A generalization of Steinberg theory and an exotic moment map
aa r X i v : . [ m a t h . R T ] M a r A GENERALIZATION OF STEINBERG THEORYAND AN EXOTIC MOMENT MAP
LUCAS FRESSE AND KYO NISHIYAMA
Abstract.
For a reductive group G , Steinberg established a map from the Weyl groupto the set of nilpotent G -orbits by using moment maps on double flag varieties. Inparticular, in the case of the general linear group, it provides a geometric interpretationof the Robinson-Schensted correspondence between permutations and pairs of standardtableaux of the same shape.We extend Steinberg’s approach to the case of a symmetric pair ( G, K ) to obtain twodifferent maps, namely a generalized Steinberg map and an exotic moment map .Although the framework is general, in this paper we focus on the pair (
G, K ) =(GL n ( C ) , GL n ( C ) × GL n ( C )). Then the generalized Steinberg map is a map from partial permutations to the pairs of nilpotent orbits in gl n ( C ). It involves a generalization ofthe classical Robinson-Schensted correspondence to the case of partial permutations.The other map, the exotic moment map, establishes a combinatorial map from theset of partial permutations to that of signed Young diagrams, i.e., the set of nilpotent K -orbits in the Cartan space (Lie( G ) / Lie( K )) ∗ .We explain the geometric background of the theory and combinatorial algorithmswhich produce the above mentioned maps. Introduction G be a connected reductive algebraic group over C . Let T ⊂ G be a maximaltorus and let B ⊂ G be a Borel subgroup containing T . In [19] (see also [10]), Steinberggave a geometric correspondence between the Weyl group W := W ( G, T ) and the nilpo-tent orbits in the Lie algebra g := Lie( G ) in terms of the flag variety B := G/B (or ratherthe product
B × B ). Let us review it shortly.The action of G on B gives rise to a Hamiltonian action on the cotangent bundle T ∗ B ,which is a symplectic variety. Thus we get a G -equivariant moment map µ B : T ∗ B → g ∗ ∼ = g (see, e.g., [2]). Hereafter we identify g with g ∗ via a fixed nondegenerate invariantbilinear form on g . In fact, the image of µ B is contained in the nilpotent cone N = N g . Mathematics Subject Classification.
Key words and phrases.
Steinberg variety; conormal bundle; exotic moment map; nilpotent orbits;double flag variety; Robinson-Schensted correspondence; partial permutations.L. F. is supported in part by the ANR project GeoLie ANR-15-CE40-0012.K. N. is supported by JSPS KAKENHI Grant Number
Let us consider the double flag variety
B × B with the diagonal G -action. Then, againits cotangent bundle T ∗ ( B × B ) is a Hamiltonian G -variety, and we get a moment map µ B×B : T ∗ ( B × B ) → g . Note that, by functoriality, µ B×B is just a sum of two momentmaps on each T ∗ B . We denote the null fiber of the moment map by Y = µ − B×B (0), and callit a conormal variety (or Steinberg variety). We summarize the situation in the diagrambelow. T ∗ ( B × B ) pr / / pr (cid:15) (cid:15) µ B×B % % ❑❑❑❑❑❑ T ∗ B µ B (cid:15) (cid:15) T ∗ B µ B / / N ⊂ g µ B×B = µ B ◦ pr + µ B ◦ pr , Y = µ − B×B (0) = T ∗ B × N T ∗ B . It is interesting to note that µ B : T ∗ B → N is a resolution of singularities (the
Springerresolution ), so the Steinberg variety is a fiber product of two copies of this resolution.From general facts on conormal varieties (see Section 3), we conclude that the conormalvariety Y is equidimensional and its irreducible components are parametrized by the G -orbits in B × B , which are in one-to-one correspondence with the Weyl group W due tothe Bruhat decomposition (just note that ( B × B ) /G ∼ = B \ G/B ).The map π := µ B ◦ pr maps every irreducible component of Y to an irreducible G -stable closed subset in the nilpotent variety N g (using π := µ B ◦ pr leads to the sameresult). Since N g has only finitely many orbits (due to the Jacobson–Morozov Lemmacombined with Malcev’s Theorem), this image is the closure of a single nilpotent G -orbit.In this way, we finally obtain a map from the Weyl group to the set of nilpotent G -orbits:St : W → N g /G, which we call the Steinberg map .When G = GL n ( C ), the Weyl group W coincides with the symmetric group S n and thenilpotent orbits N g /G are classified in terms of the Jordan normal form, i.e., a partition λ ⊢ n . Let us denote the set of partitions of n by P ( n ). Then the Steinberg map inthis case gives a map from S n to P ( n ) which yields a geometric interpretation of theRobinson-Schensted correspondence [20]. A more detailed review of Steinberg’s theory isgiven in Section 4 below.1.2. One of our main goals in this paper is to extend Steinberg’s approach to the caseof a symmetric pair ( G, K ), where K is the subgroup of fixed points of an involution θ ∈ Aut( G ). The subgroup K is called a symmetric subgroup, and it is automaticallyreductive. Such pairs ( G, K ) are completely classified and the quotient space
G/K iscalled a symmetric space (see, e.g., [8]). In what follows, we assume for simplicity that K is connected (otherwise one should replace K with its connected component of theidentity). Note that K is connected, e.g., if G is semisimple and simply connected (see[18, § GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 3
Let P and Q be parabolic subgroups of G and K respectively. We define a double flagvariety for the symmetric pair as a product of two flag varieties X := K/Q × G/P.
The group K acts diagonally on X . As in Section 1.1, the Hamiltonian action of K on the cotangent bundle T ∗ X gives rise to a moment map µ X : T ∗ X → k ∗ ∼ = k , where k := Lie( K ) = g θ , and we define a conormal variety Y := µ − X (0) as the null fiber of thismoment map.In general there are infinitely many K -orbits in X , though in many interesting casesthe number of orbits is finite. In this case, we say that X is of finite type . A number ofexamples of double flag varieties of finite type are given in [14]; in the case where P or Q is a Borel subgroup, full classifications of double flag varieties of finite type are obtainedin [7].The condition that X is of finite type implies a lot of good properties of the conormalvariety Y . Proposition 1.1.
Assume that (1.1) X has a finite number of K -orbits.Then, the conormal variety Y is equidimensional and its irreducible components are inone-to-one correspondence with the K -orbits of X . More precisely, whenever O ⊂ X is a K -orbit, the closure of the conormal bundle T ∗ O X is an irreducible component of Y . For the proof, see Proposition 3.1.
Assumption 1.2.
Hereafter we assume condition (1.1) , i.e., we assume that the doubleflag variety X = K/Q × G/P is of finite type.
A clear difference of the present situation from that of Steinberg’s theory is the lack ofsymmetry in the definition of the double flag variety X .Let us denote the Cartan decomposition by g = k ⊕ s , where k is the Lie algebra of K or equivalently the (+1)-eigenspace of the involution θ (we denote the differential of θ ∈ Aut( G ) by the same letter) and s is the ( − s is not a Liealgebra, but only a vector subspace isomorphic to the tangent space of G/K at the basepoint eK . We summarize the situation in the diagram below. T ∗ X pr / / pr (cid:15) (cid:15) µ X (cid:30) (cid:30) ❃❃❃❃❃❃❃❃❃❃ T ∗ ( G/P ) µ G/P (cid:15) (cid:15) g ( · ) θ (cid:15) (cid:15) ( · ) − θ / / s T ∗ ( K/Q ) µ K/Q / / k g = k ⊕ s (where s = g − θ )with projection maps ( · ) θ and ( · ) − θ , µ X = µ K/Q ◦ pr + ( · ) θ ◦ µ G/P ◦ pr , Y = µ − X (0) . LUCAS FRESSE AND KYO NISHIYAMA
In the above diagram, there appear two maps π k := µ K/Q ◦ pr : Y → N k ⊂ k and π g := µ G/P ◦ pr : Y → N g ⊂ g , which do not play the same role. Actually π k = − ( · ) θ ◦ π g . We call π k the “symmetrizedmoment map”.Next we define π s := ( · ) − θ ◦ π g : Y → s , where ( · ) − θ denotes the projection to s along the Cartan decomposition g = k ⊕ s . Wecall π s the “exotic moment map”. Note that π s = π g + π k .In general the image of the exotic moment map π s is not necessarily contained in thenilpotent variety N s := N g ∩ s . Let us assume :(1.2) the image of π s is contained in the nilpotent variety N s .All the maps considered so far are K -equivariant. Note that both nilpotent varieties N k and N s consist of finitely many K -orbits (see, e.g., [3]). Then for any irreduciblecomponent of Y , its image by π k (resp. π s ) contains a dense nilpotent K -orbit. Conclusion.
Altogether, under assumptions (1.1) and (1.2) , we get two maps (1.3) Ξ k : X /K −→ N k /K and Ξ s : X /K −→ N s /K. We also call Ξ k a “symmetrized moment map” and Ξ s an “exotic moment map” by abuseof terminology. These maps generalize the Steinberg map W ∼ = ( B × B ) /G → N g /G of Section 1.1,which is recovered by considering the particular case where θ = id G (so that K = G and s = 0) and P = Q = B . Remark 1.3.
It is interesting to note that the situation considered by Steinberg is alsorecovered if we consider the symmetric pair (
G, K ) = ( G × G , ∆ G ), where G is aconnected reductive group and ∆ G stands for the diagonal embedding of G into G × G ,and we put P = B × G and Q = ∆ B , where B is a Borel subgroup of G .In practice, for a particular (e.g., classical) symmetric pair ( G, K ), a combinatorialdescription of the sets of nilpotent orbits N k /K or N s /K is known (see [3]). Then anatural problem is to find a combinatorial parametrization of the orbits X /K and todescribe the maps of (1.3) in an explicit way. Let us summarize this into a program: Program.
Let X = K/Q × G/P be the double flag variety corresponding to a symmetricpair ( G, K ) and a pair of parabolic subgroups ( P, Q ) as above, and assume the conditions (1.1) and (1.2) . (A) Find a combinatorial parametrization of the orbits X /K ; (B) Compute the maps Ξ k : X /K → N k /K and Ξ s : X /K → N s /K of (1.3) explicitly. GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 5
G, K ) = (GL n ( C ) , GL n ( C ) × GL n ( C ))and to establish new combinatorial bijections involving partial permutations . We considereverything over C , and sometimes we simply write GL n or M n instead of GL n ( C ) orM n ( C ) respectively. In fact, all the results in this paper are still valid if we replace C byany algebraically closed field of characteristic zero.Let us choose the parabolic subgroups P and Q as follows. • P = P S := (cid:26)(cid:18) a c b (cid:19) : a, b ∈ GL n , c ∈ M n (cid:27) = Stab G ( C n ⊕ { } ) is a maximalparabolic subgroup (Siegel parabolic subgroup); • Q = B K ⊂ K is a Borel subgroup.Thus K/B K is the full flag variety of K = GL n × GL n (i.e., the double flag varietyof GL n ) while G/P S ∼ = Gr n ( C n ) can be identified with the Grassmann variety of n -dimensional subspaces of C n . In this respect, our double flag variety is isomorphic to F ( C n ) × F ( C n ) × Gr n ( C n ) on which K = GL n × GL n acts.Note that In the situation under consideration, conditions (1.1) and (1.2) are satisfied.
This assertion follows from [14, Table 3] and [5, Proposition 4.2].Let us explain in more detail our strategy to the steps (A) and (B) in the program forthe double flag variety X = K/B K × G/P S above.(A) In Section 8, we give a parametrization of the K -orbits in the double flag variety X .It is equivalent to parametrize the B K -orbits in the Grassmann variety G/P S ∼ = Gr n ( C n ).To this end, we consider the set ( T n ) ′ of (2 n ) × n matrices of rank n of the form ω = (cid:18) τ τ (cid:19) ,associated to a pair of partial permutations τ , τ ∈ T n . Here a partial permutation of[ n ] = { , , . . . , n } means an injective map from a subset J ⊂ [ n ] to [ n ]. As in the caseof permutation, we can associate a matrix in M n with a partial permutation τ , which wealso denote by τ by abuse of notation. See Definition 6.2 for details. Let us denote theimage of the matrix ω by Im ω , which is an n -dimensional subspace of C n generated bythe column vectors of ω . Thus Im ω represents a point in Gr n ( C n ).Here is a classification of the set of orbits X /K by partial permutations. Theorem 1.4 (see Theorem 8.1) . Every B K -orbit of Gr n ( C n ) is of the form B K · (Im ω ) for some ω ∈ ( T n ) ′ . Moreover, ω and ω ′ represent the same B K -orbit if and only if theycoincide up to column permutation. Thus we get ( T n ) ′ / S n ≃ X /K . LUCAS FRESSE AND KYO NISHIYAMA (B) Sections 9–10 are devoted to the calculation of the maps Ξ k : X /K → N k /K andΞ s : X /K → N s /K of (1.3). Now by the parametrization of X /K (Theorem 1.4 above),we can regard these maps as Ξ k :( T n ) ′ / S n → P ( n ) , (1.4) Ξ s :( T n ) ′ / S n → P ± (2 n ) , (1.5)where P ± (2 n ) denotes the set of signed Young diagrams of size 2 n with signature ( n, n )parametrizing the set of nilpotent K -orbits N s /K . We still call Ξ k and Ξ s the symmetrized and exotic moment maps respectively, though they are all reduced down to combinatorialway.The description of these maps are quite involved. We give a combinatorial descriptionfor the K -orbits of X which correspond to the matrices ω of the form ω = (cid:18) τ n (cid:19) where τ is a partial permutation. However, the problem for more degenerate orbits remains open.Our main results regarding step (B) can be summarized as follows. Theorem 1.5 (Theorems 9.1 and 10.4) . For ω = (cid:18) τ n (cid:19) ∈ ( T n ) ′ , there are combinato-rial algorithms which describe Ξ k ( ω ) and Ξ s ( ω ) . The algorithm for Ξ k ( ω ) reduces to theclassical Robinson-Schensted algorithm if τ is a permutation. Image of the conormal variety.
The setting being as in Section 1.3, we alsoconsider the image of the whole conormal variety by the maps π k and π s . The images π k ( Y )and π s ( Y ) are K -stable subsets of N k and N s , respectively, which are not closed in general(see Remark 11.3). We define the nilpotent varieties N X , k := π k ( Y ) and N X , s := π s ( Y ). Theorem 1.6.
We have N X , k = N k , while the “exotic nilpotent variety” N X , s is notirreducible. Specifically, if n = 1 , then N X , s = N s (it has two irreducible components; seeRemark 5.3). If n ≥ , then N X , s has exactly three irreducible components described asfollows. (a) Assume that n is even. Then the components of N X , s are the closures of the K -orbitsparametrized by the signed Young diagrams Λ + := + − + − · · · − + − + − · · · − , Λ := + − + − · · · −− + − + · · · + , Λ − := − + − + · · · + − + − + · · · + where the rows have length n . In this case the variety N X , s is equidimensional of dimension n − n ) . (b) Assume that n is odd. Then the components of N X , s are the closures of the K -orbitsparametrized by Λ + := + − + − · · · − + − + − + − · · · − , Λ := + − + − · · · + − + − + · · · − , Λ − := − + − + · · · + − + − + − + · · · + GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 7 where the rows have lengths n − , n, n + 1 . The variety N X , s is not equidimensionaland, it has two components of dimension n − n ) + 1 and one component of dimension n − n ) . In the theorem, we use the parametrization of the K -orbits of N s by signed Youngdiagrams; see [3], [15], or Section 5 below. Theorem 1.6 is proved in Section 11.1.5. To determine the maps X /K → N k /K and X /K → N s /K in (1.3) we consideranother version of the Steinberg maps, which might be a more straightforward general-ization of the original Steinberg theory and is of independent interest. Let us explainit.Note that B K is of the form B K = B × B , for Borel subgroups B , B of GL n . Weconsider an action of B × B on the space of n × n matrices M n , one factor acting on theleft and the other acting on the right. Namely the explicit action is given by( b , b ) · x := b xb − ∀ ( b , b ) ∈ B K , ∀ x ∈ M n . Since the action of GL n × GL n on M n is clearly spherical, there are only finitely many B × B orbits on M n . By means of Gaussian eliminations, it is easy to obtain a completesystem of representatives of the double coset space B \ M n /B , which turns out to be theset of partial permutations T n (Proposition 6.3). The conormal variety for this action isgiven by Y M n = { ( x, y ) ∈ M n × M n : xy ∈ n , yx ∈ n } , where n i stands for the nilradical of Lie( B i ); see Proposition 6.1. By the general theory(Proposition 3.1), this conormal variety is equidimensional and its irreducible componentsare parametrized by partial permutations. The image of each component of Y M n by themoment map µ M n : ( x, y ) ( xy, − yx ) ∈ M n determines a pair of nilpotent GL n -orbits, or in other words, a pair of partitions ( λ, µ ) ∈ P ( n ) . This yields a map from partial permutations to pairs of partitions:Φ : T n −→ P ( n ) , τ ( λ, µ ) . We call the map Φ a generalized Steinberg map .The result summarized in the next theorem is also one of our main results, whichgeneralizes the Robinson-Schensted correspondence to the case of partial permutations.The theorem is actually independent of the theory of the double flag varieties, and we areexpecting a further generalization to general reductive groups. Theorem 1.7 plays a keyrole in our calculations of the maps Ξ k and Ξ s of Theorem 1.5.Let us prepare one more notation to state the theorem. For each r (0 ≤ r ≤ n ),we consider triples of the form ( T , T ; ν ) where T , T are standard tableaux of shapesdenoted λ, µ ∈ P ( n ) respectively and ν ∈ P ( r ) is a partition such that λ \ ν and µ \ ν are column strips (or vertical strips). In particular, ν is in the intersection of λ and µ .Let us denote the set of such triples by Υ r . LUCAS FRESSE AND KYO NISHIYAMA
Theorem 1.7 (Theorems 7.4 and 7.6) . The map Φ can be described through an explicitcombinatorial algorithm. This algorithm establishes a bijection between the set of partialpermutations T n and the set of triples S nr =0 Υ r defined above. If the partial permutationis a full permutation, the bijection reduces to the classical Robinson-Schensted correspon-dence. Namely τ ∈ S n corresponds to ( T , T ; λ ) where shape( T ) = shape( T ) = λ (sothat r = n ). K -orbits on G/B for classical symmetric pairs (
G, K ) of type A. This particular casecan be considered as a special case described in Section 1.2 if we take Q = K and P = B .Her description of the problem is very close to ours in spirit.In [21], based on the theory of mirabolic character sheaves in [4], Travkin gave a gen-eralization of Robinson-Schensted-Knuth algorithm for a triple flag variety of the form F ( C n ) × F ( C n ) × P ( C n ), which can be also recognized as a special case of the settingof Section 1.2 if we take G = GL n × GL n , K = ∆GL n (the diagonal embedded GL n into G ), P = B × B (a Borel subgroup of G ), and Q = P mir (the mirabolic maximalparabolic subgroup of K ≃ GL n ). Travkin actually considered the diagonal GL n actionon F ( C n ) × F ( C n ) × C n and, in his terminology, the orbits are parametrized by a certainset of pairs ( w, β ), where w ∈ S n and β ⊂ [ n ] is related to a decreasing sequence in w . Hisparameter set can actually be identified with the set of partial permutations T n , and themirabolic Robinson-Schensted-Knuth correspondence establishes a bijection between T n and the set of triples S nr =0 Υ r . However, this bijection appears to be totally different fromours. It is interesting to compare these two different bijections and to get a geometricinterpretation. See Remark 7.9 for further discussion.Rosso generalized the correspondence to the case of partial flag varieties and gave ageometric interpretation of the classical Robinson-Schensted-Knuth correspondence [16].In [9], Henderson and Trapa also gave a generalization of Robinson-Schensted cor-respondence for the symmetric pair ( G, K ) = (GL n , Sp n ), with Q ⊂ K mirabolic and P = B ⊂ G a Borel subgroup. Their approach is very close to ours. See also Remark 7.12.Finally, in our previous paper [5], we considered the case where ( G, K ) = (GL n , GL p × GL q ) ( p + q = n ); P is the stabilizer of a k -dimensional subspace of C n (a maximalparabolic subgroup of G ) and Q = Q × GL q , where Q is a mirabolic subgroup of GL p .In some sense [5] can be taken as a first trial, and the present paper largely deepens it.1.7. Organization of the paper.
The paper is divided into three parts. In the firstpart, we review the background on the Robinson-Schensted algorithm (Section 2), momentmaps and conormal varieties (Section 3), the Steinberg map (Section 4). In Section 5, wereview the basic facts and prepare the notation related to the symmetric pair (
G, K ) oftype AIII studied in this paper.In the second part of the paper, we consider the problem outlined in Section 1.5.In Section 6, we consider an action of the product of Borel subgroups on the space
GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 9 of n × n matrices given by left and right multiplications. We show that the partialpermutations serve as a complete set of representatives for the orbits. In Section 7, wedefine a generalized Steinberg map and provide an explicit combinatorial algorithm onpartial permutations to calculate this map, which generalizes the Robinson-Schenstedcorrespondence on permutations. We establish an identity between the number of orbits(i.e., partial permutations) and the dimension of a certain induced representation of S n (see Corollary 7.10). This identity suggests the existence of a geometric interpretation ofthe construction of such representations (see Conjecture 7.11).The final part of the paper is devoted to the main results outlined in Sections 1.3–1.4.Precise statements and proofs of these results are given in Sections 8 (parametrization ofthe orbit set X /K ), 9 and 10 (calculation of the maps Ξ k and Ξ s ). Section 11 containsthe proof of Theorem 1.6.The setting is recalled at the beginning of each section. An index of notation can befound at the end of the paper.1.8. Acknowledgements.
We thank Anthony Henderson and George Lusztig for usefulremarks which improve the manuscript. L.F. thanks Aoyama Gakuin University for warmhospitality during his visit in February 2017. This work was initiated in this period. K.N.thanks Institut ´Elie Cartan de Lorraine in Universit´e de Lorraine for warm hospitalityduring his visit in June and July, 2018. Key ingredients of this work were obtained in thisperiod.
Part Preliminaries on miscellanea and a review of Steinberg theory
In this part we review known facts, which we need in the rest of the paper. The contentof each section varies independently, and the notation or settings are often different fromsection to section. 2.
Robinson-Schensted correspondence
In this first preliminary section we summarize the background on Young diagrams andtableaux; we refer to [6] for more details.2.1.
Partitions, Young diagrams, tableaux.
We use the following conventions forYoung diagrams and tableaux.Let P ( n ) denote the set of partitions of n , i.e., nonincreasing sequences of positiveintegers λ = ( λ , . . . , λ k ) such that λ + . . . + λ k = n .A partition can be represented by a Young diagram , also denoted by λ , which is an arrayof empty boxes, left-justified, whose rows have lengths λ , . . . , λ k . We do not distinguishbetween the notions of partition and Young diagram.We call Young tableau of shape λ a numbering of the boxes of λ by pairwise distinctentries, which are in increasing order along rows (from left to right) and columns (from top to bottom). We write λ = shape( T ) whenever T is a Young tableau of shape λ , e.g., T = ⇒ shape( T ) = = (3 , , , ∈ P (8) . If the set of entries of a Young tableau T is [ n ] := { , , . . . , n } , we call T a standardtableaux . The set of standard tableaux of shape λ ∈ P ( n ) is denoted as STab( λ ).2.2. Row-insertion and column-insertion algorithms.
Given a Young tableau T and a number a distinct from all the entries of T , we denote by ( T ← a ) (resp. ( a → T ))the Young tableau obtained from T by inserting the entry a according to the followingalgorithm: • If a is larger than any entry in the first row (resp. column) of T , then insert a ina new box at the end of the first row (resp. column). • Otherwise, let a be the smallest entry in the first row (resp. column) of T whichis > a ; substitute a by a and insert a in the subtableau of T starting with thesecond row (resp. column), according to the same rule.We call this procedure row insertion (resp. column insertion ).For instance, for T as above, we get( T ←
4) = and (4 → T ) = . Given a list ( a , . . . , a ℓ ) of pairwise distinct numbers, letRowInsert( a , . . . , a ℓ ) := (( · · · (( ∅ ← a ) ← a ) · · · ) ← a ℓ )and ColumnInsert( a , . . . , a ℓ ) := ( a ℓ → ( · · · → ( a → ( a → ∅ )) · · · )) . We get the following equality (see, e.g., [11, Theorem 4.1.1]):RowInsert( a , . . . , a ℓ ) = ColumnInsert( a ℓ , . . . , a ) . The Robinson-Schensted correspondence.
For the later use, we formulate thecorrespondence in a slightly general way.Given a pair of Young tableaux (
T, S ) of the same shape, a number a which is distinctfrom all the entries of T , and a number b which is bigger than all the entries of S , wedenote by(2.1) ( T, S ) ← ( a, b )the pair of Young tableaux ( T ′ , S ′ ) such that • T ′ = ( T ← a ), GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 11 • S ′ is obtained from S by putting the entry b in a new box at the same position asthe unique box of shape( T ′ ) \ shape( T ).Let a < . . . < a ℓ and b < . . . < b ℓ be two sequences of numbers. Given a bijection w : { b , . . . , b ℓ } → { a , . . . , a ℓ } , we define a pair of Young tableaux using the insertion(2.1) successively(RS ( w ) , RS ( w )) := ( ∅ , ∅ ) ← ( w ( b ) , b ) ← ( w ( b ) , b ) ← · · · ← ( w ( b ℓ ) , b ℓ ) . Thus RS ( w ) = RowInsert( w ( b ) , . . . , w ( b ℓ )) and RS ( w ) encodes the development of theshape. By definition, the tableaux RS ( w ) and RS ( w ) have the same shape with entries { a , . . . , a ℓ } and { b , . . . , b ℓ } respectively. Moreover, we have (cf. [6, § ( w − ) , RS ( w − )) = (RS ( w ) , RS ( w )) . Since an element w ∈ S n is a bijection from [ n ] to itself, we can apply the procedureabove. Then the map(2.2) S n ∋ w (RS ( w ) , RS ( w )) ∈ a λ ∈ P ( n ) STab( λ ) × STab( λ )establishes a bijection between the symmetric group and the set of pairs of standardtableaux of the same shape. This bijection is referred to as the Robinson-Schensted cor-respondence .2.4. Jeu de taquin.
For two partitions ν = ( ν , . . . , ν ℓ ) ∈ P ( m ) and λ = ( λ , . . . , λ k ) ∈ P ( n ), we write ν ⊂ λ if ℓ ≤ k and ν i ≤ λ i for all i ∈ { , . . . , ℓ } . Then the set of boxes λ \ ν is called a skew diagram . A skew tableau of shape λ \ ν is a numbering of the boxes of λ \ ν by pairwise distinct integers which are in increasing order along rows and columns.Such a skew tableau T can be transformed into a Young tableau by the procedure of jeude taquin (or slidings ): • Choose any inside corner of T , i.e., a box c of ν which is adjacent to a box of T and such that { c } ∪ ( λ \ ν ) is also a skew diagram. • Sliding step: one or both of the boxes on the right or below c is contained in T .Then choose the smaller entry (if there are two) and slide it into the box c . • If the box c that has just been emptied has also a neighbor on the right or below,then apply the sliding step to c . Repeat this procedure until the emptied box hasno neighbor on the right nor below. • Then one obtains a skew tableau whose number of inside boxes is smaller. Repeatthe whole procedure, until one obtains a skew tableau without inside boxes, i.e.,a Young tableau.The Young tableau rect( T ) so-obtained is called the rectification of T . Note that theresult is independent of the choice of inside corners. For instance, if we denote by a dot the choice of inside corner, we proceed T = •
34 71 6 98 sliding −−−−→ • −−−−→ • −−−−→ = rect( T ) . If T, S are two Young tableaux whose entries are disjoint, we define T ∗ S as therectification of the skew tableau obtained by putting S at the top right to T , e.g., ∗ = rect = . Note that T ∗ a = ( T ← a ) and a ∗ T = ( a → T ) , and more generally T ∗ ℓ ... ℓ s = T ← ℓ s ← ℓ s − ← · · · ← ℓ and m ... m s ∗ T = m s → m s − → · · · → m → T whenever a , ℓ < . . . < ℓ s , m < . . . < m s are not entries of T .3. Moment maps and conormal varieties
Conormal variety.
Let H be a connected algebraic group acting on a smoothalgebraic variety X . The cotangent bundle T ∗ X has a structure of symplectic variety andthe H -action induces a Hamiltonian action of H on T ∗ X . The corresponding momentmap is denoted by µ X : T ∗ X → h ∗ , where h ∗ denotes the algebraic dual of the Lie algebraof H ([2, Proposition 1.4.8]).The null fiber of the moment map Y X := µ − X (0)is a union of Lagrangian subvarieties of T ∗ X that we call conormal variety . Let us seesome of the beautiful nature of conormal varieties. In our situation above, the momentmap µ X can be described explicitly as follows (see [2, § µ X : T ∗ X = { ( x, ξ ) : ξ ∈ ( T x X ) ∗ } −→ h ∗ , ( x, ξ ) ξ ◦ dρ x where ρ x : H → X , h h · x is the orbit map and dρ x : h → T x X is its differential. Proposition 3.1.
We consider the cotangent bundle π X : T ∗ X → X and its restrictionto the conormal variety Y X . (1) For each H -orbit O ⊂ X , the restriction π − X ( O ) ∩ Y X → O coincides with theconormal bundle T ∗ O X → O . In particular π − X ( O ) ∩ Y X = T ∗ O X is a smooth, irreducible,Lagrangian subvariety of T ∗ X , and hence it has dimension dim X . GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 13 (2)
The conormal variety is a union of the conormal bundles: Y X = F O ∈ X/H T ∗ O X . (3) Consequently, if X has a finite number of H -orbits, then Y X is equidimensional ofdimension dim X , and each irreducible component of Y X is of the form T ∗ O X for a unique H -orbit O ⊂ X .Proof. It suffices to prove (1). The rest of the statements are clear. For ( x, ξ ) ∈ T ∗ X , wehave ( x, ξ ) ∈ π − X ( O ) ∩ Y X ⇐⇒ x ∈ O and µ X ( x, ξ ) = 0 ⇐⇒ x ∈ O and ∀ η ∈ h , ξ ( dρ x ( η )) = 0 ⇐⇒ x ∈ O and ξ | T x O = 0 ⇐⇒ ( x, ξ ) ∈ T ∗ O X where we use that the map dρ x : h → T x X is surjective onto T x ( H · x ) (see, e.g., [17,Theorem 4.3.7]), hence π − X ( O ) ∩ Y X = T ∗ O X . The other assertions made in (1) followfrom the properties of the conormal bundle T ∗ O X → O . (cid:3) Moment map for double flag variety.
Let G be a connected reductive algebraicgroup with Lie algebra g . Hereafter we identify g with its algebraic dual g ∗ via a fixednondegenerate G -invariant bilinear form h· , ·i on g .For a parabolic subgroup P ⊂ G , the partial flag variety G/P can also be viewed asthe set of parabolic subalgebras p ⊂ g which are conjugate to p := Lie( P ). The tangentspace T p ( G/P ) coincides with the quotient space g / p . Its dual ( T p ( G/P )) ∗ = ( g / p ) ∗ identifies with the space { ξ ∈ g ∗ : ξ | p = 0 } , which itself corresponds (through theinvariant form h· , ·i ) to the nilpotent radical nil ( p ). In this way the cotangent bundle isgiven as T ∗ ( G/P ) = { ( p , ξ ) ∈ ( G/P ) × g ∗ : ξ | p = 0 } ∼ = { ( p , x ) ∈ ( G/P ) × g : x ∈ nil ( p ) } and the action of G on G/P gives rise to the moment map µ G/P : T ∗ ( G/P ) −→ g ∗ , ( p , ξ ) ξ or equivalently, ( p , x ) x. In this paper, we consider a double flag variety of the form X = G/P × K/Q for a(connected) symmetric subgroup K = G θ ⊂ G , defined by an involution θ ∈ Aut( G ), anda parabolic subgroup Q ⊂ K . In this situation, we assume that the bilinear form h· , ·i isalso θ -invariant (the bilinear form can always be chosen in this way).We identify X with the collection of pairs of parabolic subalgebras ( p , q ) where p ⊂ g is G -conjugate to p and q ⊂ k := Lie( K ) is K -conjugate to q := Lie( Q ). Then thecotangent bundle is described as T ∗ X = { ( p , q , ξ, η ) ∈ ( G/P ) × ( K/Q ) × g ∗ × k ∗ : ξ | p = 0 , η | q = 0 }∼ = { ( p , q , x, y ) ∈ ( G/P ) × ( K/Q ) × g × k : x ∈ nil ( p ) , y ∈ nil ( q ) } and the diagonal action of K on X gives rise to a moment map µ X : T ∗ X → k ∗ , ( p , q , ξ, η ) ξ | k + η or, equivalently, ( p , q , x, y ) x θ + y with x θ := ( x + θ ( x )) /
2. The conormal variety Y X = µ − X (0) can be described as Y X = { ( p , q , ξ, η ) ∈ ( G/P ) × ( K/Q ) × g ∗ × k ∗ : ξ | p = 0 , η | q = 0 , η = − ξ | k } hence we get an isomorphism Y X ∼ = { ( p , q , ξ ) ∈ ( G/P ) × ( K/Q ) × g ∗ : ξ | p = 0 , ξ | q = 0 }∼ = { ( p , q , x ) ∈ ( G/P ) × ( K/Q ) × g : x ∈ nil ( p ) , x θ ∈ nil ( q ) } . In the following we often identify these isomorphic varieties without mention.3.3.
Moment maps for rational representations.
Let V be a finite dimensional H -module. We denote by ( η, v ) ηv the action of h on V obtained by differentiation. Thecotangent space T ∗ V = V × V ∗ is endowed with a Hamiltonian action of H given by h · ( v, ξ ) = ( hv, ξ ◦ h − ). The corresponding moment map is given by µ V : T ∗ V −→ h ∗ , ( v, ξ ) (cid:8) η ξ ( ηv ) (cid:9) . So the conormal variety Y V is expressed as(3.1) Y V = µ − V (0) = { ( v, ξ ) ∈ V × V ∗ : ∀ η ∈ h , ξ ( ηv ) = 0 } . The Steinberg map
Let us explain in more detail the construction of the Steinberg map, outlined in Section1.1. We follow the approach of [10], which is slightly different from Steinberg’s originalconstruction [19].The flag variety B = G/B can be identified with the set of all Borel subalgebras b ′ ⊂ g .As explained in Section 3.2, the cotangent bundle T ∗ ( B × B ) and the moment map µ B×B can be described as µ B×B : T ∗ ( B × B ) = { ( b ′ , b ′ , x , x ) : x i ∈ nil ( b ′ i ) } −→ g , ( b ′ , b ′ , x , x ) x + x and the conormal variety is given by Y = { ( b ′ , b ′ , x ) ∈ B × B × g : x ∈ nil ( b ′ ) ∩ nil ( b ′ ) } . In this case, Y is often referred to as the Steinberg variety .On the other hand, every G -orbit of B × B takes the form Z w := G ( b , w b ) for a uniqueWeyl group element w ∈ W . Here B is a (fixed) Borel subgroup containing the maximaltorus T , and we denote b = Lie( B ) and n = nil ( b ). Hereafter, we use the notation w b = Ad( w )( b ) and w n = Ad( w )( n ). The orbit Z w gives rise to the conormal bundle(4.1) T ∗Z w ( B × B ) = { ( b ′ , b ′ , x ) ∈ Y : ( b ′ , b ′ ) ∈ Z w } = G · { ( b , w b , x ) : x ∈ n ∩ w n ) } GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 15 whose closure is an irreducible component of Y according to Proposition 3.1. Everycomponent of Y is of this form.The projection map π : ( b ′ , b ′ , x ) x is G -equivariant and closed. It therefore maps T ∗Z w ( B × B ) onto the closure of a nilpotent orbit O w ∈ N /G . Note that O w is alsocharacterized as the unique nilpotent orbit which intersects the space n ∩ w n along adense open subset. The so-obtained mapSt : W ∼ = ( B × B ) /G −→ N /G, w w is the Steinberg map introduced in Section 1.1.Note that the bijection W ∼ → ( B × B ) /G , w G ( b , w b ) is not canonical, as it dependson the choice of a Borel subgroup B ⊂ G which contains the maximal torus T (thoughit becomes canonical if W is replaced by the abstract Weyl group; see [2, Proposition3.1.29]).In the case of G = GL n , we always consider the Steinberg map corresponding to theBorel subgroup B = B + n of upper triangular matrices. In this case, the Weyl group W = S n is the symmetric group, whereas the nilpotent orbits O λ ∈ N /G are encoded bythe partitions λ ∈ P ( n ). Notation.
For a partition λ = ( λ , . . . , λ k ) ∈ P ( n ), we denote by O λ ⊂ M n the subsetconsisting of the nilpotent matrices which have k Jordan blocks of sizes λ , . . . , λ k . Thesubsets O λ (for λ ∈ P ( n )) are exactly the nilpotent orbits in gl n .In this way, we get a combinatorial incarnation of the Steinberg map, which we stilldenote by St : S n → P ( n ) by abuse of notation. This map can be described in terms ofthe Robinson-Schensted algorithm (see Section 2.3). Theorem 4.1 ([20]) . Assume that G = GL n and B = B + n is the subgroup of uppertriangular matrices. Then, for any permutation w ∈ S n , we have St( w ) = shape(RowInsert( w (1) , . . . , w ( n ))) = shape(RS i ( w )) ( i ∈ { , } ) . For λ ∈ P ( n ) , the fiber St − ( λ ) is the set of permutations w which correspond to a pairof standard tableaux in STab( λ ) × STab( λ ) via the Robinson-Schensted correspondence. The symmetric pair of type AIII
Symmetric pair of type AIII (tube type).
Here we denote G = GL n and g = gl n . We consider an involution θ ∈ Aut( G ) given by θ ( g ) = ιgι − where ι = (cid:18) n − n (cid:19) . Its differential yields an involution θ : g → g , which can be defined exactly in the sameway in matrix expression. The symmetric subgroup K := G θ can be described as K = (cid:26)(cid:18) a b (cid:19) : a, b ∈ GL n (cid:27) = { g ∈ GL n : g ( V + ) = V + , g ( V − ) = V − }∼ = GL n × GL n where we denote V + := C n × { } n and V − := { } n × C n so that we get a direct sumdecomposition C n = V + ⊕ V − .The Cartan decomposition g = k ⊕ s is given by k := Lie( K ) = (cid:26)(cid:18) α β (cid:19) : α, β ∈ M n (cid:27) , s := g − θ = (cid:26)(cid:18) γδ (cid:19) : γ, δ ∈ M n (cid:27) . We denote the projections along this direct sum decomposition as g → k , x = (cid:18) x x x x (cid:19) x θ := (cid:18) x x (cid:19) and g → s , x x − θ := (cid:18) x x (cid:19) . The nilpotent varieties N k and N s . We denote by N k and N s the closed subsetsof nilpotent elements in k and s , respectively. Each one of these nilpotent sets has afinite number of K -orbits that are parametrized as follows. We also indicate the closurerelations among orbits.The decomposition of N k into K -orbits is given by N k = [ ( λ,µ ) ∈ P ( n ) O λ × O µ where, by abuse of notation, O λ ×O µ stands for the set of elements (cid:0) α β (cid:1) with α ∈ O λ and β ∈ O µ . Thus we get a bijective parametrization N k /K ∼ = P ( n ) of nilpotent K -orbits.We recall the definition of the dominance order on partitions of n (or Young diagrams).Let λ ≤ k denote the number of boxes in the first k columns of λ . We set λ (cid:22) λ ′ if λ ≤ k ≥ λ ′≤ k ∀ k ≥ . Then, we have O λ ⊂ O λ ′ if and only if λ (cid:22) λ ′ , and consequently O λ × O µ ⊂ O λ ′ × O µ ′ if and only if λ (cid:22) λ ′ and µ (cid:22) µ ′ .For describing the K -orbits of N s , we need further notation. Definition 5.1. (a) A signed Young diagram (of signature ( n, n )) is a Young diagram ofsize 2 n whose boxes are filled in with n symbols + and n symbols − so that: • two consecutive boxes of the same row have opposite signs, so that each row is asequence of alternating signs; • we identify two such fillings up to permutation of rows, in particular we can stan-dardize the filling in such a way that, among rows which have the same length,the rows starting with a + are above those starting with a − (if there is any). GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 17
Let P ± (2 n ) denote the set of signed Young diagrams of size 2 n . For Λ ∈ P ± (2 n ), theshape of Λ is an element of P (2 n ), denoted by shape(Λ).(b) We introduce the dominance order on signed Young diagrams. Let ≤ k (+) (resp. ≤ k ( − )) denote the number of +’s (resp. − ’s) contained in the first k columns of Λ.Given Λ , Λ ′ ∈ P ± (2 n ), we set Λ (cid:22) Λ ′ if ≤ k (+) ≥ ′≤ k (+) and ≤ k ( − ) ≥ ′≤ k ( − ) ∀ k ≥ . Note that Λ (cid:22) Λ ′ implies shape(Λ) (cid:22) shape(Λ ′ ) where the latter relation is the dominanceordering. For instance, + − + − + − + − ++ − + −− + − + − ++ − + −− + − + − , + − + − ++ − + − + − + −− (cid:22) + − + − ++ − + −− + − + − . (c) Given a signed Young diagram Λ, we denote by O Λ the set of nilpotent elements x ∈ s which have a Jordan basis ( ε c ) indexed by the boxes c ∈ Λ such that • the vector ε c belongs to the subspace V + (resp. V − ) whenever the box c is filledin with a + (resp. a − ); • if c belongs to the first column of Λ, then x ( ε c ) = 0; otherwise, then x ( ε c ) = ε c ′ ,where c ′ is the box on the left of c .For instance, if Λ = − + − + − + then we obtain x = ∈ O Λ , x : e − → e +1 → e − → e − → e +2 → e +3 → { e ± i } ≤ i ≤ is the standard basis of V + = C ⊕ { } , resp. V − = { } ⊕ C . Proposition 5.2 ([15]) . (a) The subsets O Λ (for Λ ∈ P ± (2 n ) ) are exactly the K -orbitsof the nilpotent set N s . (b) The signed Young diagram Λ such that x ∈ O Λ is also characterized by ≤ k ( ± ) = dim(ker x k ) ∩ V ± ∀ k ≥ . (c) O Λ ⊂ O Λ ′ if and only if Λ (cid:22) Λ ′ . Remark 5.3. (a) It follows from Proposition 5.2 that the variety N s is not irreducible(contrary to N k ). Indeed N s has exactly two irreducible components which are the closuresof the K -orbits corresponding to the horizontal signed Young diagrams + − + − · · · and − + − + · · · (of size 2 n ) . (b) If Λ ∈ P ± (2 n ) is a signed Young diagram with at most n columns, then it is easyto see that we have Λ (cid:22) Λ , Λ (cid:22) Λ + , or Λ (cid:22) Λ − , where Λ , Λ + , Λ − are the signed Youngdiagrams given in Theorem 1.6 (a) when n is even and Theorem 1.6 (b) when n is oddrespectively. Thus { x ∈ N s : x n = 0 } ⊂ O Λ ∪ O Λ + ∪ O Λ − if n ≥ Part A generalized Steinberg map arising from the action of a pair ofBorel subgroups on the space of n × n matrices A partial permutation on the set [ n ] := { , , · · · , n } is an injective map from a (possiblyempty) subset J ⊂ [ n ] to [ n ]; equivalently it can be viewed as a degenerate permutationmatrix. The partial permutations form a semigroup denoted by T n .In Section 6, we consider the simultaneous action of a pair of Borel subgroups by (leftand right) multiplication on the space of n × n matrices, and we show that T n is a completeset of representatives for the orbits. Note that the considered action extends the Bruhatdecomposition of the group GL n , and in particular T n naturally contains the group ofpermutations S n .In Section 7, we use this action to give a bijective correspondence between T n and a setof pairs of tableaux with additional partition; see Theorems 7.4–7.6. This correspondencenaturally extends the original Robinson-Schensted correspondence for permutations.6. Action of a pair of Borel subgroups on the space of n × n matrices In this part of the paper, we consider two Borel subgroups B , B of GL n . Let b , b ⊂ gl n be the corresponding Borel subalgebras and let n , n be their respective nilradicals.We assume that B , B contain the standard torus of GL n .Let us consider an action of the group B × B on the space of n × n matrices M n ,which is given by( b , b ) · x := b xb − ∀ ( b , b ) ∈ B × B , ∀ x ∈ M n . As explained in Section 3.3, this action gives rise to a conormal variety Y M n ⊂ T ∗ M n =M n × M ∗ n , which is stable by the Hamiltonian action of the group B × B on the cotangentbundle. GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 19
Proposition 6.1.
Identifying M ∗ n with M n through the trace form h x, y i := Tr( xy ) , theconormal variety Y M n is identified with the variety Y M n = { ( x, y ) ∈ M n × M n : xy ∈ n , yx ∈ n } endowed with the action of B × B obtained by restriction of the following action on M n × M n : (6.1) ( b , b ) · ( x, y ) = ( b xb − , b yb − ) ∀ ( b , b ) ∈ B × B , ∀ ( x, y ) ∈ M n × M n . Proof.
While identifying the cotangent bundle T ∗ M n = M n × M ∗ n with the space M n × M n through the trace form, the action of B × B on T ∗ M n translates into the action onM n × M n given in (6.1), because for y ∈ M n ∼ = M ∗ n and ( b , b ) ∈ B × B we have h y, ( b , b ) − · z i = Tr( yb − zb ) = Tr( b yb − z ) = h b yb − , z i ∀ z ∈ M n . By differentiating the action of B × B on M n we obtain the infinitesimal action of Liealgebras: ( β , β ) · x = β x − xβ ∀ ( β , β ) ∈ b × b , ∀ x ∈ M n . Then by (3.1), for any ( x, y ) ∈ M n × M n ∼ = M n × M ∗ n , we have( x, y ) ∈ Y M n ⇐⇒ ∀ ( β , β ) ∈ b × b , h ( β , β ) · x, y i = 0 ⇐⇒ ∀ ( β , β ) ∈ b × b , Tr( β xy − xβ y ) = 0 ⇐⇒ ∀ β ∈ b , ∀ β ∈ b , Tr( β xy ) = Tr( yxβ ) = 0 ⇐⇒ xy ∈ b ⊥ (= n ) and yx ∈ b ⊥ (= n )where the notation ⊥ refers to the orthogonal space with respect to the trace form onM n . The proof of the proposition is complete. (cid:3) The action of B × B on the space M n has a finite number of orbits, as shown bythe following statement. (This also follows from the general theory of spherical varieties,knowing that this action is the restriction of an action of GL n × GL n , and it has an opendense orbit by virtue of the Bruhat decomposition.) The orbits are parametrized by theset of so-called partial permutations. In Corollary 6.4 we deduce a description of theirreducible components of the conormal variety. Definition 6.2.
We call a matrix τ ∈ M n a partial permutation if each row (resp. column)of τ has at most one nonzero entry, equal to 1. (Equivalently τ is obtained from apermutation matrix by erasing some 1’s, replaced by 0’s.) Let T n ⊂ M n denote the subsetof partial permutations. Proposition 6.3.
The set T n of partial permutations is a complete set of representativesof the B × B -orbits in M n . In other words every B × B -orbit of M n is of the form O τ := B τ B = { b τ b : b ∈ B , b ∈ B } for a unique partial permutation τ ∈ T n . Proof.
By assumption B and B are Borel subgroups of GL n which contain the standardtorus of diagonal matrices. Thus there are permutations σ , σ ∈ S n such that B i = σ i Bσ − i for i ∈ { , } , where B ⊂ GL n is the Borel subgroup of upper-triangular matrices.Since τ σ τ σ − is a bijection on the set of partial permutations, we may assume withoutloss of generality that B = B = B .A matrix remains in the same B × B -orbit as a ∈ M n whenever it is obtained from a ∈ M n by adding to a given row (resp. column) a sum of rows (resp. columns) situatedbelow it (resp. on its left) or by multiplying a row (a column) by a nonzero scalar.Applying Gauss elimination, it follows that every orbit BaB contains a matrix of theform τ ∈ T n .We observe that the maps a d i,j ( a ) := rank ( a k,ℓ ) i ≤ k ≤ n, ≤ ℓ ≤ j (for i, j ∈ { , . . . , n } ) areconstant on the B × B -orbits. Since we have d i,j ( τ ) = d i,j ( τ ′ ) for any ( i, j ) only if τ = τ ′ ,it follows that every orbit contains exactly one representative of the form τ ∈ T n . (cid:3) Corollary 6.4.
The conormal variety Y M n of Proposition 6.1 is equidimensional and itsirreducible components are parametrized by the partial permutations. More precisely everyirreducible component is of the form Y τ := T ∗ O τ M n = ( B × B ) · { ( τ, y ) : y ∈ M n , τ y ∈ n , yτ ∈ n } for a unique τ ∈ T n .Proof. This follows from Propositions 3.1, 6.1, and 6.3. (cid:3) Generalized Steinberg map and Robinson-Schensted correspondencefor partial permutations
The setting and the notation are the same as in Section 6, except that we assume forsimplicity B = B = B and n = n = n where B ⊂ GL n is the Borel subgroup of upper triangular matrices and n ⊂ M n is thesubspace of strictly upper triangular matrices. The goal of this section is to define andcalculate a generalized Steinberg map on the set of partial permutations and, to this end,it is indeed preferable to standardize the notation.7.1. The map Φ on partial permutations. We consider the map ϕ = ( ϕ , ϕ ) : Y M n −→ n × n , ( x, y ) ( xy, yx ) . Note that ϕ is B × B -equivariant, where B × B acts on n × n by the adjoint action on eachfactor.For any irreducible component Y τ ⊂ Y M n , the set ϕ ( Y τ ) ⊂ n × n is irreducible. Thereforethere exist a pair of nilpotent orbits O λ and O µ of M n which intersect ϕ ( Y τ ) densely. Inother words, we have (GL n × GL n ) · ϕ ( Y τ ) = O λ × O µ . GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 21
This yields a mapΦ : T n ∼ = Irr( Y M n ) −→ P ( n ) × P ( n ) , τ ( λ, µ ) . Let us denote Φ ( τ ) = λ and Φ ( τ ) = µ for the first and the second component of Φ,respectively. We call maps Φ , Φ , Φ generalized Steinberg maps .The next lemma immediately follows from the definition. Lemma 7.1.
With the above notation, O λ (resp. O µ ) is characterized as the nilpotentorbit which meets any of the sets ϕ ( Y τ ) , ϕ ( T ∗ O τ M n ) , { τ y : y ∈ M n such that ( τ y, yτ ) ∈ n × n } (resp. ϕ ( Y τ ) , ϕ ( T ∗ O τ M n ) , { yτ : y ∈ M n such that ( τ y, yτ ) ∈ n × n } )along dense open subsets. The map Φ on permutations. The decomposition of M n into B × B -orbits isevidently an extension of the Bruhat decomposition of G = GL n : G = G σ ∈ S n BσB ⊂ M n = G τ ∈ T n Bτ B.
Here the permutations σ ∈ S n are viewed as permutation matrices, in particular they areinvertible matrices. Thus, for ( x, y ) ∈ T ∗ BσB M n , the element x ∈ BσB ⊂ G is an invertiblematrix, hence the matrices xy and yx = x − ( xy ) x are G -conjugate. This readily impliesthat they generate the same nilpotent orbit, and we get Φ ( σ ) = Φ ( σ ) whenever σ is apermutation. In contrast, for a partial permutation τ , we have Φ ( τ ) = Φ ( τ ) in general.See Example 7.8 below.Let us compare the conormal bundles for the diagonal action of G on B × B and forthe action of B × B on G via the left and right multiplications.For σ ∈ S n , if we put O σ = BσB ⊂ G ⊂ M n and Z σ := G · ( b , σ b ) ⊂ B × B , thecorresponding conormal bundles are T ∗ O σ G = T ∗ O σ M n = ( B × B ) · { ( σ, y ) : σy ∈ n , yσ ∈ n } = ( B × B ) · { ( σ, y ) : σy ∈ n ∩ σ n } and T ∗Z σ ( B × B ) = G · { ( b , σ b , z ) : z ∈ n ∩ σ n } (see (4.1)). The fibers are the very same and we get G · ( ϕ ( T ∗ O σ G )) = G · ( ϕ ( T ∗ O σ G )) = G · ( n ∩ σ n ) = π ( T ∗Z σ ( B × B ))with π the projection given in Section 4. Thus these sets have the same dense nilpotentorbit O λ . We summarize this into the following proposition. Proposition 7.2.
For any permutation σ ∈ S n , we have Φ( σ ) = (St( σ ) , St( σ )) , where St : S n → P ( n ) is the Steinberg map for GL n . Remark 7.3.
Let σ be a permutation and set λ = Φ ( σ ) = Φ ( σ ). Then we have ϕ ( T ∗ O σ G ) = B · ( n ∩ σ n ) =: V σ . The set V σ := V σ ∩O λ is called an orbital variety (and V σ is its closure). It is an irreduciblecomponent of the variety O λ ∩ n and every component of O λ ∩ n is of the form V σ ′ forsome σ ′ ∈ S n such that Φ ( σ ′ ) = Φ ( σ ′ ) = λ ; see [12].Note also that ϕ ( T ∗ O σ G ) = V σ − .In fact, even for a partial permutation τ ∈ T n , we obtain that the (closures of the)images of the conormal bundle T ∗ O τ M n by ϕ and ϕ are closures of orbital varieties. SeeCorollary 7.5 below.7.3. Calculation of the map Φ for partial permutations.Notation. (a) As in the previous subsections, B ⊂ GL n denotes the Borel subgroupof upper triangular matrices and n ⊂ M n is the subspace of strictly upper triangularmatrices.(b) We can view a partial permutation τ ∈ T n as a map τ : { , . . . , n } → { , , . . . , n } such that τ − ( i ) ≤ i = 0; the corresponding matrix has 1 as an entry in theposition ( τ ( j ) , j ) whenever τ ( j ) = 0 and 0’s elsewhere. The map τ can also be written inthe form(7.1) τ = (cid:18) j · · · j r m · · · m n − r i · · · i r · · · (cid:19) , which means that τ ( j k ) = i k for k ∈ { , . . . , r } and τ ( m t ) = 0 for t ∈ { , . . . , n − r } , andwe will call (cid:18) j · · · j r i · · · i r (cid:19) the nondegenerate part of τ , which is a bijection between the sets J ( τ ) := { j , . . . , j r } and I ( τ ) := { i , . . . , i r } . We also write M ( τ ) := { m , . . . , m n − r } (the “kernel” of τ ) and L ( τ ) := { , . . . , n } \ I ( τ ) (the “coimage” of τ ).(c) Observe that the transpose of a partial permutation is a partial permutation, namelyfor τ in (7.1), the transpose is given by t τ = (cid:18) i · · · i r ℓ · · · ℓ n − r j · · · j r · · · (cid:19) , where { ℓ , . . . , ℓ n − r } = L ( τ ). In particular the nondegenerate part of t τ is the inverse ofthe nondegenerate part of τ . GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 23
In the following statement we use the notation in Section 2 related to the Robinson-Schensted algorithm and the operation ∗ on Young tableaux. Theorem 7.4.
Consider a partial permutation τ = (cid:18) j · · · j r m · · · m s i · · · i r · · · (cid:19) ∈ T n ( r = rank τ and r + s = n ) . Let σ = (cid:18) j · · · j r i · · · i r (cid:19) be the nondegenerate part of τ ; it yields a pair of Young tableaux (RS ( σ ) , RS ( σ )) defined through the Robinson-Schensted algorithm. Assume that m < · · · < m s . Let { ℓ , . . . , ℓ s } := { , . . . , n } \ { i , . . . , i r } and assume that ℓ < · · · < ℓ s .Then, the image of the generalized Steinberg map Φ( τ ) = (Φ ( τ ) , Φ ( τ )) ∈ P ( n ) isgiven by Φ ( τ ) = shape (cid:16) RS ( σ ) ∗ ℓ ... ℓ s (cid:17) , Φ ( τ ) = shape (cid:16) m ... m s ∗ RS ( σ ) (cid:17) . Proof.
We denote V ( τ ) := { τ y : y ∈ M n s.t. ( τ y, yτ ) ∈ n × n } , and similarly, V ( τ ) := { yτ : y ∈ M n s.t. ( τ y, yτ ) ∈ n × n } . By Lemma 7.1, the nilpotent orbit corresponding to Φ i ( τ ) intersects V i ( τ ) along a denseopen subset and this serves as characterization of the orbit (for i ∈ { , } ).Let us compute the spaces V ( τ ) and V ( τ ). Let e i,j stand for the elementary matrixwith 1 at the position ( i, j ) and 0’s elsewhere. It is straightforward to see that τ e i,j ∈ n ⇐⇒ (cid:26) i ∈ { m , . . . , m s } (in which case τ e i,j = 0), or i = j k and i k < j (in which case τ e i,j = e i k ,j )and e i,j τ ∈ n ⇐⇒ (cid:26) j ∈ { ℓ , . . . , ℓ s } (in which case e i,j τ = 0), or j = i k and j k > i (in which case e i,j τ = e i,j k ).This yields V ( τ ) = M ( i,j ) ∈ D C e i,j and V ( τ ) = M ( i,j ) ∈ D C e i,j where D = { ( i k , ℓ t ) : i k < ℓ t } ∪ { ( i k , i t ) : i k < i t , j k < j t } and D = { ( m k , j t ) : m k < j t } ∪ { ( j k , j t ) : j k < j t , i k < i t } . For w ∈ S n , we put D ( w ) := { ( i, j ) : 1 ≤ i < j ≤ n, w − ( i ) < w − ( j ) } . Note that the equality(7.2) M ( i,j ) ∈ D ( w ) C e i,j = n ∩ w n holds. By the classical Steinberg theory (Section 4), we already know the nilpotent orbitwhich intersects n ∩ w n densely. Thus let us see that D and D are exactly of the form D ( w ) above for some w .We may assume that j < . . . < j r . Moreover let { i ′ , . . . , i ′ r } := { i , . . . , i r } with i ′ < . . . < i ′ r and set j ′ k = σ − ( i ′ k ) for all k ∈ { , . . . , r } . Using these indices, we definepermutations w , w by(7.3) w := (cid:18) · · · r r +1 · · · ni · · · i r ℓ s · · · ℓ (cid:19) , w := (cid:18) · · · s s +1 · · · nm s · · · m j ′ · · · j ′ r (cid:19) . Then it is easy to see that D i = D ( w i ) ( i = 1 , V i ( τ ) = M ( k,ℓ ) ∈ D ( w i ) C e k,ℓ = n ∩ w i n for i ∈ { , } .By definition, the partition λ = St( w ) encodes the nilpotent orbit which intersects thespace n ∩ w n along a dense open subset (see Section 4). Therefore (7.4) implies(Φ ( τ ) , Φ ( τ )) = (St( w ) , St( w )) . By Theorem 4.1, the Steinberg map St can be computed by means of the Robinson-Schensted algorithm. Namely for w , w we deduce thatΦ ( τ ) = St( w ) = shape(RS ( w )) = shape(RowInsert( i , . . . , i r , ℓ s , . . . , ℓ ))= shape(RowInsert( i , . . . , i r ) ← ℓ s ← · · · ← ℓ )= shape (cid:16) RS ( σ ) ∗ ℓ ... ℓ s (cid:17) and Φ ( τ ) = St( w ) = shape(RS ( w )) = shape(RowInsert( m s , . . . , m , j ′ , . . . , j ′ r ))= shape(ColumnInsert( j ′ r , . . . , j ′ , m , . . . , m s ))= shape( m s → · · · → m → ColumnInsert( j ′ r , . . . , j ′ ))= shape (cid:16) m ... m s ∗ RS ( σ ) (cid:17) where we use that ColumnInsert( j ′ r , . . . , j ′ ) = RowInsert( j ′ , . . . , j ′ r ) andRowInsert( j ′ , . . . , j ′ r ) = RS (cid:0) (cid:18) i ′ · · · i ′ r j ′ · · · j ′ r (cid:19) (cid:1) = RS ( σ − ) = RS ( σ ) . GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 25
The proof is complete. (cid:3)
Recall from Remark 7.3 the notion of orbital variety.
Corollary 7.5.
Let τ be a partial permutation and let ( λ, µ ) = (Φ ( τ ) , Φ ( τ )) . Thenthe varieties ϕ ( T ∗ O τ M n ) and ϕ ( T ∗ O τ M n ) are closures of orbital varieties of O λ and O µ ,respectively. Namely we have ϕ ( T ∗ O τ M n ) ∩ O λ = V w and ϕ ( T ∗ O τ M n ) ∩ O µ = V w where w , w are the permutations defined in (7.3).Proof. By definition we have ϕ i ( T ∗ O τ M n ) = B · V i ( τ ) (for i ∈ { , } ), where V i ( τ ) is as inthe proof of Theorem 7.4. Then the claim follows from (7.4). (cid:3) The proof of Theorem 7.4 actually involves a generalization of the Robinson-Schenstedcorrespondence for partial permutations. We state it as a theorem below.Let STab( λ ) denote the set of standard tableaux of shape λ ∈ P ( n ), i.e., Youngtableaux of shape λ with entries 1 , . . . , n . Also, for a subdiagram ν ⊂ λ , we say that λ \ ν is a column strip (or a vertical strip) if the skew diagram λ \ ν contains at most one boxin each row. Theorem 7.6.
There is a bijective correspondence between the set of partial permutations T n and the set of triples G λ,µ ∈ P ( n ) r ∈{ ,...,n } (cid:8) ( T , T , ν ) ∈ STab( λ ) × STab( µ ) × P ( r ) : ν ⊂ λ, ν ⊂ µ, and λ \ ν , µ \ ν are column strips (cid:9) , which is given explicitly as τ (cid:16) RS ( σ ) ∗ ℓ ... ℓ s , m ... m s ∗ RS ( σ ) , shape(RS ( σ )) (cid:17) where σ denotes the nondegenerate part of τ (with the same notation as in Theorem 7.4).Proof. By t T we denote the transpose of a Young tableau T . For a partial permutation τ ∈ T n , the corresponding triple ( T , T , ν ) is constructed in the following way. Let usdenote the pair of intermediate tableaux by S = RS ( σ ) and S = RS ( σ ). Then thetriple is obtained by(7.5) T = S ← ℓ s ← · · · ← ℓ t T = t S ← m ← · · · ← m s ν = shape( S ) = shape( S )From [6, Proposition in § T i ) \ ν ( i ∈ { , } ) arecolumn strips, i.e., the image of the map in the statement is contained in the right-handside. Conversely the same proposition in [6] tells that for any triple ( T , T , ν ) in the right-hand side we can reverse the procedure to get a pair of Young tableaux ( S , S )and lists of entries ℓ < · · · < ℓ s and m < · · · < m s such that (7.5) holds. We denoteby { i , . . . , i r } (resp. { j , . . . , j r } ) the set of entries of S (resp. S ). By the Robinson-Schensted correspondence, there is a unique bijection w : { j , . . . , j r } → { i , . . . , i r } such that S = RS ( w ) and S = RS ( w ). In this way, we can recover the original partialpermutation τ . This shows that the map under consideration is bijective, which completesthe proof. (cid:3) Remark 7.7.
Through the correspondence described in Theorem 7.6, any permutation σ ∈ S n corresponds to a triple ( T , T , ν ) such that ν = shape( T ) = shape( T ); moreoverin this case we have ( T , T ) = (RS ( σ ) , RS ( σ )). Thus, in this way, the correspondencein Theorem 7.6 reduces to the classical Robinson-Schensted correspondence on the set ofpermutations. Example 7.8.
In Figures 1–3, we describe the correspondence τ ( T , T , ν ) of Theorem7.6 for n = 3. The set T contains 34 elements. For each partial permutation τ , weindicate the nondegenerate part σ , the pair (RS ( σ ) , RS ( σ )), and the correspondingtriple ( T , T , ν ). We display the list in three parts, according to the rank r of τ . Remark 7.9.
There might be a close relationship between the correspondence τ ( T , T , ν ) of Theorem 7.6 and Travkin’s mirabolic Robinson-Schensted-Knuth correspon-dence [21]. Let us explain this in more detail.In [21] the diagonal action of GL n on the variety GL n /B × GL n /B × C n is considered.The orbits are parametrized by the so-called marked permutations, i.e., the pairs ( w, β )formed by a permutation w ∈ S n and a subset β ⊂ [ n ] satisfying ∀ i, j ∈ [ n ] , ( i / ∈ β and j ∈ β ) = ⇒ ( i > j or w ( i ) > w ( j ));namely ( w, β ) is mapped to the GL n -orbit of the triple ( B, wB, P i ∈ β e i ), where ( e , . . . , e n )is the standard basis of C n . This is a variant of the parametrization of the GL n -orbits onGL n /B × GL n /B × P ( C n ) due to Magyar-Weyman-Zelevinsky [13].By using the conormal variety for GL n /B × GL n /B × C n and the enhanced nilpotent cone N gl n × C n (due to Achar and Henderson [1]), Travkin establishes a bijection between the setof marked permutations and the same set of triples as in Theorem 7.6. He describes thisbijection explicitly and call it the mirabolic Robinson-Schensted-Knuth correspondence.There is a natural bijection between marked permutations ( w, β ) and partial permuta-tions τ ∈ T n , which is defined by ( w, β ) τ := w | [ n ] \ β max where β max := { i ∈ β : ( j > i and j ∈ β ) ⇒ w ( j ) < w ( i ) } . Therefore, Travkin’s correspondence can also be expressed as an explicit bijection betweenpartial permutations τ ∈ T n and triples ( T , T , ν ) as in Theorem 7.6. This bijection seemsto be quite different from ours, thus giving rise to a non-identical map T n → T n on partial GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 27 permutations. We have no indication whatsoever on a direct, combinatorial descriptionnor on a possible geometric interpretation of this map.We mention the following representation theoretic interpretation of the fibers of themap Φ described in Theorem 7.4. Let ρ ( n ) λ denote the irreducible representation of S n corresponding to the partition λ ∈ P ( n ). In the next statement, the notation ⊠ standsfor the outer tensor product. Corollary 7.10.
For every pair of partitions ( λ, µ ) ∈ P ( n ) × P ( n ) , the number of ele-ments in the fiber Φ − (( λ, µ )) is equal to n X r =0 (cid:16) multiplicity of ρ ( n ) λ ⊠ ρ ( n ) µ in Ind S n S r × S n − r ( C S r ⊠ sgn ⊗ ) (cid:17) · dim( ρ ( n ) λ ⊠ ρ ( n ) µ ) where C S r denotes the regular representation of S r := S r × S r (defined by left and rightmultiplication) and sgn denotes the signature representation of S n − r .Proof. We have C S r = L ν ∈ P ( r ) ρ ( r ) ν ⊠ ( ρ ( r ) ν ) ∗ and sgn ⊗ = ρ ( n − r )(1 n − r ) ⊠ ( ρ ( n − r )(1 n − r ) ) ∗ . By the Pieriformula (a special case of the Littlewood-Richardson rule; see the formula (5) in § § S n S r × S n − r ( ρ ( r ) ν ⊠ ρ ( n − r )(1 n − r ) ) = M λ ∈ P ( n ) s.t. λ \ ν is column strip ρ ( n ) λ hence the multiplicity m r ( λ, µ ) in the decompositionInd S n S r × S n − r ( C S r ⊠ sgn ⊗ ) = M ( λ,µ ) ∈ P ( n ) ( ρ ( n ) λ ⊠ ρ ( n ) µ ) ⊕ m r ( λ,µ ) coincides with the number of partitions ν ∈ P ( r ) (subdiagrams of λ and µ ) such that λ \ ν and µ \ ν are column strips. We also know that dim ρ ( n ) λ = | STab( λ ) | since STab( λ )gives a basis of ρ ( n ) λ via the construction of Specht module. Hence, if we putΥ r ( λ, µ ) := { ( T , T , ν ) ∈ STab( λ ) × STab( µ ) × P ( r ) : λ \ ν, µ \ ν are column strips } , we obtain n X r =0 m r ( λ, µ ) dim ρ ( n ) λ ⊠ ρ ( n ) µ = (cid:12)(cid:12)(cid:12) n [ r =0 Υ r ( λ, µ ) (cid:12)(cid:12)(cid:12) = | Φ − (( λ, µ )) | where Theorems 7.4 and 7.6 are used. The proof is complete. (cid:3) The corollary only tells that the number of partial permutations and the dimension ofthe S n -module specified above exactly match. However, in the classical Steinberg theory,the corresponding representation is the regular representation of the symmetric group,and it coincides with the Springer representation on each fiber. Thus we propose thefollowing conjecture. Conjecture 7.11 (Generalization of the Springer representation) . There exists a geomet-ric way to construct an action of the group S n × S n on the top Borel-Moore homologyspace of H BMtop ( Y M n ) , identifying the irreducible components of Y M n with a basis of theinduced representation M nr =0 Ind S n S r × S n − r (cid:16) C S r ⊠ sgn ⊗ (cid:17) . Moreover the components of Y M n parametrized by the elements in the fibers Υ r ( λ, µ ) of O λ × O µ give a natural basis of ρ ( n ) λ ⊠ ρ ( n ) µ with multiplicities. Remark 7.12.
In [9], Henderson and Trapa used a similar equality between the numberof orbits and the dimension of certain induced representation (see Proposition 3.1 in [9]).However, it is still open to realize the representations geometrically.
Part The image of the symmetrized and exotic moment maps for thedouble flag variety of type AIII
In this final part, we establish our main results outlined in Sections 1.3–1.4.8.
Parametrization of the K -orbits in the double flag variety X In the rest of the paper, we consider the setting given in Sections 1.3 and 5. In particularwe have a polarized vector space V := C n = V + ⊕ V − where V + := C n × { } n , V − := { } n × C n , and consider the group G = GL n = GL( V ). The symmetric subgroup K = { g ∈ GL n : g ( V + ) = V + , g ( V − ) = V − } ∼ = GL n × GL n is the stabilizer of the polarization above. Finally, we consider the double flag variety X = K/B K × G/P S where • B K ⊂ K is a Borel subgroup, and there is no loss of generality in assumingthat B K = (cid:26)(cid:18) b b (cid:19) : b , b ∈ B (cid:27) ∼ = B × B with B = B + n ⊂ GL n the Borelsubgroup of upper triangular matrices; • P S := (cid:26)(cid:18) a c b (cid:19) : a, b ∈ GL n , c ∈ M n (cid:27) ⊂ G is a Siegel parabolic subgroup.Hence X can be identified with the direct product X = K/B K × Gr n ( C n ) = F ( V + ) × F ( V − ) × Gr n ( C n )where F ( V ± ) denotes the variety of complete flags of the subspace V ± while Gr n ( C n )stands for the Grassmann variety of n -dimensional subspaces of C n . GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 29
In the present section, we describe the parametrization of the K -orbits in the doubleflag variety X . It is easy to see that there is a bijectionGr n ( C n ) /B K ∼ −−→ X /K ∼ = ( K/B K × Gr n ( C n )) /K,V mod B K ( B K , V ) mod K. Thus we are reduced to parametrize the set of B K -orbits in Gr n ( C n ), which is achievedin the following theorem. Here again the partial permutations T n studied in Section 6appear. We identify τ ∈ T n with the corresponding matrix in M n . Theorem 8.1.
Let ( T n ) ′ denote the set of (2 n ) × n matrices of the form (cid:18) τ τ (cid:19) , with τ , τ ∈ T n , and which are of rank n (i.e., τ − (0) ∩ τ − (0) = ∅ ). The group S n acts on theset ( T n ) ′ from the right by the multiplication of the permutation matrices. For ω ∈ ( T n ) ′ ,let us denote by V ω := Im ω the n -dimensional subspace generated by the columns of thematrix ω . Then, the map ( T n ) ′ −→ Gr n ( C n ) , ω = (cid:18) τ τ (cid:19) V ω induces a bijection ( T n ) ′ / S n ∼ −−→ Gr n ( C n ) /B K ∼ = X /K. To prove the theorem, we use the following lemma.
Lemma 8.2.
Let B ′ ⊂ GL n be a Borel subgroup which contains the standard torus. Forany partial permutation τ ∈ T n , there is a permutation w ∈ S n such that τ wB ′ ⊂ B ′ τ w. Proof.
There is no loss of generality in assuming that B ′ ⊂ GL n is the subgroup of lowertriangular matrices. Given τ ∈ T n , by permuting the columns of τ , we can find w ∈ S n such that τ w = (cid:18) · · · r r + 1 · · · ni · · · i r · · · (cid:19) with i < · · · < i r , where r = rank τ . The group B ′ is generated by the torus T of diagonal matrices andthe elementary matrices of the form u j ( α ) := 1 n + αe j +1 ,j for j ∈ { , . . . , n − } . When t = diag( t , . . . , t n ) is a diagonal matrix, we have τ wt = diag( t ′ , . . . , t ′ n ) τ w ∈ B ′ τ w where t ′ i k = t k and t ′ ℓ = 1 for ℓ / ∈ { i , . . . , i r } .When u = u j ( α ) we have τ wu = (cid:26) τ w if r ≤ j ≤ n − τ w + αe i j +1 ,j = (1 n + αe i j +1 ,i j ) τ w if 1 ≤ j ≤ r − τ wu ∈ B ′ τ w in both cases. Altogether we conclude that τ wB ′ ⊂ B ′ τ w . The proofof the lemma is complete. (cid:3) Proof of Theorem 8.1.
First we show that Gr n ( C n ) is the union of the B K -orbits throughthe points V ω ∈ Gr n ( C n ) for ω ∈ ( T n ) ′ . Any point in Gr n ( C n ) is of the form V a := Im a for a certain (2 n ) × n matrix a of rank n . We write a ∼ a ′ whenever V a and V a ′ belongto the same B K -orbit. Let us consider a = (cid:18) a a (cid:19) ∈ M n,n ( C ) of rank n . Recall that B K = B × B where B ⊂ GL n is the Borel subgroup of upper triangular matrices. ByProposition 6.3 we can write a = b τ b for some ( b , b ) ∈ B × B and some partialpermutation τ ∈ T n . Furthermore, by Lemma 8.2, there is a permutation w ∈ S n suchthat τ wB ⊂ Bτ w . Setting a ′ := a b − w we have a = (cid:18) a a (cid:19) = (cid:18) b τ wa ′ (cid:19) w − b ∼ (cid:18) τ wa ′ (cid:19) . Invoking again Proposition 6.3, we find a pair ( b ′ , b ′ ) ∈ B × B and a partial permutation τ ∈ T n such that a ′ = b ′ τ b ′ . Moreover since τ wB ⊂ Bτ w , we can find b ′′ ∈ B suchthat τ wb ′− = b ′′ τ w . Whence a ∼ (cid:18) τ wa ′ (cid:19) = (cid:18) τ wb ′ τ b ′ (cid:19) ∼ (cid:18) τ wb ′− b ′ τ (cid:19) = (cid:18) b ′′ τ wb ′ τ (cid:19) ∼ (cid:18) τ wτ (cid:19) ∈ ( T n ) ′ . Hence all the B K -orbits have representatives of the form V ω for some ω ∈ ( T n ) ′ .Clearly, if ω = ω ′ w with w ∈ S n , then we have V ω = V ω ′ . Hence the map( T n ) ′ / S n → Gr n ( C n ) /B K , ω B K · V ω is well defined and surjective. It remains to show that this map is injective.Let ( ε ± , . . . , ε ± n ) be the standard basis of V ± and set V ± i = h ε ± , . . . , ε ± i i C for i =0 , , . . . , n . Thus B K is the stabilizer of the pair of complete flags ( V ± , V ± , . . . , V ± n ) ∈ F ( V ± ). Note that, for ω, ω ′ ∈ ( T n ) ′ , we have V ω = V ω ′ mod B K = ⇒ dim( V + i + V − j ) ∩ V ω = dim( V + i + V − j ) ∩ V ω ′ , ∀ i, j. (8.1)Since rank ω = n by assumption, every column of ω is nonzero. In fact every column of ω has at most two nonzero coefficients and is of the form ε + i , ε + i + ε − j , or ε − j for some i, j (regarding ε + i , ε − j as 2 n -sized column matrices). According to these three cases, we have • ε + i occurs as a column of ω if and only if dim V + i ∩ V ω > dim V + i − ∩ V ω ; • ε − j occurs as a column of ω if and only if dim V − j ∩ V ω > dim V − j − ∩ V ω ; • ε + i + ε − j occurs as a column of ω if and only ifdim( V + i + V − j ) ∩ V ω > dim( V + i − + V − j ) ∩ V ω = dim( V + i + V − j − ) ∩ V ω = dim( V + i − + V − j − ) ∩ V ω . This observation combined with (8.1) tells us that V ω = V ω ′ mod B K = ⇒ ω and ω ′ have the same columns (up to the order)= ⇒ ω = ω ′ w for some w ∈ S n . GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 31
The proof of the theorem is now complete. (cid:3) Symmetrized moment map X /K → N k /K The purpose of this section is to compute the symmetrized moment map X /K → N k /K of (1.3) in the case of the symmetric pair ( G, K ) = (GL n , GL n × GL n ) under consideration(see Section 1.3). We use the same notation as in Sections 5 and 8, in particular G = GL n and K = n (cid:18) a b (cid:19) : a, b ∈ GL n o = { g ∈ G : g ( V ± ) = V ± } ∼ = GL n × GL n where V + = C n × { } n and V − = { } n × C n , so that V + ⊕ V − = C n . Their Lie algebrasare denoted by g = gl n and k = Lie( K ) = n (cid:18) α β (cid:19) : α, β ∈ M n o ∼ = gl n × gl n . Recall the projection( · ) θ : g −→ k , x = (cid:18) x x x x (cid:19) x θ = (cid:18) x x (cid:19) along the Cartan decomposition. Since the nilpotent cone N k is the direct product N gl n ×N gl n , the nilpotent K -orbits in N k are of the form O λ × O µ with a pair of partitions( λ, µ ) ∈ P ( n ) : N k /K = {O λ × O µ : λ, µ ∈ P ( n ) } ∼ = P ( n ) × P ( n ) . Our double flag variety is identified with X = K/B K × Gr n ( C n ) ∼ = F ( V + ) × F ( V − ) × Gr n ( C n ) , where B K = B × B is the Borel subgroup of K corresponding to the subgroup B ⊂ GL n of upper-triangular matrices. From Theorem 8.1 we have a parametrization of the set of K -orbits X /K :(9.1) ( T n ) ′ / S n ∼ −−→ X /K, ω Z ω := K · ( B K , V ω )where ( T n ) ′ is the set of full rank matrices of the form ω = (cid:18) τ τ (cid:19) ( τ , τ ∈ T n ), and V ω = Im ω ∈ Gr n ( C n ).As before, we identify the flag variety K/B K (resp. Gr n ( C n ) ∼ = G/P S ) with thecollection of all Borel subalgebras b ′ K ⊂ k (resp. parabolic subalgebras p ′ ⊂ g conjugateto Lie( P S )). With this identification, Z ω is regarded as the K -orbit through the pair( b × b , p ω ) where b := Lie( B ) ⊂ gl n is the subalgebra of upper-triangular matrices, and p ω := { x ∈ gl n : x ( V ω ) ⊂ V ω } . We further denote n := nil ( b ) and u ω := nil ( p ω ) = { x ∈ gl n : Im x ⊂ V ω ⊂ ker x } , the nilradicals of b and p ω respectively. Recall the conormal variety(9.2) Y = { ( b ′ K , p ′ , x ) ∈ K/B K × G/P S × g : x θ ∈ nil ( b ′ K ) , x ∈ nil ( p ′ ) } , which is a union of conormal bundles T ∗ Z ω X = { ( b ′ K , p ′ , x ) ∈ Y : ( b ′ K , p ′ ) ∈ Z ω } (9.3) = K · { ( b × b , p ω , x ) ∈ X × M n : x ∈ u ω , x θ ∈ n × n } over the K -orbits Z ω (see Section 3.2), and the map π k : Y → k defined by π k ( b ′ K , p ′ , x ) = x θ . Then the symmetrized moment map X /K → N k /K in (1.3) induces a map betweenparameter sets of orbits:Ξ k : ( T n ) ′ −→ P ( n ) × P ( n ) , ω ( λ, µ )where ( λ, µ ) is the pair of partitions such that π k ( T ∗ Z ω X ) = O λ × O µ . Though it is difficultto give a combinatorial algorithm to describe Ξ k for all ω ’s, we have an efficient algorithmfor “generic” ones using Theorem 7.4. Namely we prove the following theorem. Theorem 9.1.
Whenever ω is of the form ω = (cid:18) τ n (cid:19) with a partial permutation τ ∈ T n ,we have Ξ k ( ω ) = Φ( τ ) where Φ : T n → P ( n ) × P ( n ) is the generalized Steinberg mapdescribed in Theorem 7.4. To prove the theorem, we use the following characterization of the map Ξ k (whichimmediately follows from (9.3) and the definition of the map Ξ k ). Lemma 9.2. If Ξ k ( ω ) = ( λ, µ ) , then O λ × O µ is characterized as the nilpotent orbit of N k which intersects the subspace V ( ω ) := { x θ : x ∈ u ω } ∩ ( n × n ) along a dense open subset.Proof of Theorem 9.1. Let us characterize the elements of the space V ( ω ). Put x = (cid:16) x x x x (cid:17) ∈ M n , and note that x θ = (cid:16) x x (cid:17) . Then x belongs to the nilpotent radical u ω if and only if Im x ⊂ Im ω ⊂ ker x , hence x ∈ u ω ⇐⇒ x (cid:18) τ n (cid:19) = 0 (cid:0) n − τ (cid:1) x = 0 ⇐⇒ x τ + x = 0 x τ + x = 0 x − τ x = 0 x − τ x = 0(9.4) ⇐⇒ x = (cid:18) τ x − τ x τx − x τ (cid:19) . GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 33
This yields V ( ω ) = n (cid:18) τ y − yτ (cid:19) : y ∈ M n such that ( τ y, yτ ) ∈ n × n o . By Lemma 7.1, the K -orbit O λ × O µ ∼ = n (cid:18) y y (cid:19) : y ∈ O λ , y ∈ O µ o , where ( λ, µ ) = (Φ ( τ ) , Φ ( τ )) , intersects V ( ω ) along a dense open subset. According to Lemma 9.2, this implies Ξ k ( ω ) =(Φ ( τ ) , Φ ( τ )) = Φ( τ ). (cid:3) Exotic moment map X /K → N s /K We keep the notation and setting of Section 9. In this section the Cartan space s = n (cid:18) γδ (cid:19) : γ, δ ∈ M n o plays an important role. We define the projection ( · ) − θ : g → s along the Cartan decom-position g = k ⊕ s , i.e., x = (cid:18) x x x x (cid:19) x − θ = (cid:18) x x (cid:19) . We define the map π s : Y −→ s , ( b ′ K , p ′ , x ) x − θ for the conormal variety Y in (9.2). As already pointed out in Section 1.3, we know thatthe image of this map is contained in the nilpotent variety N s , which consists of finitelymany K -orbits parametrized by P ± (2 n ), the set of signed Young diagrams of size 2 n (seeSection 5 for definition).For ω ∈ ( T n ) ′ , we have the K -orbit Z ω in X (see (9.1)). As before, we take the conormalbundle T ∗ Z ω X ⊂ Y over Z ω and conclude that the set π s ( T ∗ Z ω X ) is an irreducible, K -stable,closed subvariety of N s . Hence it coincides with the closure of a unique K -orbit O Λ . Inthis way, we get a map Ξ s : ( T n ) ′ −→ P ± (2 n ) , ω Λ , which is the combinatorial incarnation of the exotic moment map X /K → N s /K in (1.3).In the theorem below, we compute Ξ s ( ω ) for a “generic” ω = (cid:18) τ n (cid:19) as in the case ofTheorem 9.1. We use the following combinatorial definition. Definition 10.1.
Let ( T , T ) be a pair of Young tableaux of the same shape λ withentries from { , . . . , n } . Let ℓ < · · · < ℓ s (resp. m < · · · < m s ) be a list of entries in { , . . . , n } which do not appear in the tableau T (resp. T ). We define a skew tableau S = m ... m s ∗ T △ T ∗ ℓ ... ℓ s by the following algorithm: • Define a tableau b T := T ← ℓ s ← · · · ← ℓ by row insertion. Then its shape b λ contains at most one extra box in each row comparing to λ . Let b T be the tableauof the same shape b λ = shape( b T ), obtained as follows. Place T in the subshape λ ⊂ b λ . Then add to T extra boxes with the entries n + 1 , . . . , n + s from top tobottom. • Let T := m s → · · · → m → b T be a tableau obtained by column insertion. Itsshape λ contains at most one extra box in each row comparing to b λ . Let µ ( s ) denote the vertical Young diagram of size s . Then, there is a unique skew tableauof shape λ \ µ ( s ) whose rectification by jeu de taquin is the tableau b T . Define S as this skew tableau.It follows from [6, Proposition in § Lemma 10.2.
Assume that ( T , T ) = (RS ( w ) , RS ( w )) is the pair of Young tableauxcorresponding to the bijection w : j k i k . Let S be the skew tableau obtained in Definition10.1. Then S is obtained from the Robinson-Schensted tableau RS (cid:18) m s · · · m j · · · j r n + 1 · · · n + s − s · · · − i · · · i r ℓ s · · · ℓ (cid:19) by erasing the boxes of entries − , . . . , − s .Proof. This follows from [6, Proposition 1 in § (cid:3) Example 10.3.
For instance, for T = , T = , ( ℓ , ℓ ) = (2 , m , m ) =(1 , n = 7, we get b T = , b T = , T = hence ∗ T △ T ∗ = . GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 35
Theorem 10.4.
Let ω ∈ ( T n ) ′ be of the form ω = (cid:18) τ n (cid:19) for a partial permutation τ ∈ T n . We write τ = (cid:18) j · · · j r m · · · m s i · · · i r · · · (cid:19) with m < · · · < m s , where r = rank τ and s = n − r . Let { ℓ < · · · < ℓ s } := { , . . . , n } \ { i , . . . , i r } . Let usdenote the nondegenerate part of τ by σ := (cid:18) j · · · j r i · · · i r (cid:19) . Then the image Ξ s ( ω ) ∈ P ± (2 n ) of the exotic moment map is characterized as fol-lows. (1) For k > even, the number of + ’s (resp. − ’s) contained in the first k columns of Ξ s ( ω ) coincides with the number of boxes in the first k columns of the tableau RS ( σ ) ∗ ℓ ... ℓ s (cid:0) resp. m ... m s ∗ RS ( σ ) (cid:1) . (2) For k > odd, the number of + ’s contained in the first k columns of Ξ s ( ω ) coin-cides with the number of boxes contained in the first k columns of the skew tableau (seeDefinition 10.1 for notation) m ... m s ∗ RS ( σ ) △ RS ( σ ) ∗ ℓ ... ℓ s . (3) For k > odd, the number of − ’s in the first k columns of Ξ s ( ω ) is equal to s + (number of boxes in the first k columns of RS i ( σ )) ( i ∈ { , } ) . Example 10.5. (a) If τ = σ is a permutation, then RS ( τ ) , RS ( τ ) are standard Youngtableaux of the same shape, and the signed Young diagram Ξ s ( ω ) is obtained by dupli-cating this common shape and filling in the rows and columns with alternated +’s and − ’s. For instance, τ = 1 n = ⇒ shape(RS i ( τ )) = ( n ) = · · · = ⇒ Ξ s ( ω ) = + − + − · · ·− + − + · · · . See Section 10.2 for more details. (b) Assume that τ = (cid:18) · · · n · · · n − (cid:19) . In this case we get(RS ( σ ) , RS ( σ )) = ( · · · n –1 , · · · n ) , RS ( σ ) ∗ n = ∗ RS ( σ ) = · · · n , and ∗ RS ( σ ) △ RS ( σ ) ∗ n = · · · · n (the latter tableau is a skew tableau whose first column contains no box), henceΞ s ( ω ) = − + − · · · + − + − · · · + if n is even(two rows of length n ) or Ξ s ( ω ) = − + · · · + − + − + · · · + if n is odd.(rows of lengths n + 1 , n − n = 3, we have computed the signed Young diagram Ξ s ( ω ) for each matrix ofthe form ω = (cid:18) τ (cid:19) with τ ∈ T . These signed Young diagrams are listed below inFigures 1–3 at the end of this article.The rest of this section is devoted to the proof of Theorem 10.4. Let us begin withsome preliminary lemmas.10.1. A characterization of the image of the exotic moment map Ξ s . As in Section9, we prepare a lemma, which characterizes the K -orbit corresponding to the signed Youngdiagram Ξ s ( ω ) ∈ P ± (2 n ). Recall that n ⊂ gl n stands for the subalgebra of strictly uppertriangular matrices. Lemma 10.6. (a)
For ω ∈ ( T n ) ′ , put Λ = Ξ s ( ω ) ∈ P ± (2 n ) . Then the nilpotent K -orbit O Λ ⊂ N s is characterized as the K -orbit which intersects W ( ω ) := { x − θ : x ∈ u ω such that x θ ∈ n × n } along a dense open subset. (b) Moreover, if ω is of the form ω = (cid:18) τ n (cid:19) with τ ∈ T n , then we have W ( ω ) = n (cid:18) − τ yτy (cid:19) : y ∈ M n such that ( τ y, yτ ) ∈ n × n o . Proof.
By (9.3), we have T ∗ Z ω X = K · { ( b × b , p ω , x ) : x ∈ u ω , x θ ∈ n × n } , hence we get π s ( T ∗ Z ω X ) = K · W ( ω ) = O Λ by the definition of the map Ξ s . This shows part (a) of the statement. Part (b) followsfrom the calculation made in the proof of Theorem 9.1 (see (9.4)). (cid:3) GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 37
The permutation case.
We first determine Ξ s ( ω ) in the case where ω involves apermutation τ . Notation.
Given a partition λ = ( λ , . . . , λ s ) ∈ P ( n ), we denote by Λ[2 λ ] the signedYoung diagram of size 2 n obtained by duplicating λ , i.e., each row of λ of length λ i givesrise to two rows of Λ[2 λ ] of length λ i , one starting with + and the other starting with − .For instance: λ = = ⇒ Λ[2 λ ] = + − + −− + − ++ − + −− + − ++ − . In other words, with the notation of Section 5.2, the signed Young diagram Λ[2 λ ] ischaracterized as follows: λ ]) ≤ k (+) = λ ]) ≤ k ( − ) = λ ≤ k , ∀ k ≥ . Lemma 10.7.
Assume that ω = (cid:18) τ n (cid:19) where τ is a permutation. Put λ = shape(RS i ( τ )) (for i ∈ { , } ). Then we have Ξ s ( ω ) = Λ[2 λ ] .Proof. By Lemma 10.6, we have to determine the signed Young diagram Λ parametrizingthe K -orbit of N s containing (cid:18) − τ yτy (cid:19) whenever y ∈ M n is a generic element of thespace { y ∈ M n : ( τ y, yτ ) ∈ n × n } . For such an element y , we know from Theorem 7.4 that the Jordan form of a := yτ isgiven by λ . Using that the matrix τ is a permutation (thus invertible), we can write (cid:18) τ −
00 1 n (cid:19) (cid:18) − τ yτy (cid:19) (cid:18) τ
00 1 n (cid:19) = (cid:18) − yτyτ (cid:19) = (cid:18) − aa (cid:19) =: x hence x also belongs to the K -orbit O Λ . Note also that, for all k ≥
0, we have x k = ( − k (cid:18) a k a k (cid:19) and x k +1 = ( − k (cid:18) − a k +1 a k +1 (cid:19) . Hence for all k ≥ ≤ k ( ± ) = dim ker x k ∩ V ± = dim ker a k = λ ≤ k and therefore Λ = Λ[2 λ ] as asserted. (cid:3) A lemma for a partial permutation.
Let ω = (cid:18) τ n (cid:19) where τ is a partialpermutation. We consider an element y ∈ M n which is generic in the space { y ∈ M n : ( τ y, yτ ) ∈ n × n } so that the signed Young diagram Λ := Ξ s ( ω ) parametrizes the K -orbit O Λ ⊂ N s whichcontains the element x := (cid:18) − τ yτy (cid:19) (see Lemma 10.6). The proof of the following lemma is straightforward (the second part ofeach claim (a) and (b) follows from the definition of the orbit O Λ ; see Definition 5.1 (c)). Lemma 10.8. (a) x k = ( − k (cid:18) ( τ y ) k
00 ( yτ ) k (cid:19) for all k ≥ , hence ≤ k (+) = dim ker( τ y ) k and ≤ k ( − ) = dim ker( yτ ) k . (b) Similarly x k +1 = ( − k (cid:18) − ( τ y ) k +1 τ ( yτ ) k y (cid:19) for all k ≥ , hence ≤ k +1 (+) = dim ker (cid:0) ( yτ ) k y (cid:1) and ≤ k +1 ( − ) = dim ker (cid:0) ( τ y ) k +1 τ (cid:1) . Proof of Theorem 10.4 (1) . We can take y ∈ { y ∈ M n : ( τ y, yτ ) ∈ n × n } genericso that (cid:18) − τ yτy (cid:19) belongs to the K -orbit O Λ ⊂ N s for Λ := Ξ s ( ω )(see Lemma 10.6) and τ y (resp. yτ ) belongs to the nilpotent orbit O λ (resp. O µ ) for ( λ, µ ) := Φ( τ )(see Lemma 7.1). In view of Lemma 10.8 (a), it follows that the number of +’s (resp. − ’s)in the first k columns of Λ, for k even, coincides with the number of boxes in the first k columns of λ (resp. µ ). Thus Theorem 10.4 (1) follows from Theorem 7.4.10.5. Proof of Theorem 10.4 (2) . We consider the map ψ : M n −→ M n + s , y (cid:18) y (cid:19) and the permutation b τ := (cid:18) m · · · m s j · · · j r n + 1 · · · n + ss · · · i ′ · · · i ′ r ℓ ′ s · · · ℓ ′ (cid:19) ∈ S n + s where we set i ′ k := i k + s and ℓ ′ k := ℓ k + s . GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 39
Lemma 10.9.
Let λ := shape(RS i ( b τ )) ( i ∈ { , } ) for b τ as above and let ν be the one-column Young diagram with s boxes. Then the skew tableau S := m ... m s ∗ RS ( σ ) △ RS ( σ ) ∗ ℓ ... ℓ s in Theorem 10.4 (2) is of shape λ \ ν .Proof. Let T i = RS i ( σ ) for i ∈ { , } . Let ( b T , b T ) be the pair of tableaux obtained afterthe first step of the definition of the skew tableau S . By definition of the operation △ ,the shape of S is obtained as(10.1) shape (cid:0) m s → · · · → m → b T (cid:1) \ ν. By definition of the algorithm, for i ∈ { , } we have b T i = RS i ( w ) where w := (cid:18) j · · · j r n + 1 · · · n + si · · · i r ℓ s · · · ℓ (cid:19) . Let j ′ , . . . , j ′ n be the elements of the set { j , . . . , j r , n + 1 , . . . , n + s } ordered in such away that w ( j ′ ) < · · · < w ( j ′ n ); equivalently b τ ( j ′ )(= s + 1) < · · · < b τ ( j ′ n ). Then we haveRS ( b τ ) = RS ( b τ − ) = RowInsert( m s , . . . , m , j ′ , . . . , j ′ n )= ( m s → · · · → m → RowInsert( j ′ , . . . , j ′ n ))= ( m s → · · · → m → b T ) . Comparing this equality with (10.1) completes the proof of the lemma. (cid:3)
Lemma 10.10. (a)
The map ψ restricts to a bijection { y ∈ M n : ( τ y, yτ ) ∈ n × n } ∼ −−→ { z ∈ M n + s : ( b τ z, z b τ ) ∈ b n × b n } where b n ⊂ M n + s stands for the subspace of strictly upper triangular matrices. (b) For any y ∈ M n such that ( τ y, yτ ) ∈ n × n , we have ψ (( yτ ) k y ) = ( ψ ( y ) b τ ) k ψ ( y ) for all k ≥ . Proof.
To show part (a), it suffices to prove the following claims: { z ∈ M n + s : ( b τ z, z b τ ) ∈ b n × b n } ⊂ Im ψ ;(10.2) for y ∈ M n and b y := ψ ( y ), ( τ y, yτ ) ∈ n × n ⇐⇒ ( b τ b y, b y b τ ) ∈ b n × b n .(10.3)Let us show (10.2). Let z ∈ M n + s such that ( b τ z, z b τ ) ∈ b n × b n . For i ∈ { , . . . , s } , usingthat b τ z ∈ b n , we have z n + i,j = ( b τ z ) ℓ ′ s − i +1 ,j = 0 whenever 1 ≤ j ≤ ℓ ′ s − i +1 . Note that { ℓ ′ s − i +1 + 1 , . . . , n + s } ⊂ { ℓ ′ s − k +1 : 1 ≤ k < i } ∪ { i ′ , . . . , i ′ r } . Since z b τ ∈ b n , wealso have z n + i,i ′ k = ( z b τ ) n + i,j k = 0 for all k ∈ { , . . . , r } (since j k ≤ n ) and z n + i,ℓ ′ s − k +1 = ( z b τ ) n + i,n + k = 0 whenever 1 ≤ k < i. Altogether this implies that(10.4) z n + i,j = 0 for all i ∈ { , . . . , s } , all j ∈ { , . . . , n + s } . Let j ∈ { , . . . , s } . Since b τ z ∈ b n , we have z j k ,j = ( b τ z ) i ′ k ,j = 0 for all k ∈ { , . . . , r } (since i ′ k ≥ s + 1)and z m s − k +1 ,j = ( b τ z ) k,j = 0 whenever j ≤ k ≤ s .For k ∈ { , . . . , j − } , using that z b τ ∈ b n , we get z m s − k +1 ,j = ( z b τ ) m s − k +1 ,m s − j +1 = 0 (since m s − k +1 > m s − j +1 ) . Since { , . . . , n } = { j , . . . , j r } ∪ { m , . . . , m s } , altogether we obtain(10.5) z i,j = 0 for all i ∈ { , . . . , n } , all j ∈ { , . . . , s } .From (10.4) and (10.5), we conclude that (10.2) holds true.Next, let us show (10.3). Note that the matrix corresponding to b τ is of the form b τ = (cid:18) α τ β (cid:19) for some matrices α ∈ M s,n , β ∈ M n,s . This yields(10.6) b τ b y = (cid:18) γ τ y (cid:19) and b y b τ = (cid:18) yτ δ (cid:19) for some γ ∈ M s,n and δ ∈ M n,s . Whence the equivalence( b τ b y, b y b τ ) ∈ b n × b n ⇐⇒ ( τ y, yτ ) ∈ n × n which establishes (10.3).Let us prove part (b) by induction in k ≥
0. The case k = 0 is trivial. So assume that ψ (( yτ ) k y ) = ( ψ ( y ) b τ ) k ψ ( y ). Set b y = ψ ( y ). Using the second equality in (10.6), we get( b y b τ ) k +1 b y = (cid:18) yτ δ (cid:19) (cid:18) yτ ) k y (cid:19) = (cid:18) yτ ) k +1 y (cid:19) whence the equality ( b y b τ ) k +1 b y = ψ (( yτ ) k +1 y ). (cid:3) Proof of Theorem 10.4 (2) . We set b ω := (cid:18) b τ n + s (cid:19) , where b τ ∈ S n + s is the permutationgiven above. By Lemmas 10.7 and 10.8 (b) applied to b ω and b n × b n , for a generic b y in thespace { b y ∈ M n + s : ( b τ b y, b y b τ ) ∈ b n × b n } , we havedim ker (cid:0) ( b y b τ ) k b y (cid:1) = (cid:0) shape(RS ( b τ )) (cid:1) ≤ k +1 ∀ k ≥ . GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 41
By Lemma 10.10 (a), we have b y = ψ ( y ) for y ∈ M n generic in the space { y ∈ M n :( τ y, yτ ) ∈ n × n } , and by Lemma 10.8 (b) we may suppose thatdim ker (cid:0) ( yτ ) k y (cid:1) = (cid:0) Ξ s ( ω ) (cid:1) ≤ k +1 (+) ∀ k ≥ . Moreover in view of Lemma 10.10 (b) we have dim ker (cid:0) ( yτ ) k y (cid:1) = dim ker (cid:0) ( b y b τ ) k b y (cid:1) − s for all k . By Lemma 10.9, this yields the equality stated in Theorem 10.4 (2). (cid:3) Proof of Theorem 10.4 (3) . We consider the sets I := { i , . . . , i r } and J := { j , . . . , j r } , and the increasing bijections w I : I → { , . . . , r } and w J : J → { , . . . , r } .The bijection σ : J → I gives rise to a permutation τ ′ := w I σw − J ∈ S r . Let us consider linear maps ξ : M r → M n , z z with z j,i = (cid:26) z w J ( j ) ,w I ( i ) if ( j, i ) ∈ J × I ,0 if ( j, i ) / ∈ J × I and φ : M n → M r , y y ′ with y ′ w J ( j ) ,w I ( i ) = y j,i for all ( j, i ) ∈ J × I .Let n ′ ⊂ M r denote the subspace of strictly upper triangular matrices. Lemma 10.11. (a) φ ◦ ξ = id M r ; (b) For z ∈ M r and z = ξ ( z ) , we have: ( τ ′ z, zτ ′ ) ∈ n ′ × n ′ = ⇒ ( τ z, zτ ) ∈ n × n ;(c) For y ∈ M n and y ′ = φ ( y ) , we have: ( τ y, yτ ) ∈ n × n = ⇒ ( τ ′ y ′ , y ′ τ ′ ) ∈ n ′ × n ′ and t ( τ ( yτ ) k yτ ) = ξ ( t ( τ ′ ( y ′ τ ′ ) k y ′ τ ′ ) ) for all k ≥ .Proof. First note that( φ ◦ ξ ( z )) i,j = ( ξ ( z )) w − J ( i ) ,w − I ( j ) = z w J ( w − J ( i )) ,w I ( w − I ( j )) = z i,j for all i, j ∈ { , . . . , r } ,whence (a). Before showing parts (b) and (c), we note that if z = φ ( y ) then( τ y ) i k ,i ℓ = y j k ,i ℓ = z w J ( j k ) ,w I ( i ℓ ) = ( τ ′ z ) w I ( i k ) ,w I ( i ℓ ) , (10.7) ( yτ ) j k ,j ℓ = y j k ,i ℓ = z w J ( j k ) ,w I ( i ℓ ) = ( zτ ′ ) w J ( j k ) ,w J ( j ℓ ) , and(10.8) ( τ yτ ) i k ,j ℓ = y j k ,i ℓ = z w J ( j k ) ,w I ( i ℓ ) = ( τ ′ zτ ′ ) w I ( i k ) ,w J ( j ℓ ) (10.9)for all k, ℓ ∈ { , . . . , r } .Let us show part (b). Assume that ( τ ′ z, zτ ′ ) ∈ n ′ × n ′ . Note that ( τ z ) i,j = 0 if i / ∈ I (due to the definition of I ) or if j / ∈ I (due to the definition of z ). Similarly ( zτ ) i,j = 0 if( i, j ) / ∈ J × J . By (a) we have z = φ ( z ). By (10.7), for i, j ∈ I such that i ≥ j , we get( τ z ) i,j = ( τ ′ z ) w I ( i ) ,w I ( j ) = 0 since w I ( i ) ≥ w I ( j ) (because w I is increasing) and τ ′ z ∈ n ′ . By (10.8) we get similarly ( zτ ) i,j = ( zτ ′ ) w J ( i ) ,w J ( j ) = 0 whenever i, j ∈ J satisfy i ≥ j .Altogether we have shown that ( τ z, zτ ) ∈ n × n .Let us show part (c). Assume that ( τ y, yτ ) ∈ n × n . Then, (10.7) and (10.8) yield( τ ′ y ′ ) i,j = ( τ y ) w − I ( i ) ,w − I ( j ) = 0 and ( y ′ τ ′ ) i,j = ( yτ ) w − J ( i ) ,w − J ( j ) = 0whenever i, j ∈ { , . . . , r } are such that i ≥ j . Whence ( τ ′ y ′ , y ′ τ ′ ) ∈ n ′ × n ′ . It remains toshow the second assertion in part (c). By definition of I, J , we have(10.10) ( τ ( yτ ) k yτ ) i,j = 0 for all k ≥ , if ( i, j ) / ∈ I × J. Next fix ( i, j ) ∈ I × J and let us show the formula(10.11) ( τ ( yτ ) k yτ ) i,j = ( τ ′ ( y ′ τ ′ ) k y ′ τ ′ ) w I ( i ) ,w J ( j ) by induction on k ≥
0. The case k = 0 follows from (10.9). Assuming that formula(10.11) holds for k , by using (10.10) and (10.7), we see that( τ ( yτ ) k +1 yτ ) i,j = n X ℓ =1 ( τ y ) i,ℓ ( τ ( yτ ) k yτ ) ℓ,j = X ℓ ∈ I ( τ y ) i,ℓ ( τ ( yτ ) k yτ ) ℓ,j = r X ℓ =1 ( τ ′ y ′ ) w I ( i ) ,ℓ ( τ ′ ( y ′ τ ′ ) k y ′ τ ′ ) ℓ,w J ( j ) = ( τ ′ ( y ′ τ ′ ) k +1 y ′ τ ′ ) w I ( i ) ,w J ( j ) . This establishes (10.11). Finally relations (10.10) and (10.11) yield the desired equality t ( τ ( yτ ) k yτ ) = ξ ( t ( τ ′ ( y ′ τ ′ ) k y ′ τ ′ ) ) for all k ≥ (cid:3) Proof of Theorem 10.4 (3) . Let y ∈ M n be an element which is generic in the space { y ∈ M n : ( τ y, yτ ) ∈ n × n } , so that(10.12) (cid:0) Ξ s ( ω ) (cid:1) ≤ k +1 ( − ) = dim ker (cid:0) τ ( yτ ) k yτ (cid:1) (by Lemma 10.8 (b)). By Lemma 10.11 we may assume that y ′ := φ ( y ) is generic in thespace { z ∈ M r : ( τ ′ z, zτ ′ ) ∈ n ′ × n ′ } and, by Lemmas 10.7 and 10.8 (b), we may assumethat(10.13) dim ker (cid:0) τ ′ ( y ′ τ ′ ) k y ′ τ ′ (cid:1) = (cid:0) shape(RS ( τ ′ )) (cid:1) ≤ k +1 . In addition, by Lemma 10.11 (c), we have(10.14) dim ker (cid:0) τ ( yτ ) k yτ (cid:1) = dim ker (cid:0) τ ′ ( y ′ τ ′ ) k y ′ τ ′ (cid:1) + s. Finally note that the Young tableaux RS ( τ ′ ) and RS ( σ ) are of the same shape, becausewe have τ ′ = w I σw − J where w I , w J are increasing bijections. Then, part (3) of Theorem10.4 follows from (10.12), (10.13), and (10.14). (cid:3) GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 43
Proof of Theorem 1.6
In this section we focus on the images of the conormal variety Y by the maps π k : Y →N k and π s : Y → N s . Our goal is to prove Theorem 1.6, which describes the irreduciblecomponents of the nilpotent varieties N X , k := π k ( Y ) and N X , s := π s ( Y ).Recall that the conormal variety can be described as Y = { ( b ′ K , p ′ , x ) ∈ K/B K × G/P S × g : x ∈ nil ( p ′ ) , x θ ∈ nil ( b ′ K ) } . Here
K/B K (resp. G/P S ) is identified with the set of Borel subalgebras b ′ K ⊂ k (resp.parabolic subalgebras p ′ ⊂ g conjugate to p S ). Note that S b ′ K ∈ K/B K nil ( b ′ K ) coincides withthe nilpotent cone N k of k , while S p ′ ∈ G/P S nil ( p ′ ) = G · nil ( p S ) ⊂ g = M n is the subsetof nilpotent matrices of square zero (the closure of the Richardson orbit correspondingto p S ). This implies that the images of Y by the maps π k : ( b ′ K , p ′ , x ) x θ and π s :( b ′ K , p ′ , x ) x − θ can be described as π k ( Y ) = ((cid:18) a b (cid:19) ∈ N k : ∃ y, z ∈ M n such that (cid:18) a yz b (cid:19) = 0 ) ;(11.1) π s ( Y ) = ((cid:18) yz (cid:19) ∈ s : ∃ a, b ∈ M n nilpotent, such that (cid:18) a yz b (cid:19) = 0 ) . (11.2)Given a, b, y, z ∈ M n , note that the equality (cid:18) a yz b (cid:19) = 0 is equivalent to the followingcondition:(11.3) a + yz = b + zy = ay + yb = za + bz = 0 . Since π k ( Y ) (resp. N X , k ) is a K -stable subset of N k , it is a union of nilpotent K -orbits of theform O λ × O µ for pairs of partitions ( λ, µ ) ∈ P ( n ) × P ( n ). Similarly, since π s ( Y ) (resp. N X , s ) is a K -stable subset of N s , it is a union of K -orbits of the form O Λ correspondingto certain signed Young diagrams Λ ∈ P ± (2 n ). Lemma 11.1. (a) O λ × O µ ⊂ π k ( Y ) = ⇒ O µ × O λ ⊂ π k ( Y ) ; (b) O Λ ⊂ π s ( Y ) = ⇒ O Λ ⊂ π s ( Y ) , where we denote by Λ the signed Young diagramobtained from Λ by switching the + ’s and the − ’s.Proof. The property follows by observing that the quadruple ( a, b, y, z ) satisfies (11.3) ifand only if ( b, a, z, y ) satisfies (11.3), and then by invoking (11.1)–(11.2). (cid:3)
Notation. (a) Let Λ ∈ P ± (2 n ) be a signed Young diagrams with k rows. Let Λ i [+)(resp. Λ i [ − )) be the number of +’s (resp. − ’s) contained in the i -th row of Λ but not inthe rightmost box of the row. Let Λ[+) and Λ[ − ) be the partitions corresponding to thelists of numbers (Λ [+) , . . . , Λ k [+)) and (Λ [ − ) , . . . , Λ k [ − )) after rearranging the terms innonincreasing order and erasing the terms equal to zero if necessary. For instance,Λ = + − + − + − + − + − ++ − + −− + − + − + − = ⇒ Λ[+) = and Λ[ − ) = . (b) We consider partitions which satisfy the following condition:(11.4) the partition λ = ( λ , . . . , λ k ) satisfies (cid:26) λ i − − λ i ∈ { , } ∀ i ∈ { , . . . , ⌊ k ⌋} , if k is odd then λ k = 1.If x is a nilpotent matrix whose Jordan normal form is encoded by the partition µ =( µ α , . . . , µ α ℓ ℓ ) with numbers µ > . . . > µ ℓ and multiplicities α , . . . , α ℓ ≥
1, then theJordan normal form of x is encoded by the partition ( ⌈ µ ⌉ α , ⌊ µ ⌋ α , . . . , ⌈ µ ℓ ⌉ α ℓ , ⌊ µ ℓ ⌋ α ℓ ).This readily implies that(11.5) a partition λ ∈ P ( n ) encodes the Jordan normal form of a nilpotent matrixof the form x if and only if λ satisfies (11.4). Lemma 11.2.
Let Λ ∈ P ± (2 n ) be a signed Young diagram. (a) If the K -orbit O Λ is contained in π s ( Y ) , then the partitions Λ[+) and Λ[ − ) satisfycondition (11.4). (b) Any x ∈ π s ( Y ) satisfies x n = 0 if n is even and x n +1 = 0 if n is odd. Thus, if O Λ iscontained in π s ( Y ) , then Λ has at most n (resp. n + 1 ) columns if n is even (resp. odd).Proof. (a) Assume that O Λ ⊂ π s ( Y ). Take (cid:18) yz (cid:19) ∈ O Λ . By (11.2), there existnilpotent matrices a, b ∈ M n such that the relations in (11.3) hold. The subspace Im z isstabilized by the matrix zy . The last equality in (11.3) implies that Im z is also stabilizedby b , and the equality b + zy = 0 yields zy | Im z = − ( b | Im z ) . Note that the Jordan normalform of the nilpotent endomorphism zy | Im z : Im z → Im z corresponds to the partitionΛ[ − ). It therefore follows from (11.5) that the partition Λ[ − ) satisfies (11.4). A similarargument (or Lemma 11.1) implies that Λ[+) also satisfies (11.4).(b) Any element x ∈ π s ( Y ) is of the form x = (cid:18) yz (cid:19) with y, z ∈ M n and such thatthere exist nilpotent matrices a, b ∈ M n satisfying (11.3). Let m be any even number suchthat m ≥ n . Then x m = (cid:18) yz (cid:19) m = (cid:18) ( yz ) m
00 ( zy ) m (cid:19) = ( − m (cid:18) a m b m (cid:19) = 0 . The proof is complete. (cid:3)
GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 45
Proof of Theorem 1.6.
Recall that the K -orbits of X are parametrized by the elements ω ∈ ( T n ) ′ , and each orbit Z ω gives rise to a conormal bundle T ∗ Z ω X ⊂ Y . Thus π k ( T ∗ Z ω X ) ⊂ N X , k and π s ( T ∗ Z ω X ) ⊂ N X , s . For a matrix ω = (cid:18) τ n (cid:19) corresponding to a partial permutation τ ∈ T n , Theorems9.1 and 10.4 describe the K -orbits which are dense in π k ( T ∗ Z ω X ) and π s ( T ∗ Z ω X ). Choosing τ = 1 n , we get (by Theorem 9.1)Ξ k ( ω ) = (shape(RS (1 n )) , shape(RS (1 n ))) = (( n ) , ( n )) , hence O ( n ) × O ( n ) = π k ( T ∗ Z ω X ) ⊂ N X , k . Since the K -orbit O ( n ) × O ( n ) is dense in N k , we already obtain N X , k = N k .It remains to consider N X , s . For n = 1, the equality N X , s = N s easily follows from(11.2). Hereafter we assume that n ≥
2. Choosing τ = 1 n , we get (by Theorem 10.4, andin view of Example 10.5 (a))Ξ s ( ω ) = + − + − · · ·− + − + · · · = Λ (as in Theorem 1.6) hence O Λ = π s ( T ∗ Z ω X ) ⊂ N X , s . Choosing τ as in Example 10.5 (b), we have Ξ s ( ω ) = Λ − (the signed Young diagram ofTheorem 1.6 (a)–(b)) hence O Λ − ⊂ N X , s . From Lemma 11.1 (b), we deduce O Λ + ⊂ N X , s with Λ + as in Theorem 1.6 (a)–(b). Altogether this yields O Λ + ∪ O Λ ∪ O Λ − ⊂ N X , s . It remains to show the reversed inclusion.First assume that n is even. In this case, the signed Young diagrams Λ , Λ + , and Λ − are described in Theorem 1.6 (a). For any K -orbit O Λ ⊂ π s ( Y ), the corresponding signedYoung diagram Λ has at most n columns by Lemma 11.2 (b), hence we have Λ (cid:22) Λ ,Λ (cid:22) Λ + , or Λ (cid:22) Λ − (see Remark 5.3 (b)), and we get O Λ ⊂ O Λ + ∪ O Λ ∪ O Λ − . Weconclude that the inclusion N X , s ⊂ O Λ + ∪ O Λ ∪ O Λ − holds in this case.Finally assume that n is odd. Then, the signed Young diagrams Λ , Λ + , and Λ − aredescribed in Theorem 1.6 (b). Let O Λ be a K -orbit contained in π s ( Y ). By Lemma11.2 (b), the signed Young diagram Λ has at most n + 1 columns. If Λ has at most n columns, then O Λ ⊂ O Λ + ∪ O Λ ∪ O Λ − (see Remark 5.3 (b)). It remains to consider thecase where Λ has n + 1 columns, i.e., the first row of Λ has length n + 1. Say that the lastbox of this row contains the symbol + (the other case is similar), thus Λ [ − ) = n +12 . IfΛ is not Λ − , then the second row of Λ has length < n − − ; inboth cases we get Λ [ − ) < n − , hence the signed Young diagram Λ does not satisfy (11.4),so that Lemma 11.2 (a) yields a contradiction. Therefore Λ + and Λ − are the only signedYoung diagrams with exactly n + 1 columns whose corresponding K -orbits are contained in π s ( Y ). Altogether, we obtain the desired inclusion N X , s = π s ( Y ) ⊂ O Λ + ∪ O Λ ∪ O Λ − .The proof of the theorem is complete. (cid:3) Remark 11.3. (a) Theorem 1.6 shows that the set π k ( Y ) is dense in N k , however this setis not closed (thus the map π k : Y → N k is not surjective) unless n ≤
3. (For n ≤
3, it isstraightforward to see that π k ( Y ) = N k .)For n ≥
4, let us see that a K -orbit of the form O λ × O (1 n ) (for λ ∈ P ( n )) is notcontained in π k ( Y ) whenever λ >
3. Indeed, take a ∈ O λ and b = 0 ∈ O (1 n ) . Assumethat O λ × O (1 n ) ⊂ π k ( Y ). Then, in view of (11.1), there are matrices y, z ∈ M n satisfying(11.3). Whence a = − a ( yz ) = ( yb ) z = 0 , so that λ ≤ π s : Y → s is not closed, unless n ≤
2. Indeed, for n ≥ O Λ corresponding to the signed Young diagramΛ = + − + − +... − ... ∈ P ± (2 n )is not contained in π s ( Y ), whereas Theorem 1.6 shows that O Λ ⊂ π s ( Y ). Index of notation B , N = N g , St1.2 K , k , s , N k , N s , X , Y , π k , π s P ( n ), shape( T )2.2 ( T ← a ), ( a → T ), RowInsert, ColumnInsert2.3 RS ( w ), RS ( w )2.4 T ∗ S Y X , µ X nil § O λ G , K , V + , V − , k , s , x θ , x − θ N k , N s , λ ≤ k , (cid:22) , P ± (2 n ), ≤ k (+), ≤ k ( − ), O Λ , B K , P S (type AIII) § n , Y M n , T n , O τ , Y τ § B , n , Φ = (Φ , Φ ) § T n ) ′ § Z ω , Ξ k §
10 Ξ s , △ λ ] GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 47
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E-mail address : [email protected] Department of Physics and Mathematics, Aoyama Gakuin University, Fuchinobe 5-10-1,Sagamihara 229-8558, Japan
E-mail address : [email protected] GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 49 τ σ (RS ( σ ) , RS ( σ )) ( T , T , ν ) Φ( τ ) = Ξ k (( τ )) Ξ s (( τ ))( ) = τ , , , , + − + − + − ( ) = τ , , , , + −− ++ − ( ) = τ , , , ( ) = τ , , , ( ) = τ , , , ( ) = τ ,
123 123 , , , +++ −−− Figure 1.
The correspondence τ ( T , T , ν ) for T (rank τ = 3); themaps Φ, Ξ k , and Ξ s τ σ (RS ( σ ) , RS ( σ )) ( T , T , ν ) Φ( τ ) = Ξ k (( τ )) Ξ s (( τ ))( ) ( ) , , , , − + − + − + ( ) ( ) , , , , + − + − + − ( ) ( ) , , , ( ) ( ) , , , , − + −− ++ ( ) ( ) , , , ( ) ( ) , , , , + −− + − + ( ) ( ) , , , ( ) ( ) , , , ( ) ( ) , , , ( ) ( ) ,
13 1 32 , , − + − ++ − ( ) ( ) ,
23 1 23 , , ( ) ( ) ,
13 1 23 , , ( ) ( ) ,
23 1 32 , , ( ) ( ) ,
23 123 , , , − +++ −− ( ) ( ) ,
13 123 , , ( ) ( ) ,
12 1 32 , , , ( ) ( ) ,
12 1 23 , , ( ) ( ) ,
12 123 , , , Figure 2.
The correspondence τ ( T , T , ν ) for T (rank τ = 2); themaps Φ, Ξ k , and Ξ s GENERALIZATION OF STEINBERG THEORY AND AN EXOTIC MOMENT MAP 51 τ σ (RS ( σ ) , RS ( σ )) ( T , T , ν ) Φ( τ ) = Ξ k (( τ )) Ξ s (( τ ))( ) ( ) , , , , − + − + − + ( ) ( ) , , , ( ) ( ) , , , ( ) ( ) , , , ( ) ( ) , , , , − + − ++ − ( ) ( ) , , , ( ) ( ) , , , , ( ) ( ) , , , ( ) ( ) , , , , ( ) ∅ ∅ , ∅ , , ∅ , − + − + − + Figure 3.
The correspondence τ ( T , T , ν ) for T (rank τ ≤ k , and Ξ, and Ξ