aa r X i v : . [ m a t h . R T ] F e b B n − -BUNDLES ON THE FLAG VARIETY, I MARK COLARUSSO AND SAM EVENS Introduction

In this paper, we consider certain solvable subgroups acting on the ﬂag variety withﬁnitely many orbits. The subgroups we consider arise naturally in the complex Gelfand-Zeitlin integrable system and in the theory of Gelfand-Zeitlin modules. The goal of thispaper, and its sequel, is to provide a combinatorial description of the orbits and in par-ticular to understand monoidal actions on the orbits. We then hope to use these resultsto understand representations of the Lie algebra realized via the Beilinson-Bernstein cor-respondence using local systems on these orbits.In more detail, let G n = GL ( n ) or SO ( n ), and embed G n − as a symmetric subgroup of G n − , up to center in the GL ( n )-case. We note that the pairs ( G n , G n − ) are (up to centerand isogeny) the multiplicity free symmetric pairs. We let B n − denote a Borel subgroupof G n and recall that B n − acts on the ﬂag variety of Borel subgroups B n of G n withﬁnitely many orbits. Our main result is that B n − -orbits on B n ﬁbre over B n − -orbits ona generalized ﬂag variety of G n − with ﬁbre given by B m − -orbits on B m for some m < n .More precisely, we construct a class of parabolic subgroups of G n , which we call specialparabolic subgroups . Theorem 1.1.

For each B n − -orbit Q on B n , there is a special parabolic R of G n so that R ∩ G n − is a parabolic subgroup of G n − , and (1) there is a ﬁbre bundle Q → Q , where Q is a B n − -orbit on G n − /R ∩ G n − ; (2) the ﬁbre of Q → Q may be identiﬁed with a B m − -orbit Q on B m .If we denote the above B n − -orbit by Q = O ( Q , Q ) , then O ( Q , Q ) = O ( Q ′ , Q ′ ) if andonly if Q i = Q ′ i for i = 1 , . Our theorem allows for an inductive description of B n − -orbits on the ﬂag variety B n in terms of the special parabolics R , Weyl group data, and data derived inductively fromthe same problem for a smaller index. For SO (2 l + 1) and SO (2 l ), the special parabolicsare indexed by integers i = 1 , . . . , l + 1 and are given by taking parabolics containing theupper triangular matrices corresponding to the choice of simple roots { α j : i ≤ j ≤ l } using the usual ordering of simple roots. For GL ( n ), the special parabolics are given by Mathematics Subject Classiﬁcation.

Key words and phrases. K -orbits on ﬂag variety, algebraic group actions. choosing a consecutive subset of simple roots and a particular type of element of the Weylgroup.In a sequel to this paper, we use our results to give a combinatorial description of the B n − -orbits on B n . Further, we use the action of a monoid using both simple roots of G n − and of G n to show that every B n − -orbit arises from a zero dimensional B n − -orbitusing the monoid action. As a consequence of work of Richardson and Springer [RS90],we use this result to determine the closure relation between B n − -orbits in terms of themonoid action.In the GL ( n )-case, Hashimoto [Has04] describes the B n − -orbits on B n , and discussespart of the monoid action that we consider in this paper. Our work advances on [Has04]in that our approach also describes orbits both in general linear and orthogonal cases inan essentially uniform manner, elucidates the bundle structure, and giving a completedescription of the closure relation. It would also be interesting to relate our results tothe work of Gandini and Pezzini [GP18], who study the general case of orbits of solvablesubgroups of G with ﬁnitely many orbits on the ﬂag variety of G . In particular, it wouldbe of interest to understand some of the invariants they consider in their more generalcontext in our situation.In Section 2 of this paper, we introduce notation and recall standard facts about orbitsand embeddings of G n − in G n for later use. In Section 3, we establish the ﬁbre bundlestructure for B n − -orbits and use it to give an inductive method to describe B n − -orbitson B n . In Section 4, we discuss the monoid action and compute it in some cases. Theremaining cases are treated in the sequel to this paper using a more explicit combinatorialdescription of the orbits.We would like to thank Friedrich Knop, Jacopo Gandini and Guido Pezzini for usefuldiscussions which aided in the development of our results. During the preparation ofthis paper, The ﬁrst author was supported in part by the National Security Agency grantH98230-16-1-0002 and the second author was supported in part by the Simons Foundationgrant 359424. 2. Preliminaries

We discuss some notation and standard results that will be used throughout the paper.2.1.

Notation.

By convention, all algebraic groups and Lie algebras discussed in thispaper have complex coeﬃcients. If an algebraic group M is deﬁned, we denote by thecorresponding Gothic letter m its Lie algebra, and if a Lie subalgebra m of the Lie algebraof a complex Lie group G is given, we let M denote the connected subgroup of G withLie algebra m . For an algebraic group M and g ∈ M , we denote by Ad( g ) both theadjoint action of M on its Lie algebra, and the conjugation action of M on itself. Asa convention, we let B m denote the ﬂag variety of Borel subalgebras of the Lie algebra m , and let B n = B g n for g n = gl ( n ) or so ( n ). For ( V, β ) a vector space with symmetric n − -BUNDLES ON THE FLAG VARIETY, I 3 bilinear form β and subspace U , we let U ⊥ := { v ∈ V : β ( u, v ) = 0 ∀ u ∈ U } . Throughoutthe paper, we let ı denote a ﬁxed choice of √− Realizations.

We use the conventions for realizing so ( n ) from Chapters 1 and 2 of[GW98] (see also [CE19]). More explicitly, we let so ( n ) be the symmetry Lie algebra ofthe non-degenerate, symmetric bilinear form β on C n given by β ( x, y ) = x T J y , where J is the automorphism of C n such that J ( e i ) = e n +1 − i for the standard basis e , . . . , e n of C n and i = 1 , . . . , n . Note that β ( e j , e n +1 − j ) = 1 for j = 1 , . . . , n . We let SO ( n ) be theconnected algebraic group with Lie algebra so ( n ) . Then the subalgebra h n consisting ofdiagonal matrices in so ( n ) is a Cartan subalgebra, and the subalgebra b + consisting ofupper triangular matrices in so ( n ) is a Borel subalgebra, and refer to these as the standardCartan and Borel subalgebras. We refer also to h n and b + to denote the diagonal andupper triangular matrices in gl ( n ) . For g = gl ( n ) or so ( n ), any parabolic subalgebra p ⊃ b + we refer to as a standard parabolic subalgebra. We also let Φ = Φ( g , h n ) denotethe roots of g with respect to the Cartan subalgbra h n . If α ∈ Φ, we let g α denote theroot space corresponding to the root α .We recall the real rank one subalgebras for gl ( n ) and so ( n ). For g = gl ( n ), let t =diag[1 , . . . , , − ∈ GL ( n ), and let θ ( x ) = txt − for x ∈ g . Then the ﬁxed set g θ = gl ( n − ⊕ gl (1), regarded as block matrices, and g n = gl ( n − ⊕ g = so (2 l + 1), let t be an element of the Cartan subgroup with Lie algebra h l +1 with the property that Ad( t ) | g αi = id for i = 1 , . . . , l − t ) | g αl = − id. Considerthe involution θ l +1 := Ad( t ). Then k := g θ l +1 ∼ = so (2 l ), realized as block matrices withthe l + 1st rows and columns equal to 0 (see [Kna02], p. 700). Note that h l +1 = h l ⊂ k .For later use, let σ l +1 be the diagonal matrix with respect to the basis { e , . . . , e l +1 } suchthat σ l +1 · e j = e j for j = l +1 and σ l +1 · e l +1 = − e l +1 and note that θ = θ l +1 = Ad( σ l +1 ).In the case g = so (2 l ), consider the involutive linear map σ l : C l → C l such that σ l ( e l ) = e l +1 and σ l ( e i ) = e i for i = l or l + 1 . Deﬁne an involution θ of g by θ = Ad( σ l ).Then k := g θ l ∼ = so (2 l − θ l is the involution induced by the diagram automorphisminterchanging the simple roots α l − and α l (see [Kna02], p. 703). If we realize h l asdiagonal matrices of the form x = diag[ x , . . . , x l , − x l , . . . , − x ] and deﬁne for i = 1 , . . . , l , ǫ i ∈ h ∗ l by ǫ i ( x ) = x i , then the induced action of θ l is via θ l ( ǫ l ) = − ǫ l and θ l ( ǫ i ) = ǫ i for i = 1 , . . . , l −

1. We will omit the subscripts 2 l + 1 and 2 l from θ when g is understood.We also denote the involution of the corresponding classical group G by θ . For G = GL ( n ), G θ ∼ = GL ( n − × GL (1) is connected, while for G = SO ( n ), the group G θ ∼ = S ( O ( n − × O (1)) is disconnected. We let K := ( G θ ) be the identity component of G θ .Then K ∼ = SO ( n − K has Lie algebra k = g θ . Remark 2.1.

For SO ( n ) , note that we can realize the subgroups SO ( n − and SO ( n − as follows. For n either odd or even, note that the − eigenspace ( C n ) − σ n is onedimensional and is nondegenerate for the restriction of β . If ( C n ) − σ n = C v n , then since M. COLARUSSO AND S. EVENS the perpindicular of an SO ( n − -stable subspace is SO ( n − -stable, it follows that (2.1) SO ( n −

1) = { g ∈ SO ( n ) : g · v n = v n } . In the case n = 2 l, let V = C e l + C e l +1 . Then (2.2) SO (2 l −

2) = { g ∈ SO (2 l ) : g · v = v ∀ v ∈ V } . Here SO (2 l − is realized as determinant one orthogonal transformations of the orthog-onal complement V ⊥ to V . In the case n = 2 l + 1 , one similarly observes that (2.3) SO (2 l −

1) = { g ∈ SO (2 l + 1) : g · e l +1 = e l +1 , g · ( e l − e l +2 ) = e l − e l +2 . } For later use, we also note that if α , . . . , α l are the simple roots for SO (2 l + 1) , then thesimple roots for SO (2 l ) are given by α , . . . , α l − , α l − + 2 α l . We may take as root vectorsfor the last two simple roots of the Lie algebra so (2 l ) the vectors e α l − = E l − ,l − E l +2 ,l +3 and e α l = E l − ,l +2 − E l,l +3 and then their sum e α l − + e α l is the root vector for for so (2 l − for the simple root α l − . Description of K -orbits on B n in the multiplicity free setting. Note that thecases ( g , g n − ) = ( gl ( n ) , gl ( n − so ( n ) , so ( n − g has multiplicity free branching lawwhen regarded as a representation of a symmetric subalgebra k . By abuse of notation, weuse k to denote g n − and K to denote G n − . This property is well-known to be equivalentto the property that a Borel subgroup B K of K has ﬁnitely many orbits on B g , the varietyof Borel subalgebras of g (see for example, Proposition 3.1.3 of [CE19]). Here K is theconnected subgroup of Int( g ) with Lie algebra k .We will need to use an explicit description of the K -orbits on B g . We present thismostly without proof, and note that it follows from explicit study of the Richardson-Springer discussion of orbits in [RS90], and is worked out explicitly for GL ( n ) in Section4.4 of [CE14] (see especially Example 4.16, and Examples 4.30 to 4.32) and for SO ( n ) insection 2.7 of [CE].We ﬁrst recall the monoidal action. Suppose M is a subgroup of G acting with ﬁnitelymany orbits on the ﬂag variety B g of Borel subalgebras of g , the Lie algebra of connectedreductive group G . For a simple root α for G , consider the P -bundle p : B g → P α, g ,where P α, g consists of the parabolic subalgebras of g with Lie algebra G -conjugate to b + g − α , where b is a Borel subalgebra for which α is a simple root. We call theparabolic subalgebras in P α, g simple parabolics of type α . Then if Q = M · b is a M -orbitin B g , p − p ( Q ) contains a unique open M -orbit, and we call this orbit m ( s α ) ∗ ( Q ) andnote that if m ( s α ) ∗ Q = Q , then dim( m ( s α ) ∗ Q ) = dim( Q ) + 1 . We let p α = p ( b ) andlet P α denote the corresponding parabolic subgroup of G with unipotent radical U α . Wedenote by M α the image of M ∩ P α in the group S α := P α /U α locally isomorphic to SL (2).Then one of the following occurs.(1) Complex Case: M α contains a maximal unipotent subgroup of S α and is containedin a Borel subgroup of S α . In this case, m ( s α ) ∗ Q meets p − p ( b ) in a M α -orbitisomorphic to C and the complement is a point; n − -BUNDLES ON THE FLAG VARIETY, I 5 (2) Noncompact/Real Case: M α is locally isomorphic to a maximal torus of S α . Inthis case, m ( s α ) ∗ Q meets p − p ( b ) in a M α -orbit isomorphic to C × , and if M α is atorus, the complement is a union of two M α -orbits, and otherwise, the complementis a single M α -orbit ;(3) Compact Case: M α = S α . In this case, m ( s α ) ∗ Q = Q and p − p ( b ) ∼ = P is a single M α -orbit. Deﬁnition 2.2. (1)

In the complex case above, we say that the simple root α is com-plex stable for Q if m ( s α ) ∗ Q = Q and complex unstable for Q if m ( s α ) ∗ Q = Q ;(2) In the noncompact/real case above, we say that the simple root α is noncompactfor Q if m ( s α ) ∗ Q = Q and real for Q if m ( s α ) ∗ Q = Q ;(3) In the compact case above, we say the simple root α is compact for Q , and notethat in this case m ( s α ) ∗ Q = Q. The type of a simple root α for Q depends only on Q and not on the choice of b . Weintroduce some notation needed to describe K -orbits on B g . Remark 2.3.

When M is the ﬁxed point set of an involution G , then our description ofroots by type coincides with the usual deﬁnition from real groups [RS90] . Notation 2.4.

For each simple root α , choose root vectors e α ∈ g α , f α ∈ g − α , and h α = [ e α , f α ] so that e α , f α , h α spans a subalgebra of g isomorphic to sl (2) . Consider theunique Lie algebra homomorphism φ α : sl (2) → g such that: (2.4) φ α : (cid:20) (cid:21) → e α , φ α : (cid:20) (cid:21) → f α , φ α : (cid:20) − (cid:21) → h α Let φ α : SL (2) → G be the induced Lie group homomorphism, and let (2.5) u α = φ α (cid:18) √ (cid:20) ıı (cid:21)(cid:19) . In the remainder of this section, we consider the cases where K = G n − as above,and consider the monoid action relative to the K -action on B n . We now introduce somenotation which will be used throughout the paper. Notation 2.5.

Let F = ( V = { } ⊂ V ⊂ · · · ⊂ V i ⊂ · · · ⊂ V n = C n ) be a full ﬂag in C n , with dim V i = i and V i = span { v , . . . , v i } , with each v j ∈ C n . Wewill denote this ﬂag F by F = ( v ⊂ v ⊂ · · · ⊂ v i ⊂ v i +1 ⊂ · · · ⊂ v n ) . We will also make use of the following notation for partial ﬂags in C n . Let P = ( V = { } ⊂ V ⊂ · · · ⊂ V i ⊂ · · · ⊂ V k = C n ) M. COLARUSSO AND S. EVENS denote a k -step partial ﬂag with dim V j = i j and V j = span { v , . . . , v i j } for j = 1 , . . . , k .Then we denote P as P = ( v , . . . , v i ⊂ v i +1 , . . . , v i ⊂ · · · ⊂ v i k − +1 , . . . , v i k . ) Notation 2.6.

Let T be the maximal torus of G with Lie algebra t , and let W be theWeyl group with respect to T . For an element w ∈ W , let ˙ w ∈ N G ( T ) be a representativeof w . If m ⊂ g is a Lie subalgebra which is normalized by T , then Ad( ˙ w ) m is independentof the choice of representative of w , and we denote it by w ( m ) . Similarly, we denote by w ( M ) the corresponding group. Proposition 2.7.

Let g = gl ( n ) and k = gl ( n − . (1) The number of K -orbits on B is (cid:0) n +12 (cid:1) . (2) For i = 1 , . . . , n , let w i be the cycle ( n, n − , . . . i ) in the symmetric group S n , andlet b i := w i ( b + ) . The distinct closed K -orbits on B are the K -orbits K · b i so thereare exactly n closed K -orbits. Further, the Borel subalgebra b i is the stabilizer ofthe ﬂag: (2.6) F i := ( e ⊂ · · · ⊂ e i − ⊂ e n |{z} i ⊂ e i ⊂ · · · ⊂ e n − ) . (3) The non-closed K -orbits are of the form: (2.7) Q i,j = m ( s α j − ) . . . m ( s α i ) · Q i with ≤ i < j ≤ n . The codimension of Q i,j is n − − ( j − i ) . Further, the Borelsubalgebra: (2.8) b i,j := Ad(ˆ v )( b + ) ∈ Q i,j , ˆ v := ˙ w i · u α i · ˙ s α i +1 . . . ˙ s α j − , and b i,j is the stabilizer of the ﬂag: (2.9) F i,j := ( e ⊂ · · · ⊂ e i − ⊂ e i + e n | {z } i ⊂ e i +1 ⊂ · · · ⊂ e j − ⊂ e n |{z} j ⊂ e j ⊂ · · · ⊂ e n − ) . In particular, the unique open orbit is Q ,n , and it contains the Borel subalgebra b ,n = Ad( w )Ad( u α ) s α . . . s α n − ( b + ) which stabilizes the ﬂag: (2.10) F ,n := ( e + e n ⊂ e ⊂ · · · ⊂ e n − ⊂ e n ) . The following diagram indicates the K -orbits on B , together with the order relationsgiven by closure. For general n , the diagram has the same shape, with n closed orbits, n − n − -BUNDLES ON THE FLAG VARIETY, I 7 Q Q Q Q Q , Q , Q , Q , Q , Q , We now classify the orbits of K = SO ( n −

1) on the ﬂag variety for g = so ( n ). When n = 2 l + 1 is odd, there is a bijection between B so ( n ) and the set of all maximal isotropicﬂags in C n , i.e., partial ﬂags of the form IF = ( V ⊂ · · · ⊂ V i ⊂ · · · ⊂ V l ) , with dim( V i ) = i and B ( u, w ) = 0 for all u, w ∈ V i . When n = 2 l is even, the spaceof maximal isotropic ﬂags in C n consists of two SO (2 n )-orbits. To distinguish them, wechoose a maximal isotropic ﬂag U ⊂ · · · ⊂ U l , where U i is the span of the standardbasis e , . . . , e i from Section 2.2. Then we identify B so ( n ) with the maximal isotropic ﬂags V ⊂ · · · ⊂ V l such that dim( V l ∩ U l ) ≡ l (mod 2) . As before, if V i is the span of vectors { v , . . . , v i } , we write the above maximal isotropic ﬂag as IF = ( v ⊂ v ⊂ · · · ⊂ v [ n ] ) . We consider the type B and D cases separately, and use the enumeration of simpleroots in [GW98]. Proposition 2.8.

Let g = so (2 l + 1) and k = so (2 l ) . (1) There are exactly l + 2 K -orbits on the ﬂag variety B of g . (2) We let b − := s α l ( b + ) . Exactly two K -orbits on B are closed, and they are Q + := K · b + and Q − := K · b − . Further, m ( s α l ) · Q + = m ( s α l ) · Q − = K · Ad( u α l ) b + . (3) The non-closed orbits are of the form Q i := m ( s α i +1 ) · m ( s α i +2 ) · · · m ( s α l − ) · m ( s α l ) · Q + for i = 0 , . . . , l − . Moreover, the codimension of Q i in B is i . Further, (2.11) b i = Ad(ˆ v )( b + ) ∈ Q i , ˆ v := u α l ˙ s α l − ˙ s α l − . . . ˙ s α i +1 . In particular, the unique open K -orbit contains the Borel subalgebra (2.12) b = Ad( u α l ) s α l − s α l − . . . s α ( b + ) . Proposition 2.9.

Let g = so (2 l ) and k = so (2 l − . (1) There are exactly l K -orbits in the ﬂag variety B of g . (2) The orbit Q + := K · b + is the only closed K -orbit. M. COLARUSSO AND S. EVENS (3)

Let Q i := m ( s α i ) . . . m ( s α l − ) · Q + and let (2.13) b i = Ad(ˆ v )( b + ) , ˆ v := ˙ s α l − ˙ s α l − . . . ˙ s α i ( b + ) for i = 1 , . . . , l − . Then Q i = K · b i has codimension i − in B . The distinct K -orbits are Q + , Q l − , . . . , Q .In particular, the unique open orbit is Q and contains the Borel subalgebra (2.14) b = s α l − s α l − . . . s α ( b + ) . Remark 2.10.

In case g = so (2 l ) or so (2 l +1) , the Borel subalgebra b + corresponds to themaximal isotropic ﬂag ( e ⊂ e ⊂ · · · ⊂ e l ) . By Proposition 2.9, the Borel subalgebra b inthe open K -orbit in B so (2 l ) corresponds to the maximal isotropic ﬂag ( e l ⊂ e ⊂ · · · ⊂ e l − ) . Similarly, by Proposition 2.8, it follows that the Borel subalgebra b in the open K -orbitin B so (2 l +1) corresponds to the maximal isotropic ﬂag ( u α l ( e l ) ⊂ e ⊂ e ⊂ · · · ⊂ e l − ) . Wemay choose our root vectors so that C · u α l ( e l ) = C · ( e l + ı √ e l +1 + e l +2 ) . We consider the α l root vector e = E l,l +1 − E l +1 ,l +2 and take f to be the transpose of e , and let x = [ e, f ] , andnote that { e, x, f } generates the three dimensional subalgebra s α l := φ α l ( sl (2)) . Let V bethe span of e l , e l +1 , and e l +2 , and note that V is a representation of s α l and is isomorphicto the adjoint representation via an isomorphism mapping e l +2 → f , e l +1 → − x , and e l → e. A brief computation using the composed isomorphism sl (2) → so (3) → V abovethen gives the above formula for u α l ( e l ) . Structure of B n − -orbits Recall that G = GL ( n ) or SO ( n ) and K = G n − from Section 2.2. We show that B n − -orbits on the ﬂag variety B of G have a useful bundle structure and describe thedata required to classify orbits.3.1. Fibre Bundle Structure.

Let Q K = K · b ⊂ B be a K -orbit. Let R be a parabolicsubgroup of G containing B and consider the projection π : B → R := G/R . Then themorphism(3.1) φ : K × K ∩ R R/B → π − π ( Q K ) , φ ( x, b ) = Ad( x ) · b , is an isomorphism, and the preimage φ − ( Q K ) is K × K ∩ R ( K ∩ R ) · b . This realizes both π − π ( Q K ) and Q K as bundles over K/K ∩ R. Let R = L · U be a Levi decompositionof the parabolic R . Recall that the morphism R · b → B l given by b b ∩ l is anisomorphism, where b ∈ R · b ∼ = B R . We now assume that R is θ -stable. Then by Theorem 2 of [BH00], there is a θ -stableLevi subgroup L of R , K ∩ R is a parabolic subgroup of K , and K ∩ R = ( K ∩ L ) · ( K ∩ U )is a Levi decomposition of K ∩ R . The group K ∩ R then acts on B l through its quotient K ∩ L , and it follows that we can identify Q K with K × K ∩ R ( K ∩ L ) · ( b ∩ l ) . Since K ∩ L is a symmetric subgroup of L , K ∩ L acts with ﬁnitely many orbits on B l . We now prove our basic result about K -orbits on B for the multiplicity free pairs ( G, K ). n − -BUNDLES ON THE FLAG VARIETY, I 9 Theorem 3.1.

Let Q K = K · b be a K -orbit on B . Then there exists a θ -stable parabolicsubgroup R ⊃ B with θ -stable Levi decomposition R = L · U such that the pair ( l , l ∩ k ) is of the same type as ( g , k ) up to abelian factors in the center of l and the K ∩ L -orbit ( K ∩ L ) · ( b ∩ l ) is open in the ﬂag variety B l . Thus, (3.2) Q K ∼ = K × K ∩ R ( K ∩ L ) · ( b ∩ l ) . is a K -homogenous ﬁbre bundle over the partial ﬂag variety K/K ∩ R with ﬁbre the open K ∩ L -orbit on B l .Proof. If Q K is closed, then B is θ -stable by Corollary 6.6 of [Spr85], and then l is abelianso that B l = { b ∩ l } is a point, and the assertion follows.When Q K is not closed, we consider the diﬀerent types A, B, and D separately. In typeA, by Proposition 2.7, the orbit Q K = Q i,j = K · b i,j for some i, j with 1 ≤ i < j ≤ n , and b i,j is the Borel subalgebra in Equation (2.8). Let P i,j be the standard parabolic subgroupof GL ( n ) which is the stabilizer of the ﬂag P i,j := ( e ⊂ · · · ⊂ e i − ⊂ { e i , . . . , e j } ⊂ e j +1 ⊂ · · · ⊂ e n )and corresponds to the choice of simple roots { α i , . . . , α j − } . Recall the cycle w i = ( n, n − , . . . i ) from Proposition 2.7(2) and let r = r i,j := w i ( p i,j ). Then r i,j is the Lie algebrastabilizer of the partial ﬂag w i ( P i,j ) which is(3.3) R i,j := ( e ⊂ · · · ⊂ { e i , e i +1 , . . . , e j − , e n } ⊂ e j ⊂ e j +1 ⊂ · · · ⊂ e n − . By Proposition 2.7(3), b i,j is the stabilizer of the full ﬂag in Equation (2.9). It followsthat b i,j also stabilizes the partial ﬂag R i,j in (3.3), establishing the claim. Since theCartan subalgebra h n is contained in r i,j , it follows that r i,j is θ -stable. Further, r i,j has Levi decomposition r i,j = l ⊕ u , where l ∼ = gl (1) i − ⊕ gl ( j − i + 1) ⊕ gl (1) n − j . Itfollows from (3.3) that r i,j ∩ k is a standard parabolic subalgebra of k with Levi factor l ∩ k ∼ = gl (1) i − ⊕ gl ( j − i ) ⊕ gl (1) n − j . To prove the Theorem when g = gl ( n ), it remains to show that ( K ∩ L ) · ( b i,j ∩ l ) is openin the ﬂag variety of l . In the Levi decomposition of r i,j above, gl ( j − i + 1) is identiﬁedwith endomorphisms of the span of { e i , . . . , e j − , e n } . Observe that b i,j ∩ gl ( j − i + 1) isthe stabilizer of the ﬂag e i + e n ⊂ e i +1 ⊂ · · · ⊂ e j − ⊂ e n in gl ( j − i + 1) . By part (3) of Proposition 2.7, the point b i,j ∩ gl ( j − i + 1) is in the open K ∩ GL ( j − i + 1)-orbit on B gl ( j − i +1) . Hence, b i,j ∩ l is in the open K ∩ L -orbit on the ﬂagvariety B l of l .For the type B case, the relevant symmetric pair is ( so (2 l + 1) , so (2 l )). Recall fromProposition 2.8 that since Q K is not closed, Q K = Q i with i < l where Q i = K · b i =Ad( u α l ) s α l − . . . s α i +1 ( b + ). Let r i ⊂ g be the standard parabolic subalgebra generated by b + and the negative simple root spaces g − α l , g − α l − , . . . , g − α i +1 . Thus,(3.4) r i = l ⊕ u with l ∼ = so (2( l − i ) + 1) ⊕ gl (1) i . Note that r i is θ -stable and that θ | [ l , l ] = θ l − i )+1 . Thus, r i ∩ k is a parabolic subalgebraof k with Levi factor:(3.5) l ∩ k ∼ = so (2( l − i )) ⊕ ( gl (1)) i , so that K ∩ L ∼ = SO (2( l − × GL (1) i . To see that b i ⊂ r i , note that we can choosethe representative ˙ s α j of s α j so that ˙ s α j ∈ L for j = i + 1 , . . . , l , and u α l ∈ L by Equation(2.5). Thus, the element(3.6) v := u α l ˙ s α l − . . . ˙ s α i +1 ∈ L ⊂ R. Hence, b i = Ad( v ) b + ⊂ Ad( v ) r i = r i .It remains to show that ( K ∩ L ) · ( b ∩ l ) can be identiﬁed with the open K ∩ L -orbit inthe ﬂag variety B l . Note that b + ∩ l can be identiﬁed with the standard Borel subalgebra b + , l of upper triangular matrices in l . Since the element v in Equation (3.6) is in L , wehave: b ∩ l = (Ad( v ) b + ) ∩ l = Ad( v )( b + ∩ l ) = Ad( v ) b + , l . It follows from Equations (2.12) and (3.6) that Ad( v ) b + , l ⊂ B l is a representative of theopen K ∩ L -orbit on B l .The type D case is similar to type B. In this setting, g = so (2 l ) and k = so (2 l − Q K is a K -orbit that is not closed, then by Proposition 2.9, Q K = Q i has codimension i − < l −

1. By part (3) of Proposition 2.9, we can take b = b i = s α l − s α l − . . . s α i ( b + ).Let r i ⊂ g be the standard parabolic subalgebra generated by b + and the negative simpleroot spaces g − α l , g − α l − , . . . , g − α i . Then r i has the Levi decomposition:(3.7) r i = l ⊕ u with l ∼ = so (2( l − i ) + 2) ⊕ ( gl (1)) i − . We claim that r i is θ -stable. Indeed, we saw in Section 2.2 that θ ( ǫ l ) = − ǫ l and θ ( ǫ k ) = ǫ k for k = l . It follows that the roots α i are imaginary for i = 1 , . . . , l − α l − and α l are complex θ -stable with θ ( α l − ) = α l . It then follows easily that r i has Levidecomposition (3.7) and that θ | l ss = θ l − i )+2 , whence l θ = k ∩ l ∼ = so (2( l − i )+1) ⊕ gl (1) i − ,and ( L θ ) = K ∩ L ∼ = SO (2( l − i ) + 1) × GL (1) i − . The remainder of the proof proceedsas in the previous case. Q.E.D.Remark 3.2.

In Proposition 12 of the paper [BH00] , Brion and Helminck prove a closelyrelated result which in the multiplicity free case, describes K -orbits on B as bundles over aclosed K -orbit with ﬁbre an open orbit in a smaller ﬂag variety. In the orthogonal cases,our result coincides with the result of [BH], but in the general linear case, our bundle isin many cases diﬀerent from theirs. Remark 3.3.

We call the parabolic subalgebras r i,j ⊃ b i,j and r i ⊃ b i constructed in theproof of Theorem 3.1 the special parabolic subalgebras of g . We also call the correspondingparabolic subgroups the special parabolic subgroups of G . Note that for r a special parabolic n − -BUNDLES ON THE FLAG VARIETY, I 11 subalgebra of g , the standard Borel subalgebra b n − of k is in r . Hence, r ∩ k is a standardparabolic subalgebra of k . Remark 3.4.

Theorem 3.1 fails for other real rank one symmetric pairs. For example,if g = sp (6) , and k = sp (4) ⊕ sp (2) , then there exists a K -orbit Q K = K · b where theminimal θ -stable parabolic subalgebra containing b is all of g . However, Q K is not open. For θ -stable parabolics R , K ∩ R is a parabolic of K and we let W K ∩ RK ⊂ W K bethe representatives of shortest length for the cosets in W K /W K ∩ R . Now we use the ﬁbrebundle structure of a K -orbit Q K given in (3.2) to describe the B n − -orbits Q containedin Q K as ﬁbre bundles. Theorem 3.5.

Let Q K = K · b be a K -orbit in B . Let r ⊃ b be a special parabolicsubalgebra, and let R ⊂ G be the corresponding parabolic subgroup with θ -stable Levidecomposition R = LU . Suppose Q is a B n − -orbit contained in Q K . Then Q ﬁbres overa B n − -orbit in the partial ﬂag variety K/K ∩ R of R with ﬁbre isomorphic to a B n − ∩ L -orbit contained in the open K ∩ L -orbit of the ﬂag variety B l of l . More precisely, (3.8) Q ∼ = B n − × B n − ∩ w ( R ) ( B n − ∩ w ( L )) · Ad( ˙ w ) b , where w ∈ W K ∩ RK , (3.9) B n − ∩ w ( L ) = w ( B n − ∩ L ) is a Borel subgroup of K ∩ w ( L ) , and Ad( ˙ w ) b is contained in the open K ∩ w ( L ) -orbit inthe ﬂag variety B w ( R ) ∼ = B w ( L ) of R .Further, we have a one-to-one correspondence: (3.10) { B n − − orbits on G/B } ←→ ` R special { B n − − orbits on K/ ( K ∩ R ) } × { B n − ∩ L − orbits in open K ∩ L − orbit on B l } Proof.

Consider the ﬁbre bundle Q K = K × K ∩ R ( K ∩ L ) · b in Equation (3.2). Let π | Q K : Q K → K/K ∩ R = Q K, r be the canonical projection. Let Q = B n − · b ′ be a B n − -orbit in Q K . Its image under π is a B n − -orbit in the partial ﬂag variety K/K ∩ R . Since W K ∩ RK indexes B n − -orbits in K/K ∩ R , then π ( Q ) = B n − · w ( r ) for some w ∈ W K ∩ RK , and we may assume that b ′ = w ( b ) by replacing b by a K ∩ L -conjugate. Then the ﬁbrebundle structure from Equation (3.1) identiﬁes Q as(3.11) Q ∼ = B n − × B n − ∩ w ( R ) ( B n − ∩ w ( R )) · Ad( ˙ w )( b ) . Now the group B n − ∩ w ( R ) acts on w ( R ) /w ( B ) ∼ = B w ( l ) through its image in the quotient w ( R ) /w ( U ). Denote this image by ˆ B w ( L ) . We claim that(3.12) ˆ B w ( L ) = B n − ∩ w ( L ) = w ( L ∩ B n − ) , so that ˆ B w ( L ) is a Borel subgroup in K ∩ w ( L ). To verify the claim, ﬁrst consider w ( l ) ∩ b n − .By Remark 3.3, the special parabolic r contains b n − , so b n − is a Borel subalgebra of r ∩ k . Let Φ + l ∩ k be the roots appearing in the Borel subalgebra b n − ∩ l . Since w ∈ W K ∩ RK , standard results imply that w (Φ + l ∩ k ) ⊂ Φ + b n − . Thus w ( l ∩ b n − ) ⊂ w ( l ) ∩ b n − . It followsthat w ( l ) ∩ b n − = w ( l ∩ b n − ), so that w ( l ∩ b n − ) is a Borel subalgebra of l ∩ k withcorresponding Borel subgroup w ( L ∩ B n − ) = w ( L ) ∩ B n − , yielding the second equality inEquation (3.12). To get the ﬁrst equality of Equation (3.12), we consider the Lie algebraˆ b w ( L ) = Lie( ˆ B w ( L ) ). Since h n ∩ k ⊂ w ( r ) ∩ k , we have the decomposition w ( r ) ∩ b n − = w ( l ) ∩ b n − ⊕ w ( u ) ∩ b n − . Thus, ˆ b w ( L ) = w ( l ) ∩ b n − and Equation (3.12) follows. Equation (3.8) now follows from(3.11). Note that the orbit ( B n − ∩ w ( L )) · Ad( ˙ w ) b in the ﬁbre of the bundle in (3.8) isisomorphic via translation by ˙ w − to an orbit of the standard Borel subgroup B n − ∩ L of L ∩ K contained in the open K ∩ L -orbit of B l .The one-to-one correspondence in Equation (3.10) follows easily. Q.E.D.Notation 3.6.

We use the notation of Theorem 3.5. Given a B n − − orbit Q r in K/K ∩ R and a B n − ∩ L -orbit Q l in B l , we let O ( Q r , Q l ) be the corresponding B n − -orbit on B from this theorem. Remark 3.7.

Let b ˆ v be one of the representatives for b i,j , b i , or b + for K -orbits on B discussed in Propositions 2.7, 2.8, and 2.9. Let Q be a B n − -orbit in one of the above K -orbits Q K and let b ˆ v = Ad(ˆ v ) b + . Then (3.13) Q = B n − · Ad( ˙ w )Ad( ℓ ) b ˆ v = B n − · Ad( ˙ wℓ ˆ v ) b + for a representative ˙ w of unique w ∈ W K ∩ RK and for ℓ ∈ K ∩ L such that Ad( ℓ )( b ˆ v ∩ l ) is a representative of the B n − ∩ L -orbit Q l contained in the open K ∩ L -orbit on theﬂag variety B l of l . In the notation given above in 3.6, the orbit Q r = B n − · w ( r ) , and Q l = ( B n − ∩ L )Ad( ℓ )( b ˆ v ∩ l ) . By Equation (3.10), the problem of describing the B n − -orbits on G/B reduces to thedescription of B n − -orbits on partial ﬂag varieties K/ ( K ∩ R ) via W KR ∩ K and the problemof determining B n − -orbits in the open K -orbit in B . We now address this second issue.3.2. B n − -orbits in the open K -orbit. Recall the pair (

G, K ) with G = G n and K = G n − from Section 2.2. Theorem 3.8.

Let ˜ Q K be the open K -orbit on B n . There are bijections (3.14) { B n − − orbits on e Q K } ←→ B n − \ K/B n − ←→ { B n − − orbits on B n − } , where the last equivalence is induced by the self-map g g − of K .Proof. To prove the ﬁrst equivalence, it suﬃces to ﬁnd a point b y in ˜ Q K with stabilizer B n − in K . The second equivalence is then clear. We prove the ﬁrst equivalence byconsidering the three cases separately. n − -BUNDLES ON THE FLAG VARIETY, I 13 When g = gl ( n ), by Proposition 2.7, the open K -orbit Q ,n on B n is the orbit throughthe ﬂag ( e + e n ⊂ e ⊂ · · · ⊂ e n − ⊂ e n ). Thus, Q ,n = GL ( n − · F y , where F y := ( e n − + e n ⊂ e ⊂ e ⊂ · · · ⊂ e n − ⊂ e n ) . Suppose that g ∈ GL ( n −

1) ﬁxes F y . For j = 1 , . . . , n −

1, let U j be the span of e , . . . , e j and note that GL ( n −

1) := { g ∈ GL ( n ) : g · e n = e n , and g · U n − = U n − } . Since g ﬁxes the line C ( e n − + e n ) in the ﬂagand g · e n = e n , we see that g · e n − = e n − . Since g ∈ GL ( n − g · e j ∈ U n − for j = 1 , . . . , n −

2. But also g · e j ∈ C ( e n − + e n ) + U j for j = 1 , . . . , n −

2. Thus, for j = 1 , . . . , n −

2, the vector g · e j ∈ U n − ∩ ( C ( e n − + e n ) + U j ) = U j . Hence, g stabilizes U n − , so g ∈ GL ( n − g stabilizes the standard ﬂag ( e ⊂ e ⊂ · · · ⊂ e n − ) in U n − . Thus, g ∈ B n − . Since B n − ﬁxes F y , this proves the Theorem in case of GL ( n ).Next, consider the case g = so (2 l ) and recall the basis e , . . . , e l . By Remark 2.10,the open SO (2 l − B so (2 l ) is identiﬁed with the SO (2 l − IF y := ( e l ⊂ e ⊂ · · · ⊂ e l − ) . Let g ∈ SO (2 l − y , the stabilizerof IF y . Since g ∈ SO (2 l − g commutes with the element σ l on C l (see Section2.2). Since g stabilizes the ﬂag IF y , we see that g · e l = ae l for some a ∈ C × . Since g commutes with σ l , it follows that g · e l +1 = ae l +1 , and since β ( e l , e l +1 ) = 1, the scalar a = ±

1. But g ∈ SO (2 l − g ﬁxes the vector e l − e l +1 . Thus, a = 1and by Equation (2.2), g ∈ SO (2 l − , so that SO (2 l − y = SO (2 l − y . But B l − is clearly in SO (2 l − y so since SO (2 l − y is solvable and B l − is a Borel subgroupof SO (2 l − , it follows that B l − = SO (2 l − y = SO (2 l − y , proving the requiredassertion.Now let g = so (2 l + 1) and recall the basis e , . . . , e l +1 from 2.2. By Remark 2.10, theopen orbit is the SO (2 l )-orbit through the point B so (2 l +1) corresponding to the maximalisotropic ﬂag ( e l + √ ıe l +1 + e l +2 ) ⊂ e ⊂ e ⊂ · · · ⊂ e l − ) . We can choose an element s of SO (2 l ) in the diagonal torus that s · ( e l + √ ıe l +1 + e l +2 ) = ( ie l + √ ıe l +1 + − ie l +2 ), sothat the open orbit is the SO (2 l )-orbit through the point y in B so (2 l +1) corresponding tothe maximal isotropic ﬂag IF y = ( ie l + √ ıe l +1 − ie l +2 ⊂ e ⊂ e ⊂ · · · ⊂ e l − ). Note thatby a small calculation using the root vectors in b l − , we see that B l − is contained inthe stabilizer SO (2 l ) y and let g ∈ SO (2 l ) y . Since g ∈ SO (2 l ), then g · e l +1 = e l +1 . Since g stabilizes the line in the ﬂag, we see that g · ( e l − e l +2 ) = e l − e l +2 , so that g ∈ SO (2 l − B l − = SO (2 l − y . Q.E.D.Remark 3.9.

For later use, we make explicit the correspondence between B n − -orbitson the open K -orbit e Q K in B n and B n − -orbits on B n − given in Equation (3.14). Let Q = B n − · Ad( k )˜ b ⊂ e Q K = K · ˜ b , where ˜ b is the stabilizer of the (isotropic) ﬂag F y ( IF y )given in the proof of Theorem 3.8. Denote the corresponding B n − -orbit in B n − by Q op .Note that Q op = B n − · Ad( k − ) · b n − . We can realize this correspondence geometricallyas follows. Let q : K → e Q K ∼ = K/B n − and π : K → B n − be the natural projections, andlet ψ : K → K be the inversion map, i.e., ψ ( k ) = k − . Then (3.15) Q op = π ( ψ ( q − ( Q ))) . More generally, if

Y ⊂ e Q K is any B n − -stable subvariety, it corresponds to a B n − -stablesubvariety of B n − : (3.16) Y op = π ( ψ ( q − ( Y ))) . Using (3.16), we note that (3.17) dim Y op = dim Y + dim b n − − dim b n − . Note the easily veriﬁed numerical identity (3.18) dim b n − − dim b n − = rank ( k ) . Equation (3.17) then becomes (3.19) dim Y op = dim Y − rank ( k ) . The monoid action for B n − -orbits on B n In this section, we use Theorems 3.5 and Theorem 3.8 to describe several monoid actionsfor the B n − -orbits on B n = B g n . One of the major points is to compute the monoid actionon an B n − -orbit O ( Q r , Q l ) in B n in terms of other monoidal actions. In later work, weapply this action to the study the order relation on B n − -orbits on B n .4.1. Left and Right Monoid Actions.

For a group A acting on a variety X , let O A ( X )denote the A -orbits on X . Let K ∆ ⊂ G × K be the diagonal copy of K in the product G × K . Note that there is a canonical bijection χ : O B n − ( B ) → O K ∆ ( B × B K ) , given by χ ( Q ) = K ∆ · ( Q, eB n − ) and we let Q ∆ denote χ ( Q ). Further, Q ∆ ∼ = K × B n − Q ,and it follows that the map Q Q ∆ preserves topological properties like closure relationsand open sets, and dim( Q ∆ ) = dim( Q ) + dim( B n − ) . In particular, K ∆ has ﬁnitely many orbits on B×B n − , and hence the set O K ∆ ( B×B n − )has 2 diﬀerent monoidal actions, coming from the monoid actions for the two diﬀerentkinds of simple roots of g × k via the generalities from Section 2.3. We decompose thesimple roots Π g × k = Π k ∪ Π g . Notation 4.1.

For a simple root α ∈ Π k , we call the monoid action Q ∆ → m ( s α ) ∗ Q ∆ a left monoid action. If α ∈ Π g , we call the monoid action Q ∆ → m ( s α ) ∗ Q ∆ a rightmonoid action. We use the equivalence between O B n − ( B ) and O K ∆ ( B × B n − ) to similarlydeﬁne left and right monoid actions m ( s α ) on O B n − ( B ) . In more detail, for α ∈ Π k , the left action uses the projection p α : G/B n × K/B n − → G/B n × K/P α where P α is the parabolic of K with Lie algebra b n − + k − α . Thus, if Q ∆ = K ∆ · ( Q, eB n − ), then p − α p α ( Q ∆ ) = K ∆ · ( Q, P α /B n − ) = K ∆ · ( P α · Q, eB n − ) . n − -BUNDLES ON THE FLAG VARIETY, I 15 It follows that for α ∈ Π k , and Q = B n − xB n /B n .(4.1) m ( s α ) ∗ Q is the unique open B n − − orbit in P α · Q = P α xB n /B n . Similar analysis shows that if Q = B n − xB n /B n and α ∈ Π g , then(4.2) m ( s α ) ∗ Q is the unique open B n − − orbit in B n − xP α /B n , where P α here is the parabolic of G whose Lie algebra is b n + g − α . For a useful generalization of the right monoid action, we recall the notion of stronglyorthogonal roots.

Deﬁnition 4.2.

Two nonproportional roots α, β ∈ Φ( g , h ) are said to be strongly orthog-onal if α ± β is not a root. Recall that by elementary arguments, strongly orthogonal roots are necessarily orthog-onal and the corresponding sl (2) subalgebras, s α and s β , commute. We also make use ofthe following notation. Notation 4.3.

Let S ⊂ Π g be a subset of the simple roots of g , and let m S ⊂ g be thestandard Levi subalgebra of g given by the subset S . We denote the corresponding standardparabolic subalgebra by p S ⊃ b + and its Levi decomposition by p S = m S ⊕ u S , and if S isunderstood, we let m = m S and u = u S . Let P S , M S , U S be the corresponding connectedsubgroups of G. Remark 4.4.

Suppose S ⊂ Π g is a subset of simple roots, let α ∈ Π g be a simple rootwhich is strongly orthogonal to the roots in S , and let S ′ = S ∪ { α } . The projection G/P S G/P S ′ is a P -bundle. Hence, if H is a subgroup of G acting on G/P S withﬁnitely many orbits, there is a right monoid action m ( s α ) on O H ( G/P S ) . Remark 4.5.

There are additional monoid actions we will use. Let α ∈ Π k , and considerthe ﬁnite set O B n − ( K/P ) where P is a standard parabolic subgroup of K . As above, thereis a bijection χ : O B n − ( K/P ) → O K ∆ ( K/B n − × K/P ) given by χ ( Q ) = K ∆ · ( eB n − , Q ) =: Q n − . This bijection satisﬁes similar properties asthe above bijection also denoted χ .(1) It follows that if α ∈ Π k is a simple root of Π k × k with root space in the ﬁrst factor,then there is another left monoid action m ( s α ) on O K ( K/B n − × K/P and hence by theequivalence of orbits induced by this version of χ , there is a left monoid action m ( s α ) on O B n − ( K/P ) . Similar analysis to the previous left monoid action veriﬁes the followingassertion. For Q ∈ O B n − ( K/P ) , (4.3) m ( s α ) ∗ Q is the unique open orbit in P α · Q. (2) There is also a right monoid action in the strongly orthogonal setting. Again, welet α ∈ Π k . The standard parabolic P ⊂ K is P = P S for S ⊂ Π k . Suppose α isstrongly orthogonal to the roots of S . Then if the root space of α ∈ Π k × k is in thesecond factor, it follows from Remark 4.4 that there is a right monoid action m ( s α ) on O K ∆ ( K/B n − × K/P ) and hence on O B n − ( K/P ) via the bijection χ . It can be shownthat if Q = B n − · xP S /P S , then (4.4) m ( s α ) ∗ Q is the unique open orbit in B n − xP S ′ /P S , where P S ′ is the standard parabolic subgroup of K corresponding to the subset of simpleroots S ′ = S ∪ { α } . Remark 4.6.

We will abuse notation by using the same symbol m ( s α ) to denote boththe left monoid action by a root α ∈ Π k on O B n − ( B ) and for the left monoid action on O B n − ( K/P ) . This may require some parsing on the part of the reader. For example, if weconsider a B n − -orbit O ( Q r , Q l ) on B , then we will prove that in certain circumstances, m ( s α ) ∗ O ( Q r , Q l ) = O ( m ( s α ) ∗ Q r , Q l ) . In this equality, on the left hand side of theequality, the monoid action is the left monoid action on O B n − ( B ) and on the right handside of the equality, the monoid action is the left monoid action on O B n − ( K/P ) . Computation of Monoid Actions.

It follows from deﬁnitions that if α ∈ Π k and Q ⊂ Q K , then m ( s α ) ∗ Q ⊂ Q K . We now examine the aﬀect of the left monoid action onthe ﬁbre bundle structure of Q = O ( Q r , Q l ) ⊂ Q K after ﬁrst recalling some notation fromRemark 3.7. Suppose Q K = K · b , where b = b ˆ v = Ad(ˆ v ) b + ∈ B is the representativegiven in Propositions 2.7, 2.8, and 2.9 respectively. Let r ⊃ b be the special parabolicsubalgebra of g associated to the K -orbit Q K , and let r = l ⊕ u be its Levi decomposition.Then Q = B n − · Ad( ˙ w )Ad( ℓ ) b , where w ∈ W K ∩ RK and ℓ ∈ K ∩ L (see Equation (3.13)).Further Q r = B n − · w ( r ) and Q l = ( B n − ∩ L ) · Ad( ℓ )( b ˆ v ∩ l ). Theorem 4.7.

Let α be a root in Π k . Let π r : G/B + → G/R be the projection. Then (4.5) π r ( m ( s α ) ∗ O ( Q r , Q l )) = m ( s α ) ∗ Q r , where the monoid action on O B n − ( K/K ∩ R ) in the right hand side is given in Remark4.5.Suppose also that α ∈ w (Φ l ) . Then α is a simple root of k ∩ w ( l ) with respect to the Borelsubalgebra b n − ∩ w ( l ) of k ∩ w ( l ) , and (4.6) m ( s α ) ∗ O ( Q r , Q l ) = O ( Q r , m ( s w − ( α ) ) ∗ Q l ) . Suppose α w (Φ l ) . Then (4.7) m ( s α ) ∗ O ( Q r , Q l ) = O ( m ( s α ) ∗ Q r , Q l ) . Proof.

By Equation (4.1), m ( s α ) ∗ Q is the open B n − -orbit in the variety Y := P α ˙ wℓB/B .By equivariance of π r , its restriction π r : Y → P α ˙ wR/R is open since quotient morphismsare open (Section 12.2 of [Hum75]). Thus, m ( s α ) ∗ Q projects to the open B n − -orbit in P α ˙ wR/R . Under the identiﬁcation Q K, r ∼ = K/K ∩ R , the latter orbit is identiﬁed with n − -BUNDLES ON THE FLAG VARIETY, I 17 the open B n − -orbit in P α ˙ w ( K ∩ R ) / ( K ∩ R ). But this orbit is exactly m ( s α ) ∗ Q r byEquation (4.3). Equation (4.5) follows.Note that the P α -orbit Y ﬁbres over P α / ( P α ∩ w ( R )) via the identiﬁcation(4.8) Y ∼ = P α × P α ∩ w ( R ) ( P α ∩ w ( L ))Ad( ˙ wℓ ) b . Indeed, the identiﬁcation

Y ∼ = P α × P α ∩ w ( R ) ( P α ∩ w ( R ))Ad( ˙ wℓ ) b is formal, and since thediagonal torus of K normalizes P α , w ( L ) , and w ( U ), it follows that the action of P α ∩ w ( R )on Ad( ˙ wℓ ) b is through P α ∩ w ( L ). Now suppose that α is a root of w ( l ) ∩ k . It follows that P α wR/R = B n − wR/R , whence m ( s α ) ∗ Q r = Q r . Further the ﬁbre bundle in Equation(4.8) may be written as(4.9) Y ∼ = B n − × B n − ∩ w ( R ) ( P α ∩ w ( L ))Ad( ˙ wℓ )( b ∩ l ) . We observed in Equation (3.9) that B n − ∩ w ( L ) is a Borel subgroup of the Levi subgroup K ∩ w ( L ) of K . Since P α ∩ w ( L ) ⊃ B n − ∩ w ( L ) and α ∈ Π k , it immediately follows that P α ∩ w ( L ) is a parabolic subgroup of K ∩ w ( L ) and α is a simple root with respect to theBorel subalgebra b n − ∩ w ( l ) of k ∩ w ( l ).Equation (4.9) implies that(4.10) m ( s α ) ∗ Q ∼ = B n − × B n − ∩ w ( R ) O , where O is the open B n − ∩ w ( R )-orbit in the variety ( P α ∩ w ( L ))Ad( ˙ wℓ )( b ∩ l ) ⊂ B w ( l ) . Now B n − ∩ w ( R ) acts on B w ( l ) through its image in the quotient w ( R ) /w ( U ) ∼ = B w ( l ) . Inthe proof of Theorem 3.5, we observed that this image is the Borel subgroup B n − ∩ w ( L )of w ( L ) ∩ K (see Equation (3.12)). Equation (4.6) now follows by translating by ˙ w − , asis prescribed in the last comments of the proof of Theorem 3.5, and from Equations (3.9)and (4.1).We now suppose that α is not a root of k ∩ w ( l ). Then P α ∩ w ( L ) = B n − ∩ w ( L ). Thus,we can rewrite Equation (4.8) as Y ∼ = P α × P α ∩ Ad( ˙ w ) R ( B n − ∩ w ( L ))Ad( ˙ wℓ ) b = P α × P α ∩ w ( R ) Ad( ˙ w ) Q l , where the last equality uses the deﬁnition of Q l . Therefore, the B n − -orbits in Y are in one-to-one correspondence with the B n − -orbits on the base P α wR/R ∼ = P α w ( K ∩ R ) / ( K ∩ R ).Thus, m ( s α ) ∗ Q = π r | − Y ( m ( s α ) ∗ Q r ) = O ( m ( s α ) ∗ Q r , Q l ) , yielding (4.7). Q.E.D.

To understand how the bundle structure of Q ⊂ Q K behaves with respect to the rightmonoid action by roots α ∈ Π g , we need some preparation. Proposition 4.8.

Let α ∈ Π g . Suppose that Q ⊂ Q K . Then (4.11) m ( s α ) ∗ Q ⊂ m ( s α ) ∗ Q K In particular, if m ( s α ) ∗ Q K = Q K , then m ( s α ) ∗ Q = Q . Proof.

Let Q = B n − xB n /B n . It follows from deﬁnitions that m ( s α ) ∗ Q K is the uniqueopen K -orbit in the variety KxP α /B n . Since B n − xP α /B n ⊂ KxP α /B n , the subvariety m ( s α ) ∗ Q K ∩ B n − xP α /B n is non-empty and open in B n − xP α /B n . Since B n − xP α /B n isirreducible, it follows from Equation (4.2) that m ( s α ) ∗ Q ∩ m ( s α ) ∗ Q K = ∅ . This impliesequation (4.11).

Q.E.D.

By Proposition 4.8, if α ∈ Π g and m ( s α ) ∗ Q K = Q K , then m ( s α ) ∗ Q ⊂ Q K . Forsuch α , we study the monoid action by m ( s α ) in terms of the ﬁbre bundle description ofthe orbit O ( Q r , Q l ) of Q . But ﬁrst we need to determine the roots α ∈ Π g for which m ( s α ) ∗ Q K = Q K . The following results are well-known and partly appear in tables onpage 92 of [Col85]. For the case of g = gl ( n ), proofs are given in Examples 4.16 and 4.30of [CE14] and for the type B and D cases, see Propositions 2.23 and 2.24 of [CE]. Proposition 4.9. (1)

Let g = gl ( n ) . First, consider the case when Q K = Q i if aclosed K -orbit as in Part (2) of Proposition 2.7. Then the roots α i − and α i arenon-compact for Q K and all other simple roots are compact. Next, consider thecase when Q K = Q i,j as in Part (3) of Proposition 2.7. Then the roots α i − and α j are complex stable for Q K , the roots α i and α j − are complex unstable, and allother simple roots are compact for Q K . (2) Let g = so (2 ℓ + 1) . If Q K is one of the two closed K -orbits given in Part (2)of Proposition 2.8, then the root α ℓ is non-compact for Q K , and all other simpleroots are compact. Consider the orbit Q K = Q i as in Part (3) of Proposition 2.8.Then the roots α i and α i +1 are complex for Q i . Further, the root α i is stable, andthe root α i +1 is unstable and all other simple roots are compact for Q K . (3) Let g = so (2 ℓ ) . In the case Q K = Q + is the unique closed orbit, then the roots α ℓ − and α ℓ are complex stable for Q + and all other simple roots are compact. Inthe case Q K = Q i as in Part (3) of Proposition 2.9, then the roots α i − and α i are complex for Q i with α i − stable and α i unstable. All other simple roots arecompact for Q i . We use Proposition 4.9 to study the relation between roots α ∈ Π g such that m ( s α ) ∗ Q K = Q K and the special parabolic subalgebras r ⊂ g constructed in Theorem 3.1. Let˜ w ∈ W be given as follows:(4.12) ˜ w = w i = ( n, n − , . . . , i + 1 , i ) if g = gl ( n ) , and Q K = Q i,j or Q i id if g = so ( n ) , and Q K = Q i or Q + s α l if g = so (2 l + 1) , and Q K = Q − , and let ˙˜ w be a ﬁxed representative of ˜ w . Then it follows from the construction of thespecial parabolic subalgebra r in Theorem 3.1 that r = ˙˜ w ( p S ), where p S is a standard n − -BUNDLES ON THE FLAG VARIETY, I 19 parabolic subalgebra of g deﬁned by the subset S ⊂ Π g given by:(4.13) S = { α i , . . . , α j − } for g = gl ( n ) , Q K = Q i,j { α i , . . . , α ℓ } for g = so (2 l ) , Q K = Q i { α i +1 , . . . , α ℓ } for g = so (2 l + 1) , Q K = Q i , and where S = ∅ and r is a Borel if Q K is closed. Proposition 4.9 implies the followingobservation. Corollary 4.10.

Let S ⊂ Π g be the subset of simple roots given in (4.13). Then (1) If α ∈ S , m ( s α ) ∗ Q K = Q K . (2) If α ∈ Π g \ S and m ( s α ) ∗ Q K = Q K , then α is compact for Q K . We can now describe the right monoid action on a B n − -orbit Q ⊂ Q K by a simpleroot α ∈ Π g with m ( s α ) ∗ Q K = Q K . Recalling notation from Theorem 4.7, we let Q = O ( Q r , Q l ) = B n − · Ad( ˙ w )Ad( ℓ ˆ v ) b + , where w ∈ W K ∩ RK and ℓ ∈ K ∩ L , with Q r = B n − · w ( r ) and Q l = ( B n − ∩ L ) · Ad( ℓ )( b ˆ v ∩ l ), and ˆ v is given in Propositions 2.7,2.8, 2.9 Theorem 4.11.

Suppose α ∈ Π g . (1) If α ∈ S , then (4.14) m ( s α ) ∗ O ( Q r , Q l ) = O ( Q r , m ( s ˜ w ( α ) ) ∗ Q l ) . (2) If α ∈ Π g \ S and m ( s α ) ∗ Q K = Q K , then ˜ w ( α ) ∈ Π k is a standard simple rootstrongly orthogonal to the roots of S , and (4.15) m ( s α ) ∗ O ( Q r , Q l ) = O ( m ( s ˜ w ( α ) ) ∗ Q r , Q l ) , where m ( s ˜ w ( α ) ) ∗ Q r is the right monoid action of the simple root ˜ w ( α ) on the B n − -orbit Q r in the partial ﬂag variety Q K, r = K/K ∩ R of k deﬁned in Remark4.5.Proof. First, suppose that α ∈ S . Then by Corollary 4.10, m ( s α ) ∗ Q K = Q K . Recall thatby Equation (4.2) m ( s α ) ∗ Q is the open B n − -orbit in the variety Z := B n − ˙ wℓ ˆ vP α /B + .Since α ∈ S , P α ⊂ P S . Further, the representative ˆ v = ˙˜ wp , where ˙˜ w is the aboverepresentative of ˜ w given by (4.12) and p ∈ M S (see Equations (2.8), (2.11), and (2.13)).It follows that the special parabolic subalgebra r = Ad(ˆ v ) pm . Thus, π r : G/B → G/R maps Z to B n − Ad( ˙ w )( r ) = Q r and hence π r : Z → Q r is a B n − -equivariant ﬁbrebundle. We call the ﬁbre F , and using Equation (3.8) and Remark 3.7, we can identify F = ( B n − ∩ w ( L ))( ˙ wℓ ˆ vP α · b + . Using Equation (3.9), we see F is isomorphic to(4.16) ( B n − ∩ L ) ℓ ˆ vP α · b + ∼ = ( B n − ∩ L ) ℓ ˆ v ( P α ∩ M S ) / ( B + ∩ M S ) . The open B n − -orbit in Z ﬁbres over Q r with ﬁbre isomorphic to the open B n − ∩ L -orbitin the variety F . Using the decomposition ˆ v = ˙˜ wp , we can rewrite F as(4.17) ( B n − ∩ L ) ℓ ˜ wp ˜ w − ( P ˜ w ( α ) ∩ L ) / (Ad( ˜ w ) B + ∩ L ) . But the open B n − ∩ L -orbit in (4.17) is exactly m ( s ˜ w ( α ) ) ∗ Q l , using Equation (4.2).Equation (4.14) follows.Let α ∈ Π g \ S with m ( s α ) ∗ Q K = Q K . By (4.2), m ( s α ) ∗ Q K is the open B n − -orbitin the variety(4.18) Y = B n − ˙ wℓ ˆ vP α · b + = B n − ˙ wℓP Ad(ˆ v ) α · b . We begin by analyzing the homogeneous space P Ad(ˆ v ) α /B . First, consider the root Ad(ˆ v ) α .We claim(4.19) Ad(ˆ v ) α = ˜ w ( α ) ∈ Π k , where ˜ w ∈ W is given by (4.12). We show (4.19) in the case where g = gl ( n ). If Q K = Q i is a closed K -orbit as in Part (2) of Proposition 2.7, then ˆ v = w i = ˜ w . Since m ( s α ) ∗ Q K = Q K , then by Proposition 4.9, α = α j , j = i − , i . It is routine to checkthat ˜ w ( α j ) = w i ( α j ) = α j − for i < j ≤ n and ˜ w ( α j ) = α j for j < i − . In either case,˜ w ( α ) ∈ Π k = { α , . . . , α n − } . Now suppose Q K = Q i,j is not closed as in Part (3) ofProposition 2.7. Then ˆ v = w i u α i +1 s α i +1 . . . s α j − by Equation (2.8). Then since α ∈ Π g \ S with m ( s α ) ∗ Q K = Q K , then it follows from Deﬁnition 4.2 and Proposition 4.9 that α is strongly orthogonal to the roots of Π m = { α i , . . . , α j − } (see Equation (4.13)). Inparticular, α is orthogonal to the roots of Π m , so Ad(ˆ v ) α = ˜ wu α i ( α ). Further, since α isstrongly orthogonal to α i , Proposiiton 6.72 of [Kna02] implies that u α i ( α ) = α , and weconclude that Ad(ˆ v ) α = ˜ w ( α ). The remainder of the argument proceeds as in the casewhere Q K = Q i is closed, and the claim in (4.19) is established in the gl ( n ) case. Theorthogonal cases are simpler and can be handled in an analogous fashion.Now it follows from Part (2) of Corollary 4.10 that α is compact for Q K . It then followsfrom our discussion in Section 2.3 that the space P ˜ w ( α ) is a single orbit of the Levi factorof P ˜ w ( α ) . Let M α ⊂ P α be the H n -stable Levi factor, so that M ˜ w ( α ) := Ad( ˜ w ) M α ⊂ K is aLevi factor of P ˜ w ( α ) . Thus, the variety Y in Equation (4.18) equals Y = B n − ˙ wℓM ˜ w ( α ) · b .Since α is strongly orthogonal to the roots of S , it follows that ˜ w ( α ) is strongly orthogonalto the roots of l . Since L ∩ K ⊂ L , it follows that the groups M ˜ w ( α ) and L ∩ K commute.Therefore,(4.20) π r ( Y ) = B n − ˙ wM ˜ w ( α ) R ∩ K/R ∩ K. Now, we claim that(4.21) B n − ˙ wM ˜ w ( α ) R ∩ K/R ∩ K = B n − ˙ wS l ∩ k , ˜ w ( α ) ( K ∩ R ) / ( K ∩ R ) , where S l ∩ k , ˜ w ( α ) is the standard parabolic subgroup of K given by the subset of simpleroots Π l ∩ k ∪ { ˜ w ( α ) } . It follows that B n − ˙ wS l ∩ k , ˜ w ( α ) ( K ∩ R ) / ( K ∩ R ) = B n − ˙ wS ˜ w ( α ) ( K ∩ R ) / ( K ∩ R ) = B n − ˙ wM ˜ w ( α ) ( K ∩ R ) / ( K ∩ R ) ∼ = π r ( Y ) . Now by Equation (4.4), m ( s ˜ w ( α ) ) ∗ Q r is the open B n − -orbit in the variety given by B n − ˙ wS l ∩ k , ˜ w ( α ) ( K ∩ R ) / ( K ∩ R ) = π r ( Y ). To complete the proof of the theorem, weobserve that the map π r endows the variety Y with the structure of a B n − -homogeneous n − -BUNDLES ON THE FLAG VARIETY, I 21 ﬁbre bundle with smooth base B n − ˙ wS l ∩ k , ˜ w ( α ) ( K ∩ R ) / ( K ∩ R ) and ﬁbre Q l , Since π r | Y is aﬂat morphism it is open by Exercise III.9.1 of [Har77]. Thus, π r ( m ( s α ) ∗ Q ) = m ( s ˜ w ( α ) ) ∗ Q r and Equation (4.15) now follows. Q.E.D.

It follows from Theorems 4.7 and 4.11 that to understand the monoid action on Q = O ( Q r , Q l ) ⊂ Q K by roots α ∈ Π g ∪ Π k such that m ( s α ) ∗ Q K = Q K , it suﬃces tounderstand the monoid action on B n − -orbits in K/ ( K ∩ R ) discussed in Remark 4.5(1)as well as the monoid action on Q l . The former is well understood, and in the next result,we use the correspondence Q Q op in Equation (3.14) to understand the latter. For thestatement of the next result, we recall the particular representative ˜ b of the open K -orbitthat is used in the proof of Theorem 3.8 to establish the correspondence in (3.14). We let e Q K = K · ˜ b where ˜ b is the Borel subalgebra of g which stabilizes the ﬂag ˜ F :(4.22) ˜ F = e n − + e n ⊂ e ⊂ e ⊂ · · · ⊂ e n − ⊂ e n if g = gl ( n ) ,e l ⊂ e ⊂ e ⊂ · · · ⊂ e l − if g = so (2 l ) ıe l + √ ıe l +1 − ıe l +2 ⊂ e ⊂ e ⊂ · · · ⊂ e l − if g = so (2 l + 1) , Theorem 4.12.

Let Q = B n − · Ad( k )˜ b be a B n − -orbit in the open K -orbit e Q K on B n ,and let Q op = B n − · Ad( k − ) b n − be the corresponding B n − -orbit on B n − as in Remark3.9.Let α ∈ Π k , then (4.23) ( m ( s α ) ∗ Q ) op = m ( s α ) ∗ Q op ⊂ B n − , where m ( s α ) ∗ Q denotes the left monoid action of s α on the B n − -orbit Q in B n , and m ( s α ) ∗ Q op denotes the right monoid action of s α on the B n − -orbit Q op in B n − .Let α = α j ∈ Π g be a simple root which is compact for ˜ Q K . Then (4.24) ( m ( s α j ) ∗ Q ) op = m ( s α j − ) ∗ Q op , where m ( s α j ) ∗ Q is the right monoid action of s α j on Q and m ( s α j − ) ∗ Q op is the leftmonoid action of s α j − on Q op and α j − ∈ Π k n − is a standard simple root.Moreover, the correspondences in Equations (4.23) and (4.24) preserve root types. Thatis to say that α is complex stable, complex unstable, non-compact, etc for Q if and onlyif α is complex stable, unstable, non-compact, etc for Q op in (4.23) and similarly for α j and α j − in (4.24).Proof. Let α ∈ Π k . Then by Equation 4.1, we know m ( s α ) ∗ Q is the open B n − -orbit inthe variety Y Q := P α kB n − /B n − ⊂ ˜ Q . Under the correspondence in Equation (3.16), wehave Y opQ = B n − k − P α /B n − . Since m ( s α ) ∗ Q op is the open B n − -orbit in B n − k − P α /B n − , Equation (4.23) now followsfrom Equation (4.2). To see the assertion about root types, consider the variety Y Q = p − ( K · ( Q, eB n − )),where p : G/B × G n − /B n − → G/P α × G n − /B n − in case α ∈ Π g and p : G/B × G n − /B n − → G/B × G n − /P n − ,α in case α ∈ Π k . For α ∈ Π k , then α is complex stablefor Q if and only if Y Q consists of two double cosets with K · ( Q, eB n − ) codimension 1in the other double coset. This is the same as the condition for α to be complex stablefor Q op . The remaining cases are similar.Now we consider the right monoid action. Let α ∈ Π g be compact imaginary for theopen K -orbit e Q K = K · ˜ b . Let ˜ b = Ad( g ) b + . By Equation (4.2), m ( s α ) ∗ Q is the open B n − -orbit in the variety B n − kgP α · b + . Since α is compact for e Q K , gP α /B + is a single K ∩ Ad( g ) P α -orbit, so that gP α /B + ∼ = ( K ∩ Ad( g ) P α ) / ( K ∩ Ad( g ) B + ) . Therefore,(4.25) B n − kgP α · b + ∼ = B n − k (Ad( g ) P α ∩ K ) b n − . Using (3.16) and (4.25), it follows that(4.26) ( B n − kgP α · b + ) op = (Ad( g ) P α ∩ K ) k − B n − · b n − . We now claim that the group Ad( g ) P α ∩ K in (4.26) is the simple parabolic subgroup of K n − corresponding to the simple root α j − . Equation (4.24) now follows from (4.26). Weverify the claim in the case where g = gl ( n ). Suppose α = α j is compact for e Q K = Q ,n .By Proposition 4.9, it follows that j = 1 , n −

1. The image of the ﬂag from Equation(4.22) is the partial ﬂag PF z denoted e n − + e n ⊂ e ⊂ · · · ⊂ e j − ⊂ { e j − , e j } ⊂ e j +1 ⊂· · · ⊂ e n − ⊂ e n . As in the proof of Theorem 3.8, we compute the stabilizer G n − ,z of thispartial ﬂag in G n − . For g in G n − ,z , since g stabilizes the line spanned by e n − + e n , wesee that g · e n − = e n − . Recalling the notation setting U k equal to the span of e , . . . , e k ,we see as before that g · U k ⊂ U k + C ( e n − + e n ) for k = 1 , . . . , n − k = j − k = j − g · U j − ⊂ U j + C ( e n − + e n ) . As in Theorem 3.8, since g ∈ G n − andhence stabilizes the subspace U n − , we see that g ∈ G n − and the claim follows since thesubgroup of G n − stablilizing the partial ﬂag ( e ⊂ · · · ⊂ { e j − , e j − } ⊂ · · · ⊂ e n − ) is thesimple parabolic subgroup of G n − corresponding to α j − . The proofs of the claim in theorthogonal cases are similar, and are left to the reader. Q.E.D.Remark 4.13.

As a consequence of the above results, we have completely understood themonoid action for simple roots of k , and for simple roots of g in the subset S associatedto the orbit Q K by (4.13) , and for simple roots α of g not in S when m ( s α ) ∗ Q K = Q K .We will treat the remaining cases in a sequel to this paper. References [BH00] Michel Brion and Aloysius G. Helminck,

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