [Deo77] (Proposition 5.3).To prove the first of the remaining three statements, note that P ˙ w ˙ s α B = P ˙ w ˙ s α ˙ w − ˙ wB = P ˙ s wα ˙ wB . The rest of the theorem now follows. (cid:3) Remark . This proof does not generalize to K \ G/P since it relies upon the Bruhatdecomposition. We could also have proved the theorem using the following more generalheuristic. Bruhat order and closure order on P \ G/B are the same since:(1) Bruhat order is induced by Bruhat order on B \ G/B and closure order and Bruhatorder for B \ G/B are the same;(2) the topology on
G/B is the quotient topology.The proof is concise, but rather than record it, we refer the reader to the proof of Theorem9.1 where the order induced from Bruhat order on K \ G/B and closure order of K \ G/P areshown to be the same. The proofs are similar.Recall that Bruhat order for P \ G/B satisfies property Z . We may restate property Z for W L \ W G : Definition 5.6.
Let u, v ∈ W G be minimal length coset representatives for W L u and for W L v . Let s be a simple reflection. Then property Z ( s, W L u, W L v ) is satisfied if whenever ℓ ( us ) ≤ ℓ ( u ) and ℓ ( vs ) ≤ ℓ ( v ), then W L u ≤ W L v ⇐⇒ W L us ≤ W L v ⇐⇒ W L us ≤ W L vs. Furthermore, an analogue of Deodhar’s description II for Bruhat order on W G from [Deo77]holds: Proposition 5.7.
Bruhat order is the unique partial order on W L \ W G such that (1) for all W L w ∈ W L \ W G , W L w ≤ W L ⇐⇒ W L w = W L ; (2) ≤ has property Z ( s, W L u, W L v ) . Again, Bruhat order for P \ G/B may be described using only simple relations and wefocus on reformulations of those relations. As for B \ G/B , our reformulations apply not justto the simple relations.5.2.
Weyl Group and Roots.
Here we describe (not necessarily simple) Bruhat relationsusing the Weyl group and roots.
Theorem 5.8.
Let w ∈ W G and let α ∈ ∆ + . Then W L w = W L ws α if wα ∈ ∆( l , t ) W L w s α −→ W L ws α if wα ∈ ∆( n , t ) W L w s α ←− W L ws α if wα ∈ ∆( n − , t ) ote that this is analogous to Bruhat order relations for B \ G/B with l analogous to t , ∆( t , t ) = {} .Remark . Compare this characterization to John Stembridge’s characterization of para-bolic Bruhat order in section 2 of [Ste02] in which W L -cosets are associated with W G -orbits inthe dual space of the Cartan subalgebra with stabilizer W L . Bruhat order then correspondsto the partial order on the root lattice.5.3. Roots and Pullbacks.
Since α ˙ w = wα , we may reformulate Bruhat order as follows: Theorem 5.10.
Let α be a positive root and w ∈ W G . Then P wB α P ws α B if α ˙ w ∈ ∆( n , t ) P wB α ← [ P ws α B if α ˙ w ∈ ∆( n − , t ) . Cross Actions.Definition 5.11.
The cross action of W on P \ G/B is the action generated by s α × P wB := P ws − α B where α is a positive root.It follows immediately: Theorem 5.12.
Let α be a positive root and w ∈ W . Then the cross action correspondingto α satisfies: P wB = s α × P wB = P ws − α B if wα ∈ ∆( l , t ) P wB α s α × P wB = P ws − α B if wα ∈ ∆( n , t ) P wB α ← [ s α × BwB = P ws − α B if wα ∈ ∆( n − , t ) . Proof.
Again, if wα ∈ ∆( l , t ), then ˙ s wα ∈ L ⊂ P . The remainder of the proof resemblesprevious arguments. (cid:3) Reduced Expressions for B \ G/B and P \ G/B
Throughout this section, wherever w = s i s i · · · s i k , let w j = s i s i · · · s i j for 1 ≤ j ≤ k .6.1. B \ G/B . An important aspect of Bruhat order is understanding decompositions of Weylgroup elements into products of simple reflections.
Definition 6.1.
Let w ∈ W G . Then the product of simple reflections w = s i s i · · · s i k is a reduced expression for w (or B -reduced expression ) if k is minimal.The following result is standard: Proposition 6.2.
Let w ∈ W G where w = s i s i · · · s i k as a product of simple reflections.Then w = s i s i · · · s i k is a reduced expression if and only if w j α i j +1 > for j = 1 , , . . . , k − . This is the equivalent definition for reduced expression that generalizes nicely to P \ G/B . .2. P \ G/B . Again, we wish to simplify the existing literature and limit the introductionof complex machinery as much as possible.
Definition 6.3.
An element w ∈ W G is P -minimal if it is a minimal length coset represen-tative for W L w . Equivalently, wα ∈ ∆( n , t ) for every α ∈ I . Remark . As discussed, W L \ W G is in bijection with the P -minimal elements of W G . Definition 6.5.
Let w ∈ W G be P -minimal. A P -reduced expression for w is a product ofsimple reflections w = s i s i · · · s i k where w j α i j +1 ∈ ∆( n , t ) for j = 1 , , . . . , k −
1. We define ℓ P ( w ) = k , the P -length of w . Lemma 6.6. If w ∈ W is P -minimal, then any B -reduced expression for w is also P -reduced.Proof. The P -minimal element has a B -reduced expression w = s i · · · s i k . Each w j − α j ispositive. Consider the equations W L w = W L s α if wα ∈ ∆( l , t ) W L w α −→ W L s α if wα ∈ ∆( n , t ) W L w α ←− W L s α if wα ∈ ∆( n − , t ) . If the expression is not P -reduced, then for some 1 ≤ j ≤ k , we have w j − α j ∈ ∆ + ( l , t ). Then w = s w j − α j w j − s i j +1 · · · s k whence W L w = W L s i · · · ˆ s i j · · · s i k , contradicting minimality of w . Thus our expression must be P -reduced. (cid:3) Proposition 6.7.
Every coset in W L \ W G has a unique P -minimal representative and every P -minimal representative has a P -reduced expression.Proof. The first statement follows immediately from the definition of P -minimal element.The second statement follows from the previous lemma. (cid:3) Remark . We can likewise define P -maximal elements: w ∈ W G is P -maximal if it is amaximal length coset representative for W L w . Equivalently, wα ∈ ∆( n − , t ) for every α ∈ I .We also have a bijection between W L \ W G and the P -maximal elements of W G . See thediscussion of P -maximal elements in section 9 where we discuss K \ G/P .7.
Exchange Property for B \ G/B and P \ G/B B \ G/B . The Exchange Property for W G is the following: Theorem 7.1.
Let w = s i s i . . . s i k ∈ W G be a reduced expression and let α ∈ Π . If wα < ,then ℓ ( w ) > ℓ ( ws α ) . The Exchange Property is the assertion that there exists some j suchthat ws α = s i s i · · · ˆ s i j · · · s i k is a reduced expression for ws α and hence s i s i · · · ˆ s i j · · · s i k s α is a reduced expression for w . P \ G/B . The Exchange Property for W L \ W G may be described similarly: Theorem 7.2.
Let w = s i s i · · · s i k ∈ W G be a P -reduced expression and let α ∈ Π . If wα ∈ ∆( n − , t ) , then there exists some j such that ws α = s i s i · · · ˆ s i j · · · s i k and it is a P -reduced expression for ws α . It follows that w = s i s i · · · ˆ s i j · · · s i k s α is a P -reduced expressionfor w . roof. We know that there exists j such that s i s i · · · ˆ s i j · · · s i k is a B -reduced expressionfor ws α . Since wα ∈ ∆( n − , t ), therefore P wB = P ws α B , dim P ws α B = dim P ws α B − ℓ ( ws α ) = k −
1. Since, as we analyze each location in any expression, roots in l fix thecorresponding orbit, roots in n increase the orbit dimension, while roots in n − decrease theorbit dimension, therefore each simple reflection in our expression must increase dimension,whence ws α = s i s i · · · ˆ s i j · · · s i k must be a P -reduced expression as well. Since w = s i · · · s i k is a P -reduced expression for w , considering length, so must w = s i · · · ˆ s i j · · · s i k s α . (cid:3) Bruhat Order on K \ G/B
Bruhat order for K \ G/B may differ in “direction” in the literature due to a preference toassociate the minimal length reduced expression with the open dense orbit since the openorbit is unique while the closed minimal dimension orbits generally are not.8.1.
Parametrizing K \ G/B . We use the parametrization of K \ G/B as presented in [Spr85],which is an excellent reference. Recall that K = G θ and B = T U is a θ -stable Borel subgroupand Levi decomposition. Notation 8.1.
Modifying Springer’s parametrization for B \ G/K , we set: • V := { x ∈ G | x − θ ( x ) ∈ N ( T ) }• V := K × T -orbits on V : ( k, t ) · x = kxt − • ˙ v ∈ V is a representative of v . Proposition 8.2. [Spr85] V is in bijection with K \ G/B . Real Forms and Root Types.
In order to discuss Bruhat order in detail, we mustdiscuss real forms and root types. A real form of the complex Lie algebra g is a real Liesubalgebra g such that g = g ⊕ i g . A less obvious way to specify a real form is to select aCartan involution θ . (Use the Cartan decomposition g = k ⊕ s and the fact that the Killingform is positive definite on i k ⊕ s .)We begin by studying the Cartan subalgebra. Lemma 8.3. ( [Kna02] ) Given any θ -stable Cartan subalgebra t and Cartan decomposition t = t c ⊕ t n of its real form, it is known that roots α ∈ ∆( g , t ) are real-valued on i t c ⊕ t n .Thus θα = − ¯ α .Remark . Recall that ¯ α ( X ) = α ( ¯ X ). Definition 8.5.
Given t a θ -stable CSA, relative to θ , α ∈ ∆( g , t ) is:(1) real if θα = − α (2) imaginary if θα = α (3) complex if θα = ± α . Definition 8.6.
Given g ∈ G , recall α g := Ad ∗ g − α : t g = Ad g t → C . Relative to g , α is: (1) real if ¯ α g = − α g (2) imaginary if ¯ α g = α g (3) complex if ¯ α g = ± α g . Notation 8.7.
Let v ∈ V and let n = ˙ v − θ ( ˙ v ), and w = nT . Since v ∈ V , it follows that θ ( w ) = w − . e study the particular case where g = ˙ v ∈ V . Lemma 8.8. ( [Spr85] ) If v ∈ V , then ˙ vT ˙ v − is a θ -stable Cartan subgroup. This allows us to describe root types using θ . Definition 8.9. If v ∈ V , then relative to v α is:(1) real if θα v = − α v (2) imaginary if θα v = α v (3) complex if θα v = ± α v .since T ˙ v is θ -stable.Equivalently, Definition 8.10. ([Spr85])
Relative to v ∈ V (or w = ˙ v − θ ( ˙ v ) T ∈ W G ) α is:(1) real if wθα = − α (2) imaginary if wθα = α (3) complex if wθα = ± α . Proposition 8.11.
The previous two definitions for real, complex, and imaginary are con-sistent.Proof.
Let T = vT v − , which is θ -stable because v ∈ V . Then − ¯ α v = θα v . Given t ∈ T , θα v ( t ) = α v ( θ − ( t )) = α ( v − θ − ( t ) v )whereas for t = v − t v ∈ T , wθα ( t ) = θα ( n − tn ) = α ( θ − ( θ ( v ) − v ) θ − ( t ) θ − ( v − θ ( v )))= α ( v − θ − ( vtv − ) v )= α ( v − θ − ( t ) v ) . Since α v ( t ) = α ( t ), we see therefore that θα v ( t ) = α v ( t ) ⇐⇒ wθα ( t ) = α ( t ) θα v ( t ) = − α v ( t ) ⇐⇒ wθα ( t ) = − α ( t ) θα v ( t ) = ± α v ( t ) ⇐⇒ wθα ( t ) = ± α ( t ) . (cid:3) It follows from these computations that:
Corollary 8.12.
For α ∈ Π , v ∈ V , w = ˙ v − θ ( ˙ v ) T ∈ W , wθα = ( θα v ) v − . We may further distinguish imaginary roots as compact or noncompact.
Definition 8.13.
Let α be an imaginary root. Normalizing the one-parameter subgroup x α appropriately, θ ( x α ( ξ )) = x θα ( c α ξ ) = x α ( c α ξ ) where c α = ± . The root α is said to be compact imaginary if c α = 1 and noncompact imaginary if c α = − efinition 8.14. Suppose the root α is imaginary relative to v ∈ V . Then, normalizing x α v appropriately, θ ( x α v ( ξ )) = x θα v ( c α v ξ ) = x α v ( c α v ξ ) where c α v = ± . We say that α is compact relative to v if c α v = 1 and noncompact if c α v = − v ∈ V and n ∈ N ( T ), then vn − ∈ V , so there is a left W G -actionon V and also on V [Spr85]. Definition 8.15.
The cross action on K \ G/B corresponds to Springer’s W G -action on V .That is, s α × K ˙ vB = K ˙ v ˙ s − α B. Suppose α ∈ Π is noncompact imaginary relative to v ∈ V . In section 6.7 of [Spr85],Springer defines the automorphism ψ ( g ) = ˙ v − θ ( ˙ v ) θ ( g ) θ ( ˙ v ) − ˙ v . We observe that this issimply θ n ( g ) := int ( n ) ◦ θ ( g ). Since θ ( n ) = n − , this is an involutive automorphism. It isnow easy to see, as Springer pointed out, that ψ descends to an involutive automorphism of G α , the subgroup corresponding to ± α , since α is imaginary relative to θ n . Springer showsthat ψ ( x α ( m )) = x α ( − m ), ψ ( x − α ( m )) = x − α ( m ), and ψ ( ˙ s α ) = ˙ s − α . Springer claims thatthere is z α ∈ G α such that z α ψ ( z α ) − = ˙ s α . We see we may choose z α = x α ( − x − α (1 / z α ψ ( z α ) − = x α ( − x − α (1 / x − α (1 / x α ( −
1) = ˙ s α . Then: Definition 8.16.
Given v ∈ V , α ∈ Π noncompact imaginary relative to v , the Cayleytransform of v through α is c α ( K ˙ vB ) = K ˙ vz − α B. The Cayley transform is known to increase orbit dimension by 1.In section 4.3 of [RS90], one finds that:Case Type of α wrt v s α × v i) real s α × v = v ii) compact imaginary s α × v = v iii) complex s α × v = v iv.I) noncompact imaginary type I s α × v = v iv.II) noncompact imaginary type II s α × v = v Also,Case Root Type of α v π − α π α O v i.I) real type I O v ∪ O v ′ where v = c α ( v ′ )i.II) real type II O v ∪ O v ′ ∪ s α × O v ′ where v = c α ( v ′ ) = c α ( s α × v ′ )ii) compact imaginary O v iii.a) complex downward O v ∪ s α × O v iii.b) complex upward O v ∪ s α × O v iv.I) noncompact imaginary type I O ∪ s α × O ∪ c α O iv.II) noncompact imaginary type II O ∪ c α O See [PSY] for more detail on the choice of nomenclature.Thus to simplify the definition of type I and type II roots, we define:
Definition 8.17.
If the simple root α is noncompact imaginary relative to v ∈ V , then α issaid to be:
1) type I relative to v if s α × v = v (2) type II relative to v otherwise.We also define types I and II for real roots. If α is real relative to v , then α is said to be:(1) type I relative to v if α is type I relative to c α ( v )(2) type II relative to v otherwise.We have a richer theory for certain type II roots. Leading towards such results, we consider: Lemma 8.18. ( [Spr85] , p. 527) Recall ˙ s α was defined using one-parameter subgroups. Then: i) if α is real: θ ( ˙ s α ) = ˙ s − α ii) if α is compact imaginary: θ ( ˙ s α ) = ˙ s α iii) if α is complex θ ( ˙ s α ) = ˙ s θα iv) if α is noncompact imaginary: θ ( ˙ s α ) = ˙ s − α . Recall that ˙ s α = φ α (cid:18) −
11 0 (cid:19) . Since the matrix (cid:18) −
11 0 (cid:19) has order 4, therefore either˙ s α = 1 or ˙ s α = 1. Notation 8.19.
Let m α = ˙ s α .We are motivated by Lemma 14.11 of [AD09] to consider the following: Proposition 8.20.
Let α be a simple root such that m α = 1 . Then: (1) ˙ s − α = ˙ s α = ˙ s − α , (2) α is type II relative to all orbits, and (3) all the roots in the Weyl group orbit of α must be of type II.Proof. Since ˙ s α = φ α (cid:18) −
11 0 (cid:19) while ˙ s − α = φ α (cid:18) − (cid:19) , we see that ˙ s − α = ˙ s − α . Con-sidering order, we obtain the first equation.The condition m α = 1 is conjugation-invariant. That is, m α g = gm α g − = 1 as well forall g ∈ G . To prove the proposition, it suffices to prove that for any v ∈ V relative to which α is noncompact imaginary, s α × K ˙ vB = K ˙ vB . Observe that θ ( ˙ s α v ) = ˙ s α v by Lemma 8.18and (1) whence ˙ s α v ∈ K . Then K ˙ v ˙ s − α B = K ˙ s − α v ˙ vB = K ˙ vB . (cid:3) Remark . The last two statements of the proposition follow directly from Lemma 14.11of [AD09] as well.8.3.
Cross Actions and Cayley Transforms.
Simple relations for Bruhat order on K \ G/B may be described by cross actions and Cayley transforms.We list the results which may be found in [RS90]. ecall: either dim π − α π α ( O v ) = dim O v or dim π − α π α ( O v ) = dim O v + 1.Case Root Type of α v dim π − α π α O v i.I) real type I samei.II) real type II sameii) compact imaginary sameiii.a) complex downward sameiii.b) complex upward +1iv.I) noncompact imaginary type I +1iv.II) noncompact imaginary type II +1Case Root Type of α v π − α π α O Other Types Bruhat Relationiii.b) complex
O ∪ s α O α complex wrt. s α O O (cid:22) Kα s α O iv.I) noncpt type I O ∪ s α O ∪ c α O α real type I wrt. c α O O (cid:22) Kα c α O α noncpt type I wrt. s α O s α O (cid:22) Kα c α O iv.II) noncpt type II O ∪ c α O α real type II wrt. c α O O (cid:22) Kα c α O Theorem 8.22. If v (cid:22) Kα v ′ , then either: • v ′ = s α × v where α is type iii.b) relative to v , or • v ′ = c α × v where α is type iv) relative to v . O v type (cid:22) Kα O ′ v type Relationshipiv.I) (cid:22) Kα i.I) O v ′ = c α O v iv.II) (cid:22) Kα i.II) O v ′ = c α O v iii.b) (cid:22) Kα iii.a) O v ′ = s α O v Is there a simple means of explaining when we have an increase or a decrease in the Bruhatorder for K \ G/B ? We will see shortly that the answer is yes.8.4.
Weyl Group and Roots.Theorem 8.23.
Let v ∈ V and α ∈ Π . Recall that w = ˙ v − θ ( ˙ v ) T ∈ W G . Simple relationsfor Bruhat order on K \ G/B may be formulated by the existence of v ′ ∈ V such that: O v (cid:22) Kα O v ′ iff wθα > and g α v k O v (cid:23) Kα O v ′ iff wθα < and g α v k .If O v (cid:22) Kα O v ′ , then v ′ = s α × v if α is complex relative to v and v ′ = c α ( v ) if α is noncompactrelative to v .We note that this description of Bruhat order is analogous to the descriptions for B \ G/B and for P \ G/B as follows. The reductive subalgebra k plays an analogous role to l in P \ G/B and to t in B \ G/B . Furthermore, in the cases B \ G/B and P \ G/B , θ = Id .Proof. Consider the following table:Case Root Type of α v dim π − α π α O Combinatorial Descriptioni.I) real same wθα = − α < wθα = − α < wθα = α > g α v ⊂ k iii.a) complex same wθα < wθα > g α v k iv.I) noncompact imaginary type I +1 wθα = α > g α v k iv.II) noncompact imaginary type II +1 wθα = α > g α v k The real and imaginary cases follow immediately from definitions. n the complex case, either wθα > wθα <
0. Since B is θ -stable, therefore θ ∆ + = ∆ + .If wθα >
0, then θ ( wθα ) >
0. Since θ ( w ) = w − , therefore θ ( wθα ) = w − α . From w − α > ℓ ( s α w ) = ℓ ( w )+1. From θ ∆ + = ∆ + , we also conclude that θ Π = Π, whence θα is a simple root. Since α is complex relative to v so that wθα = ± α , therefore s α wθα > ℓ ( s α ws θα ) = ℓ ( s α w ) + 1 = ℓ ( w ) + 2. Similarly, if wθα <
0, then θ ( wθα ) = w − α < ℓ ( s α ws θα ) = ℓ ( s α w ) − ℓ ( w ) −
2. The complex case now follows from the caseanalysis in 4.3 of [RS90]. (cid:3)
Roots and Pullbacks.Theorem 8.24.
Let v ∈ V and α ∈ Π . Simple relations for Bruhat order on K \ G/B maybe formulated by the existence of v ′ ∈ V such that: O v (cid:22) Kα O v ′ if θα v > (i.e. ∈ ∆ + ( g , t ) v ) and g α v k O v (cid:23) Kα O v ′ if θα v < and g α v k .Proof. This follows from the previous theorem and from Corollary 8.12. (cid:3) K \ G/B in More Depth.
We review the discussion of monoids in [RS90] and studyhow we may specify elements of V by the monoidal action using our combinatorial results. Definition 8.25. ([RS90], 3.10) Given the Coxeter group (
W, S ), the monoid M ( W ) hasgenerators m ( s ) ( s ∈ S ) and the relations:(1) m ( s ) = m ( s ) s ∈ S ;(2) braid relations: if s, t ∈ S are distinct, then(a) o ( st ) = 2 k : ( m ( s ) m ( t )) k = ( m ( t ) m ( s )) k (b) o ( st ) = 2 k + 1: ( m ( s ) m ( t )) k m ( s ) = ( m ( t ) m ( s )) k m ( t ). Proposition 8.26. ( [RS90] , 3.10) (1) If w = s s . . . s ℓ is a reduced decomposition of w , then m ( w ) := m ( s ) m ( s ) · · · m ( s ℓ ) ∈ M ( W ) is independent of the reduced decomposition chosen. (2) M ( W ) = { m ( w ) : w ∈ W } . (3) m ( w ) m ( s ) = (cid:26) m ( ws ) if ws > wm ( w ) if ws < w. Definition 8.27. ([RS90], 4.7) There is an action of the monoid M ( W ) on V : if O v ′ is theunique dense orbit in KvP α , then m ( s α ) v = v ′ . Thus:If O v (cid:22) Kα O v ′ then m ( s α ) v = v ′ . Otherwise, m ( s α ) v = v. The monoidal action should be thought of in the following way. When considering Weylgroup actions, s ∈ S is self-inverse, so acting twice by s should return the original element.The action of s can both raise and lower dimensions. In contrast, the monoidal action of s ∈ S on v ∈ V only changes v if a cross action or Cayley transform corresponding to s raises the dimension. Thus repeated monoidal actions of s are the same as acting once. Thisagrees with m ( s ) = m ( s ). Considering a string of simple monoidal actions, we may alwaysremove the simple elements which do not raise dimension. As for the Weyl group action on B \ G/B and on P \ G/B , any element of V can be obtained by M ( W ) acting on the closedorbits in V . otation 8.28. Let V be the set of closed orbits in V . Definition 8.29. ([RS90], 4.1) The length of an element of V is defined as follows:(1) If v ∈ V , then ℓ ( v ) = 0.(2) If v = m ( s ) u where v = u , then ℓ ( v ) = ℓ ( u ) + 1. Definition 8.30. ([RS90]) A sequence in S is s = ( s , . . . , s k ). The length of s is k and m ( s ) = m ( s k ) · · · m ( s ). Definition 8.31. ([RS90], 5.2) Given u, v ∈ V , write u α −→ v or u s α −→ v if there exists x ∈ V and a sequence s ∈ S such that(1) u = m ( s ) x and ℓ ( u ) = ℓ ( x ) + ℓ ( s );(2) v = m ( s ) m ( s α ) x and ℓ ( v ) = ℓ ( x ) + ℓ ( s ) + 1.The relation defined by u ≤ v if there exists a sequence u = v α −→ v · · · α k −→ v k = v is the standard order on V .Richardson and Springer show in [RS90] that Bruhat order on K \ G/B and standard orderare the same.The inverse of a cross action is single valued. The inverse of a type II Cayley transformis single valued while the inverse of a type I Cayley transform is double valued. We wish tounderstand how elements of K \ G/B may be identified using sequences in S . Proposition 8.32.
Given a sequence in Vv s −→ v s −→ v · · · s k −→ v k where α k is noncompact type I relative to v k − , there is a sequence u s −→ u s −→ u · · · s k − −−→ u k − = s α k × v k − s k −→ v k = c α k ( v k − ) with each α j the same types relative to v j − and to u j − (eg. α k is noncompact type I relativeto both u k − and v k − ).Proof. Begin by letting w j = v − j θ ( v j ) T ∈ W G . Since α k is noncompact imaginary relativeto v k − , therefore w k − θα k = α k . Therefore ( v k − s − α k ) − θ ( v k − s − α k ) T = s α k w k − θ ( s α k ) = s α k s w k − θα k w k − = s α k w k − = w k − . Recall Definition 8.10. Thus if β is real, imaginary,or complex relative to v and α is non-compact relative to v , then β is real, imaginary, orcomplex, respectively, relative to s α × v . Our tables before Theorem 8.22 show that if β istype I relative to v , then it is type I relative to s α × v . If β is compact relative to v (that is, d ( θint ( v )) X β = d int ( v ) X β for X β ∈ g β ) and α is non-compact type I or type II relative to v , then we see that β is compact relative to s α × v : d ( θint ( v ˙ s − α )) X β = d ( θ ( int ( v ˙ s − α v − ) int ( v )) X β = d ( int ( θ ( ˙ s α v ) θint ( v )) X β = d ( int ( ˙ s α v ) int ( v )) X β = d ( int ( v ˙ s α )) X β = d ( int ( v ˙ s − α )) X β . e see that whatever type some simple root β is relative to v k − , it is precisely the same typerelative to s α k × v k − . Recall that c α k ( s α k × v k − ) = v k as well. Since only noncompact typeI roots cause ambiguity in taking inverses of cross actions and Cayley transforms, thereforeby induction, the proposition holds. (cid:3) Thus using our simple combinatorial descriptions of simple relations in Bruhat order, it iseasy to understand:
Remark . There are two general methods of specifying any element u ∈ V up to braidrelations:(1) There is u ∈ V and a sequence u s −→ u s −→ u · · · u k −→ u k = u. This specifies u unambiguously.(2) Let the unique open dense orbit in K \ G/B be KvB . There is a sequence v = u ℓ s ℓ ←− u ℓ − s ℓ − ←−− u ℓ − · · · s k +1 ←−− u k = u. A sequence moving downwards from the open orbit does not necessarily uniquelyidentify the orbit u since the inverse Cayley transform is double valued for typeI roots. To uniquely identify u , specify a choice for each type I inverse Cayleytransform. Corollary 8.34. If u ∈ V and w , w ∈ W are minimal length satisfying m ( w ) u = m ( w ) u ,then w = w . Bruhat Order on K \ G/P
Closure Order and the Order Induced From Bruhat Order on K \ G/B . Recallthat Bruhat order on K \ G/P is defined to be closure order.
Proposition 9.1.
Closure order on K \ G/P is the same as the partial order induced fromBruhat order on K \ G/B . That is, writing
KuP ≤ KvP if there are orbit representatives u and v , respectively, such that Ku B ≤ Kv B , KuP ≤ KvP ⇐⇒ KuP (cid:22) K KvP.
Proof. ⇐ : Since KuB ⊂ π − I π I ( KuB ) ⊂ π − I π I ( KvB ) ⊂ Kv B where Kv B is the uniquedense orbit in π − I ( KvP ) (which exists, as we will see in Corollary 9.12), we see that
KuB ⊂ Kv B . ⇒ : We observe that Ku B ⊂ Kv B ⇒ KuP = π I ( Ku B ) ⊂ π I ( Kv B ) = KvP . (cid:3)
Understanding
KvP : I -Equivalence. We wish to find a simple parametrization of K \ G/P . As we will see, the key to parametrizing K \ G/P is understanding the Bruhatorder of K \ G/B restricted to the B -orbits in a P -orbit.Since P ⊃ B , therefore each P -orbit KvP can be expressed as a union of B -orbits KuB .This is an example of an I -equivalence class, defined in the preprint [PSY] on generalizedHarish-Chandra modules: efinition 9.2. Recall the map π I : G/B → G/P , the natural projection from the flagvariety to the partial flag variety of parabolic subgroups of type I . Two orbits O , O ′ in K \ G/B are I -equivalent (write O ∼ I O ′ ) if they project to the same K -orbit on G/P ; i.e. π I ( O ) = π I ( O ′ ). The I -equivalence class of O is[ O ] ∼ I = K \ π − I ( π I ( O )) . In [PSY], each I -equivalence class Kv P = ∪ O∼ I O v O is shown to be in bijection withsome double coset space v M \ L/B ∩ L . The idea of considering such a bijection is due toLusztig-Vogan, according to [Mat82], p. 313. Note that these are v M -orbits on L/B ∩ L ,the flag variety of L . The bijection permits a generalization of the following commonly usedtechnique: when computing Kazhdan-Lusztig polynomials, often, the first step is to firstmake use of polynomials arising from smaller root subsystems. For example, to compute type A polynomials, begin by finding copies of A within A . Furthermore, the subgroup v M isa spherical subgroup of L and thus there is a unique open dense orbit in v M \ L/B ∩ L . Thebijection of I -equivalence classes with double coset spaces respects Bruhat order. Therefore,each I -equivalence class has a unique maximal element since v M \ L/B ∩ L has a uniquemaximal element. These maximal elements are easy to specify combinatorially, giving usa succinct parametrization of K \ G/P . We now proceed to provide more details. Becausewe study orbits of different subgroups on different flag varieties, we use superscripts todifferentiate the different orbit types by subgroup.
Definition 9.3. ([PSY]) Consider a parabolic subgroup and Levi decomposition P = LN where L carries an involution Θ (which may not be defined on G ). A mixed subgroup of G is a subgroup of the form M = L Θ N . Remark . Mixed subgroups generalize K , B , and P as follows. Select P = G and Θ = θ , then M = K .Select P = B and Θ = Id, then M = B . Select P = P and Θ = Id, then M = P . Proposition 9.5. ( [PSY] ) There is a bijection M \ G/B ↔ L Θ \ L / B ∩ L × W L \ W G . Morespecifically, M \ G/B ↔ L Θ \ L / B ∩ L × W L W G where the fibre product is with respect to the cross action of W L on L Θ \ L / B ∩ L .Remark . Our double cosets are in bijection with a smaller K \ G/B cross a Weyl groupquotient.
Corollary 9.7.
Mixed subgroups are spherical subgroups.
Definition 9.8. ([PSY]) Cross actions and Cayley transforms on M \ G/B may be defined bymultiplying orbit representatives on the right by ˙ s − α and by z − α , respectively, as before.As mentioned, an application of the theory of mixed subgroups is a bijection betweenorbits in an I -equivalence class and mixed subgroup orbits on the flag variety for a Levisubgroup. Theorem 9.9. ( [PSY] ) Given an I -equivalence class [ O M g ] ∼ I of M -orbits on G/B , there existsa mixed subgroup g M of L and a bijection g M \ L/B ∩ L ψ −→ M \ gP/B = [ O M g ] ∼ I uch that the following diagram commutes: L > g M \ L/B ∩ L M \ gP/B.ψ ∨ > The unlabelled maps are the natural maps arising by choosing orbit representatives from L and from gL . Furthermore, ψ (cid:0) s α × O g Mℓ (cid:1) = s α × ψ (cid:0) O g Mℓ (cid:1) and ψ (cid:0) c α (cid:0) O g Mℓ (cid:1)(cid:1) = c α (cid:0) ψ (cid:0) O g Mℓ (cid:1)(cid:1) . In the case where M = K , we may set: (1) g = v ∈ V to be a representative for [ O Kv ] ∼ I of minimal dimension (2) ˜ θ = int( v − ) ◦ θ ◦ int( v )(3) J = { α ∈ S : α v is real or imaginary } ∪ { α ∈ S : α v is complex and θ ( α v ) ∈ S v } . (4) P IJ = L J N IJ the T -stable Levi decomposition of the parabolic subgroup of L corre-sponding to J (5) v M = L ˜ θJ N IJ .Remark . I -equivalence permits us to decompose any M \ G/B into unions of smallermixed subgroup double coset spaces. In particular, iterating I -equivalence to simplify com-putations does not introduce any type of subgroup beyond mixed subgroups. For this reasonand since we may develop a rich theory for mixed subgroups (parametrizing orbits and un-derstanding Bruhat order very explicitly), we choose to use g M in bijections even thoughthere are other subgroups for which bijections with the orbits in an I -equivalence class aresimpler to prove. Remark . Compare this theorem and Proposition 9.5 with Brion and Helminck’s parametriza-tion of an I -equivalence class in the symmetric case, i.e. the B -orbits in KgP , in Proposition4 of section 1.5 of [BH00]. They set V g to be { x ∈ L ∩ ˜ θ ( L ) : x − ˜ θ ( x ) ∈ N L ( T ) ∩ ˜ θ ( L ) } and N g = { n ∈ N L ( T ) : B ∩ L ∩ ˜ θ ( L ) ⊂ n ( B ∩ L ) n − } . Then L ˜ θ \ V g /T × N g /T ↔ [ O Kg ] ∼ I . The first term in the product is a smaller K \ G/B while Brion-Helminck show the secondterm to be in bijection with orbits on
L/B ∩ L of the semidirect product of the unipotentradical of ˜ θ ( P ) ∩ L with L ˜ θ . That subgroup is not usually a mixed subgroup nor a parabolicsubgroup of L . Brion and Helminck do not impose the condition that O Kg is a minimaldimension equivalence class representative.Again, we saw that mixed subgroups are spherical subgroups. Thus: Corollary 9.12.
Each I -equivalence class of orbits has a unique orbit maximal with respectto Bruhat order. Thus each P -orbit KvP contains a unique dense B -orbit.Remark . This is equivalent to Proposition 2 of section 1.2 of [BH00]. .3. Parametrizing K \ G/P . There is a simple combinatorial parametrization of the uniquedense orbits in each I -equivalence class, and hence of K \ G/P . Theorem 9.14.
Let I ⊂ Π correspond to the standard parabolic P . Then the double cosetspace K \ G/P is in bijection with V P where V P := { v ∈ V : for every α ∈ I, wθα < where w = v − θ ( v ) T } = { v ∈ V : for every α ∈ I, m ( s α ) v = v } . In other words, K \ G/P is in bijection with the I -maximal elements of V .Proof. This follows immediately from the proposition and the corollary above and our char-acterization of Bruhat order for K \ G/B . (cid:3) Remark . In comparison, P \ G/B is in bijection with W I := { w ∈ W G : wα > α ∈ I } , the P -minimal elements of W G . As discussed, P \ G/B is in bijectionwith the unique maximal length coset representatives as well, giving us a parametrizationanalogous to our parametrization of K \ G/P . We may think of K \ G/P as the P -maximalelements of K \ G/B . Proposition 9.16. (cf. Proposition 5.3) Let u, v ∈ V P . Then KuP (cid:22)
KvP ⇐⇒ KuB (cid:22)
KvB.
Proof. ⇐ : This is clear from Lemma 3.9. ⇒ : Since KvB is dense in π − I ( KvP ), KuB ⊂ π − I ( KuP ) ⊂ π − I KvP ⊂ KvB. (cid:3)
Remark . This short topological proof works for P \ G/B as well.9.4.
Behaviour of Simple Relations: Descent of the Monoidal Action.
Since Bruhatorder for K \ G/P is induced from Bruhat order on K \ G/B , which can be described usingsimple relations, one concludes that Bruhat order for K \ G/P can be described using simplerelations as well. However, the absence of a Borel subgroup among the two subgroupswith respect to which we take double cosets complicates matters somewhat, obstructing thepossibility of making a natural definition for α −→ consistent among all coset representatives. Proposition 9.18. (1) If α ∈ I , then ˙ s α , z α ∈ L ⊂ P ; thus π I ( v ) = π I ( m ( s α ) v ) for all v ∈ V. (2) If α ∈ Π \ I , π I ( v ) = π I ( m ( s α ) v ) ⇐⇒ v = m ( s α ) v. Proof.
This follows immediately from Proposition 9.9. (cid:3)
Thus we may restrict our attention to simple relations in K \ G/B for α ∈ Π \ I .We consider defining cross action to be s α × KvP = Kv ˙ s − α P . Since for w ∈ W L , KvB s α × −−→ Kv ˙ s − α B ⊂ Kv ˙ s − α PKvwB s α × −−→ Kvw ˙ s − α BKvwB s w − α × −−−−→ Kv ˙ s − α wB ⊂ Kv ˙ s − α P, we see that the cross action does not descend naturally from K \ G/B to K \ G/P . emma 9.19. For α ∈ Π \ I , L normalizes N , so W L α ⊂ ∆( n , t ) . (1) For any w ∈ W L , the coefficient of α in the expression of wα as a linear combinationof simple roots is . (2) If β ∈ W L α is a simple root, then β = α . Proposition 9.20. If w ∈ W L , α ∈ Π \ I , and v ∈ V P , then O Pm ( s wα ) v = O Pm ( s α ) v . Proof.
By Lemma 9.19 and Lemma 5.3.3 of [Yee05], there is a reduced expression of s wα ofthe form s · · · s k s α s k · · · s where the s i ∈ W L . Since v is maximal, m ( s k · · · s ) v = v . Then m ( s wα ) v = m ( s · · · s k ) m ( s α ) m ( s k · · · s ) v = m ( s · · · s k ) m ( s α ) v . Since the monoidal actionby W L preserves P -orbits, the proposition follows. (cid:3) Proposition 9.21.
Let v ∈ V P and u ∈ V with u ∼ I v . Let w ∈ W L be of minimal lengthsuch that v = m ( w ) u . If α ∈ Π \ I , then O Pm ( s α ) v = O Pm ( s w − α ) v = O Pm ( s w − α ) u .Remark . It is tempting at this point, but incorrect, to conclude that the monoidalaction of W G on K \ G/B descends naturally to a monoidal action on K \ G/P as follows: • m ( s wα ) v = m ( ws α w − ) m ( w ) u ? = m ( w ) m ( s α ) u . • O Pm ( s wα ) v = O Pm ( s α ) v and O Pm ( w ) m ( s α ) u O Pm ( s α ) u so by Proposition 9.20, O Pm ( s α ) v = O Pm ( s α ) u .However, we cannot cancel inverses in M ( W G ), so the above argument is incorrect. It is easyto find a rank two counterexample for which O Pm ( s α ) v = O Pm ( s α ) u . Proposition 9.23. If v ∈ V P , α and β ∈ Π \ I , and α = β with v α −→ m ( s α ) v and v β −→ m ( s β ) v ,then O Pm ( s α ) v = O Pm ( s β ) v . Proof.
Assume by contradiction that m ( s α ) v and m ( s β ) v belong to the same P -orbit. Thenthere exist minimal length elements w α , w β ∈ W L such that m ( w α s α ) v = m ( w α ) m ( s α ) v = m ( w β ) m ( s β ) v = m ( w β s β ) v ∈ V P . Then by Corollary 8.34, w α s α = w β s β which implies that s α s β = w − α w β ∈ W L , giving s α = s β –contradiction. (cid:3) Conclusion
It would be interesting to apply the simplifications of Bruhat order to the study ofKazhdan-Lusztig-Vogan polynomials. The theory of parabolic Kazhdan-Lusztig polynomialsappears the most likely to benefit from the simplifications.Another topic for future consideration is to further explore the philosophy of provingresults for P \ G/B and for K \ G/B by reducing to B \ G/B using our analogies for simplerelations. For example, can it be applied to develop a better understanding of the exchangeproperty and the deletion condition for K \ G/B ?Can the theory for K \ G/B be simplified by using the Tits group?Can the theories for K \ G/P and P \ G/B be made more similar by recasting results for P \ G/B using maximal length representatives rather than minimal length representatives?The reader will find more material on Bruhat order in [PSY]. In particular, it contains adescription of Bruhat order for mixed subgroups (for which parabolic subgroups and symmet-ric subgroups are a special case) and for situations where one of the subgroups with respect o which we take double cosets is twisted by conjugation. The descriptions of Bruhat orderthrough pullbacks of roots in particular carries over to the twisted case very naturally. References [AD09] Jeffrey D. Adams and Fokko DuCloux. Algorithms for the representation theory of real reductiveLie groups.
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Department of Mathematics and Statistics, University of Windsor, Windsor, Ontario,CANADA
E-mail address : [email protected]@uwindsor.ca