On generalized Steinberg theory for type AIII

Given a symmetric pair $(G,K)=(\mathrm{GL}_{p+q}(\mathbb{C}),\mathrm{GL}_{p}(\mathbb{C})\times \mathrm{GL}_{q}(\mathbb{C}))$ of type AIII, we consider the diagonal action of $K$ on the double flag variety $\mathfrak{X}=\mathrm{Grass}(\mathbb{C}^{p+q},r)\times K/B_K$ whose first factor is a Grassmann variety for $G$ and whose second factor is a full flag variety of $K$. There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization, dimensions, closure relations, and cover relations. Specifically, the orbits are parametrized by certain pairs of partial permutations. Each orbit in $\mathfrak{X}$ gives rise to a conormal bundle. As in the references [5] and [6], by using the moment map associated to the action, we define a so-called symmetrized Steinberg map, respectively an exotic Steinberg map, which assigns to each such conormal bundle (thus to each orbit) a nilpotent orbit in the Lie algebra of $K$, respectively in the Cartan complement of that Lie algebra. Our main result is an explicit description of these Steinberg maps in terms of combinatorial algorithms on partial permutations, extending the classical Robinson--Schensted procedure on permutations. This is a thorough generalization of the results in [5], where we supposed $p=q=r$ and considered orbits of special forms.

1. Introduction 1.1.A multiple flag variety and its orbital decomposition.In this paper, we consider the multiple flag variety (1.1) X = Gr(V, r) × Fl(V + ) × Fl(V − ), where • V = C p+q is equipped with a polar decomposition V = V + ⊕ V − with V + = C p × {0} q and V − = {0} p × C q ; • Gr(V, r) denotes the Grassmann variety of r-dimensional subspaces of V ; • Fl(V + ) and Fl(V − ) denote the varieties of complete flags of V + and V − , respectively.Each factor of the variety X has a natural action of and X is endowed with the resulting diagonal action of K.
The multiple flag variety X can be written in the form where G = GL(V ), P ⊂ G is a maximal parabolic subgroup, B K ⊂ K is a Borel subgroup.In this way, X is a double flag variety associated to the symmetric pair (G, K) in the sense of [10] and [8].In particular, it is known from [10] that the above variety X has a finite number of K-orbits.
In [3,4], we have initiated an analogue of Steinberg theory for double flag varieties associated to symmetric pairs such as X.Specifically, we have defined two Steinberg maps, from the set of K-orbits of X to the sets of nilpotent K-orbits of k := Lie(K) and of its Cartan complement s, respectively.
For general symmetric pairs, the calculation of the generalized Steinberg maps appears to be quite difficult.In [4], we have considered the variety X of (1.1) in the special case where p = q = r and we have computed the Steinberg maps on a special subset of Korbits parametrized by partial permutations.The present paper deals with the variety X of (1.1) and its K-orbits in full generality.
Here we summarize the main results achieved in this paper: • We describe completely the decomposition of X into K-orbits: we give a parametrization of the orbits (in terms of certain graphs), we provide a dimension formula, and describe the closure relations and the cover relations; see Section 2.2.• Our main result (Theorem 2.5) is the calculation of the two Steinberg maps mentioned above.This calculation is concrete, and it is done by means of a sophisticated combinatorial algorithm that generalizes the Robinson-Schensted correspondence; see Sections 2.3-2.4.In the following subsection, we explain the construction of the generalized Steinberg maps and we give more insight on our main result.We write x = x k + x s with (x k , x s ) ∈ k × s for the decomposition of an element x ∈ g along the Cartan decomposition.Moreover, we identify the Lie algebras g and k with their duals g * and k * through the trace form.Any partial flag F = (F 0 = 0 ⊂ F 1 ⊂ . . .⊂ F k = V ′ ) of a vector space V ′ gives rise to a parabolic subalgebra p(F ) of gl(V ′ ) and the corresponding nilradical nil(F ) defined by For a subspace W ⊂ V ′ , we denote by p(W ) and nil(W ) the (maximal) parabolic subalgebra and the nilradical associated to the partial flag (0 ⊂ W ⊂ V ′ ).
Every K-orbit O ⊂ X yields a conormal bundle T * O X that can be realized as a (locallyclosed) subvariety of Y given by T * O X = {(W, F + , F − , x) ∈ Y : (W, F + , F − ) ∈ O}.The variety Y is equidimensional of dimension dim X, and its irreducible components are precisely the closures of the various conormal bundles T * O X, since the set of orbits X/K is finite.One can find a more comprehensive introduction to the theory of conormal varieties in [4, §3] (see also [1]).
The conormal variety Y is equipped with two K-equivariant projections to k and s, namely φ k : Y → k, (W, F + , F − , x) → x k and φ s : Y → s, (W, F + , F − , x) → x s .
It immediately follows from the description of the conormal variety that the image of φ k is contained in the cone of nilpotent elements N k ⊂ k.It is shown in [3,Proposition 4.2] that (for the variety X considered in this paper) the image of φ s is also contained in the nilpotent cone N s ⊂ s.It is known that both nilpotent cones N k and N s consist of finitely many adjoint K-orbits Therefore, we can define two maps Φ k : X/K → N k /K and Φ s : X/K → N s /K in the following way: for every orbit O ∈ X/K, define Φ k (O) ∈ N k /K, resp.Φ s (O) ∈ N s /K, as the unique nilpotent K-orbit which is open and dense in the image of the conormal bundle T * O X by the projection map φ k , resp.φ s .According to the terminology introduced in [4], we will refer to Φ k as the symmetrized Steinberg map and to Φ s as the exotic Steinberg map.
In Section 2.3, we describe the maps Φ k and Φ s .In [4], in the special case where p = q = r, the images Φ k (O) and Φ s (O) are determined when an orbit O is contained in a "big cell" of X/K.Moreover, in Section 2.4, we describe the fibers of Φ k by means of a combinatorial procedure that extends the classical Robinson-Schensted correspondence; this also generalizes [4,Theorem 7.8].In this way, the results in the present paper are new and complement those in [4], as now we have a full description of the two Steinberg maps and a better understanding of them at the same time.Note that the results given in Section 2.3 regarding Φ k and for p = q = r were already announced in [5, §2] without proofs.

2.1.
Combinatorial notation on pairs of partial permutations.By T p,r we denote the set of p × r matrices whose coefficients are 0 or 1, with at most one 1 in each row and each column.(If p = r, we recover the set of partial permutation matrices considered in [4].)By T = T (p,q),r we denote the set of (p + q) × r matrices of rank r (we have r ≤ p + q) of the form where τ 1 ∈ T p,r and τ 2 ∈ T q,r .Note that the symmetric group S r acts on T by right multiplication, and we denote the quotient set by T = T/S r .In Section 2.2, we will show that the elements of T parameterize the K-orbits of X.
Graphic representation of a pair of partial permutations.
We represent any element ω ∈ T by a graph G(ω) obtained as follows: • The set of vertices consists of p "positive" vertices 1 + , . . ., p + and q "negative" vertices 1 − , . . ., q − , displayed along two horizontal lines.• Put an edge between i + and j − for every column of ω that contains exactly two 1's, in positions i (within the block τ 1 ) and p + j (within the block τ 2 ).• Put a mark at the vertex i + , respectively j − , for every column of ω that contains exactly one 1, in position i (within τ 1 ), respectively p + j (within τ 2 ).For instance, for (p, q) = (5, 3) and r = 4, We will say that a vertex is free whenever it is not a marked point nor an end point of an edge (like 1 + and 3 + in the above example).
In general, the assignment ω → G(ω) establishes a bijection between T and the set of graphs with vertices {1 + , . . ., p + } ∪ {1 − , . . ., q − }, exactly r edges or marked vertices, where every vertex is incident with at most one edge, and such that there is no edge which is incident with a marked vertex or joins two vertices of the same sign.

Numerical invariants for the graph.
The following data associated to an element ω ∈ T and its graphic representation G(ω) will play a role in the statement of our main results.
• Set the degree of a vertex of G(ω) as 0, 1, or 2, depending on whether this vertex is free, incident with an edge, or marked.We define a + (ω), respectively a − (ω), as the number of pairs of positive vertices (i + , j + ) with i < j and deg(i + ) < deg(j + ), respectively pairs of negative vertices (i − , j − ) with i < j and deg(i Let b(ω) be the number of edges of G(ω).
In Section 2.2, the numbers a ± (ω), b(ω), c(ω) appear in the dimension formula for the K-orbits of X, while the matrices R(ω) are used to describe the inclusion relations between orbit closures.
Let σ : J → I be the bijection defined by letting σ(j Note that ω is characterized by the subsets I, L, L ′ , J, M, M ′ and the bijection σ : J → I.In Section 2.3, these data are used to compute the symmetrized and exotic Steinberg maps, by means of a combinatorial algorithm. Note also that we have b(ω) = #I = #J, and c(ω) is the number of inversions of σ.
Example 2.1.Let ω be as in (2.1).Then, Note that the matrix R(ω) can also be viewed as a plane partition.

2.2.
Orbit decomposition of the multiple flag variety X. Recall that we consider the space V = C p+q endowed with the polar decomposition be the standard complete flags of V + and V − .Every (p + q) × r matrix ω determines a subspace [ω] := Im ω ⊂ V , which remains the same up to permutation of the columns of ω.In particular, every ω ∈ T determines a point [ω] in Gr(V, r), and thus a point ( As a complement of this result, we determine the cover relations in the poset ({O ω }, ⊂).We say that O ω ′ covers O ω if O ω ′ strictly contains O ω and is minimal (among the orbit closures) for this property.Equivalently, this means that O ω is an irreducible component of the boundary Theorem 2.3.The following conditions are equivalent: (1)   2, we represent the elements ω ∈ T (under the form of their graphic incarnations G(ω)) in the case where p = q = r = 2.We indicate the dimensions of the corresponding K-orbits O ω .An edge joining two parameters indicates a cover relation.dim : 6 The parameters of the K-orbits of X and the cover relations for p = q = r = 2.
(b) In Figure 1, the vertices a + , b + or c − , d − involved in an elementary move that yields a cover relation are not necessarily consecutive.For example, This corresponds to Case 1 of Figure 1 with a = c = 1, b = d = 3.Note however that in Case 5 of Figure 1, the vertices a + , b + (resp.c − , d − ) must be consecutive for having a cover relation.
The proofs of Theorems 2.2 and 2.3 are given in Section 3. One ingredient for showing the parametrization of the orbits in Theorem 2.2 (1) is that the orbits of a pair of Borel subgroups B + p × B + r ⊂ GL p (C) × GL r (C) on the space of p × r matrices are parametrized by partial permutations (Lemma 3.4).This classification of orbits is also shown in [6], where dimension formulas, closure relations, and properties of closures of orbits are described.
2.3.Description of symmetrized and exotic Steinberg maps.We turn our attention to the maps Φ k : X/K → N k /K and Φ s : X/K → N s /K defined in Section 1.2.The three orbit sets arising here can be parametrized combinatorially.
• X/K = {O ω : ω ∈ T} (see Section 2.2).For the other two orbit sets, the parametrization is well known (see, e.g., [2]): • N k is the nilpotent cone of the Lie algebra and its adjoint K-orbits O (λ,µ) are parametrized by pairs of partitions λ ⊢ p and µ ⊢ q (viewed as Young diagrams) through Jordan normal form.Specifically, the number of boxes in the first k columns of λ (resp., µ) indicates the dimension of ker a k (resp., ker d k ).
and its adjoint K-orbits O Λ are parametrized by signed Young diagrams Λ of signature (p, q).Specifically, the number of +'s (resp., −'s) in the first k columns of Λ indicates the dimension of V + ∩ ker x k (resp., V − ∩ ker x k ) for x ∈ O Λ .We give a combinatorial algorithm which describes Φ k and Φ s completely.If w : S → R is a bijection between two sets of integers, let (RS 1 (w), RS 2 (w)) denote the pair of Young tableaux associated to w via the Robinson-Schensted correspondence, so that the set of entries of RS 1 (w) (resp., RS 2 (w)) is R (resp., S) (see, e.g., [7]).
We prove this theorem in Section 4.
The tableaux RS 1 (w k,+ ) and RS 1 (w k,− ) involved in Theorem 2.5 can also be obtained as the result of the following combinatorial algorithms.We need more notation: • If T, S are Young tableaux with disjoint sets of entries, we denote by T * S the rectification by jeu de taquin of the skew tableau obtained by displaying S on the top right corner of T .For example, 1 2 4 5  3 7  6 .
If U is a third tableau whose entries do not appear in T nor S, the properties of jeu de taquin imply that (T * S) * U = T * (S * U) (see [7]), hence the notation denote the vertical Young tableaux whose entries are the elements in L, L ′ , M, M ′ , respectively.We then have: In fact, we can determine the fibers of Φ k in terms of a combinatorial correspondence which extends the Robinson-Schensted correspondence.The following theorem also generalizes [4,Theorem 7.6].We use the previous notation.In addition, we write λ ′ ⊂ • λ whenever λ ′ is a Young subdiagram of λ such that the skew diagram λ \ λ ′ is column strip (i.e., it contains at most one box in each row).Hereafter, P(n) denotes the set of partitions λ ⊢ n, also seen as Young diagrams of size |λ| = n.Theorem 2.9.There is an explicit bijection gRS : where T λ,µ is the set of 5-tuples (T 1 , T 2 ; λ ′ , µ ′ ; ν) satisfying (⋆) T 1 and T 2 are standard Young tableaux of shapes λ and µ, respectively; Specifically, to the element ω ∈ T, we associate the 5-tuple By combining Theorems 2.5 (a) and 2.9, we get a commutative diagram from which we have that gRS restricts to a bijection Proof.First, we note that the considered map is well defined: the fact that λ \ λ ′ , λ ′ \ ν, µ \ µ ′ , and µ ′ \ ν are column strips follows from [7, Proposition in §1.1], and we have where, as before, s = #L, t = #M, k = #I = #J.Next, we show that the map is bijective.Let (T 1 , T 2 ; λ ′ , µ ′ ; ν) ∈ T λ,µ .Applying twice [7, Proposition in §1.1], we find that there is a unique 6-tuple (S 1 , S 2 , L, L ′ , M, M ′ ), where S 1 , S 2 are Young tableaux of shape ν and L, L ′ , M, M ′ are sets of integers, such that , and µ ′ = shape([M] * S 2 ) (understanding that the contents of L, S 1 , L ′ are disjoint, as well as those of M, S 2 , M ′ ).Let I (resp., J) be the set of entries of S 1 (resp., S 2 ).By the Robinson-Schensted correspondence, we get (S 1 , S 2 ) = (RS 1 (σ), RS 2 (σ)) for a unique bijection σ : J → I.Then, the data I, L, L ′ , J, M, M ′ , σ determine a unique element ω ∈ T (see Section 2.1) such that gRS(ω) = (T 1 , T 2 ; λ ′ , µ ′ ; ν).
(we encode the 5-tuple as the pair of tableaux (T 1 , T 2 ) where the boxes of λ ′ , µ ′ , ν are colored).(b) In Figure 4, we give the bijection of Theorem 2.9 in the case where p = 3, q = r = 2.In this case, the set T has 34 elements.
Remark 2.11.We point out that Singh [12] has recently developed a Robinson-Schensted correspondence for partial permutations.Specifically, he has established a bijection between the set of partial permutations of size p × q and a set of triples (Λ, P, Q) consisting of a signed Young diagram of size p + q and two standard Young tableaux of sizes p and q.Note that, if r = q, the set of partial permutations of size p × q can be realized in a natural way as a subset of our set T. One can ask whether there is a relation between the correspondence in Theorem 2.9 and the bijection given in [12, Theorem A].
We refer to [11,15,16] for other generalizations of the Robinson-Schensted correspondence that arise in geometry.
We derive from Theorem 2.9 an interpretation of the cardinals of the fibers Φ λ denote the corresponding irreducible representation of S n .In this way, the (isomorphism classes of) irreducible representations ρ µ of S p × S q are parametrized by the pairs of partitions (λ, µ) ∈ P(p) × P(q).Here ⊠ stands for the outer tensor product.
For every triple of nonnegative integers (k, s, t) such that we consider the subgroup of S p × S q given by where ∆S k stands for the diagonal embedding of (1 q ) ).The induced representation Ind Sp×Sq H k,s,t ε decomposes as a sum of irreducible representations where m k,s,t (λ, µ) denotes the multiplicity of ρ µ .The next corollary is a generalization of [4,Corollary 7.10].
The intermediate induced representation Ind Hk,s,t H k,s,t ε can be written as where ν ) * is the regular representation of S k × S k .Applying twice the Pieri rule (see [7, §2.2 and §7.3]), we have for all triples (k, s, t).Note also that dim ρ µ is the number of pairs of standard Young tableaux (T 1 , T 2 ) of shape λ and µ, respectively.The claimed equality now follows from Theorem 2.9.
Corollary 2.13.The total number of K-orbits in X is given by where the sums are over triples (k, s, t) satisfying (2.8).

On the decomposition of X into K-orbits
The purpose of this section is to prove the results stated in Theorems 2.2 and 2.3, regarding the decomposition of By B + k ⊂ GL k (C) we denote the subgroup of invertible upper triangular matrices in GL k (C).Then is a Borel subgroup of K. Recall that F + 0 and F − 0 denote the standard flags of V + and V − .Thus, B K is the stabilizer of the pair (F + 0 , F − 0 ) for the action of K on the product of flag varieties Fl(V + ) × Fl(V − ).In Section 3.1, we observe that there is a one-toone correspondence between the orbit sets X/K and Gr(V, r)/B K , which preserves the inclusion relations between closures of orbits.This fact is a useful ingredient in the rest of the section.
In Section 3.2, we show the parametrization of orbits and the dimension formula stated in Theorem 2.2 (1)- (3).In Section 3.3, we describe the closure relations of orbits by proving Theorems 2.2 (4) and 2.3.In Section 3.4, we make further remarks and mention relations with the existing literature.
3.1.A preliminary lemma.We will use the following lemma.Lemma 3.1.
(1) The mapping Gr(V, r) → X, W → (W, F + 0 , F − 0 ) induces a one-toone correspondence between the orbit sets Lemma 3.1 is a consequence of the following lemma (which applies to a general connected reductive group K).As already used in Lemma 3.1, K × Q X stands for the quotient of K × X by the action of Q given by q • (k, x) = (kq −1 , qx), and [k, x] denotes the class of (k, x) in this quotient.Lemma 3.2.Let K be a connected reductive group and let Q ⊂ K be a parabolic subgroup.Let X be an algebraic variety endowed with an action of K. Consider the diagonal action of K on X := K/Q × X.Note that there is an isomorphism χ : X → K × Q X given by χ(kQ, x) = [k, k −1 x].Also we consider the closed immersion ι : X → X, x → (Q, x).
(1) There is a one-to-one correspondence (in fact an isomorphism of partially ordered sets) given by Ξ(M) = K • ι(M).The inverse bijection is given by N → ι −1 (N).Moreover, Ξ restricts to a one-to-one correspondence between the orbit sets X/Q and X/K.
Though this lemma is well known, we give a proof for the sake of completeness. Proof.
(1) The map Ξ is certainly well defined.For a Q-stable subset M ⊂ X, the inclusion For a K-stable subset N ⊂ X, given x ∈ ι −1 (N) and q ∈ Q we have We have thus shown that Ξ is a bijection, with inverse bijection given by N → ι −1 (N).Note also that the implications are clear, which show that Ξ is in fact an isomorphism of posets. Finally This subset can also be written as Since K/Q is complete, we conclude that N is closed.
More generally, using that Ξ(M ) and Ξ −1 (N ) are closed, and the fact that Ξ is an isomorphism of posets, we get Remark 3.3.In Lemma 3.2, the assumption that Q is parabolic is used only in part (3).Also note that the isomorphism χ is K-equivariant when K acts on X = K/Q × X diagonally and on K × Q X by left multiplication.

Parametrization and dimension formula -proof of Theorem 2.2 (1)-(3).
As before, we denote by B + p ⊂ GL p (C) (resp., B + r ⊂ GL r (C)) the Borel subgroup of upper triangular matrices.We need two lemmas.The first one is an analogue of [4, Proposition 6.3].It is also shown in [6, p. 390], but we give a proof for the sake of completeness.
r , and a unique τ ∈ T p,r .Proof.Through Gauss elimination, any p × r matrix a can be transformed into a matrix τ ∈ T p,r by a series of operations consisting of multiplying a row (resp., a column) by a nonzero scalar or adding to a row (resp., to a column) another row (resp., column) situated below it (resp., on its left).These operations correspond to multiplying on the left (resp., on the right) by an element of B + p (resp., B + r ).Hence the double coset B + p aB + r contains an element τ ∈ T p,r , which means that a ∈ B + p τ B + r .
This implies that two different elements τ, τ ′ ∈ T p,r cannot belong to the same double coset B + p aB + r , whence the uniqueness.The second lemma is analogous to [4, Lemma 8.2].Lemma 3.5.For every τ ∈ T p,r , there is a permutation w ∈ S r such that τ wB + r ⊂ B + p τ w.Proof.By (e ℓ 1 , . . ., e ℓ ℓ ) we denote the standard basis of C ℓ .By e ℓ i,j we denote the elementary ℓ×ℓ matrix whose (i, j) coefficient is 1 and the other coefficients are 0. By diag(t 1 , . . ., t ℓ ), we denote the diagonal matrix with coefficients t 1 , . . ., t ℓ along the diagonal.
Let k = rank τ .Let 1 ≤ i 1 < . . .< i k ≤ p be such that Im τ = e p i 1 , . . ., e p i k .We choose w ∈ S r such that τ w(e r j ) = 0 for 1 ≤ j ≤ r − k and τ w(e r r−k+j ) = e p i j for 1 ≤ j ≤ k.For every diagonal matrix t = diag(t 1 , . . ., t r ) ∈ B + r , we have τ wt = t ′ τ w ∈ B + p τ w for any diagonal matrix t ′ = diag(t ′ 1 , . . ., t ′ p ) ∈ B + p such that t ′ i j = t r−k+j for all j ∈ {1, . . ., k}.For every transvection u = 1 r + xe r j,ℓ ∈ B + r where 1 ≤ j < ℓ ≤ r, we have p τ w in each case.Since B + r is generated by the diagonal matrices and the transvections, we conclude that τ wB + r ⊂ B + p τ w.Now we are ready to prove parts (1) and (3) of Theorem 2.2.
If ω, ω ′ are two different elements of the set T, then their graphs G(ω), G(ω ′ ) must be different, hence the matrices R(ω) = (r i,j (ω)) and R(ω ′ ) = (r i,j (ω ′ )) are different.From (3.1), it follows that the orbits O ω and O ω ′ are disjoint.Therefore, X is the disjoint union of the orbits O ω for ω ∈ T. This also implies that the inclusion in (3.1) must be an equality for all ω ∈ T. This establishes Theorem 2.2 (1) and (3).

Proof of Theorem 2.2 (2). Lemma 3.1 implies
, where b + p = Lie(B + p ) ⊂ M p (C) and b + q = Lie(B + q ) ⊂ M q (C) are the subspaces of upper triangular matrices.We have dim As before, we denote by (e + 1 , . . ., e + p ), resp.(e − 1 , . . ., e − q ), the standard basis of V + = C p × {0} q , resp.V − = {0} p × C q .The linear space [ω] is spanned by the vectors e + a with a ∈ {1, . . ., p} such that the graph G(ω) has a mark at a + , the vectors e − c with c ∈ {1, . . ., q} such that there is a mark at c − , and the linear combinations e + a + e − c for all pairs (a, c) ∈ {1, . . ., p} × {1, . . ., q} such that G(ω) has an edge joining a + and c − .This implies that a matrix z = x 0 0 y ∈ b K satisfies z([ω]) ⊂ [ω] if and only if the upper triangular matrices x = (x i,j ) 1≤i,j≤p and y = (y i,j ) 1≤i,j≤q satisfy the following equations: (1) For every a ∈ {1, . . ., p} such that G(ω) has a mark at a + , we must have x i,a = 0 for all i < a such that there is no mark at i + .(2) For every c ∈ {1, . . ., q} such that G(ω) has a mark at c − , we must have y j,c = 0 for all j < c such that there is no mark at j − .(3) For every pair (a, c) ∈ {1, . . ., p} × {1, . . ., q} such that G(ω) has an edge (a + , c − ), we must have x i,a = 0 for all i < a such that i + is a free vertex (i.e.not marked nor incident with an edge) in G(ω), and we must have y j,c = 0 for all j < c such that j − is a free vertex in G(ω).( 4) For (a, c) as in (3), we must also have x i,a = y j,c for all pair (i, j) ∈ {1, . . ., a} × {1, . . ., c} such that (i + , j − ) is an edge in G(ω), i.e., for all edge which is situated on the left of (a + , c − ) or coincides with (a + , c − ) itself.Finally, we get one more equation x i,a = 0, or resp.y j,c = 0, for every edge (i + , j − ) which has a crossing with (a + , c − ), i.e., such that i < a and c < j, resp.i > a and j < c.We have listed linearly independent equations which characterize the subspace {z ∈ b K : Combining this equality with (3.2) and (3.3), we get the dimension formula stated in Theorem 2.2 (2).
Lemma 3.6.Assume that ω is obtained from ω ′ by one of the elementary moves described in Figure 1.Then, the following relations hold: Proof.We consider the cases described in Figure 1.
As before, we denote by e k i,j the k × k elementary matrix with 1 at position (i, j) and 0 elsewhere.Then, let u k i,j (t) = 1 k + te k i,j and δ k i (t) = 1 k + (t − 1)e k i,i .For t ∈ C * , we consider the matrix h t given by where A t and D t are blocks of respective sizes p × p and q × q given by in Cases 2 + , 4 + , 5 + .
Here the notation N + (resp., N − ) refers to the upper (resp., lower) subcase of Case N in Figure 1.In each case, we obtain a subset {h t } t∈C * ⊂ K such that and this shows that the inclusion O ω ⊂ O ω ′ holds.
In the case where G(ω) and G(ω ′′ ) have one common edge or mark -call it x, by removing this edge or mark together with the corresponding vertices (or vertex), and after renumbering of the vertices, we obtain subgraphs G(ω) = G(ω) \ x and G(ω ′′ ) = G(ω ′′ ) \ x associated to smaller sized matrices ω and ω′′ , and we still have ω ≺ ω′′ due to the definition of the relation .The induction hypothesis yields ω′ with ω ω′ ω′′ , whose associated graph G(ω ′ ) yields G(ω) or is yielded by G(ω ′′ ) through one of the elementary moves described in Figure 1.There is an element ω ′ ∈ T such that G(ω ′ ) = G(ω ′ ) \ x, and this element satisfies the requirements of the lemma.In conclusion, (3.4) we may assume that G(ω) and G(ω ′′ ) have no common edge nor marked vertex.
We define the set E(ω ′′ ) relative to ω ′′ in the same way.Both sets E(ω) and E(ω ′′ ) have r elements, in particular they are nonempty.
We choose an element (a 0 , c 0 ) ∈ E(ω ′′ ) with the minimal possible value of c 0 .If c 0 = 0, then c − 0 is not a free vertex in G(ω ′′ ): it is marked if a 0 = 0 or incident with an edge (a + 0 , c − 0 ) if a 0 = 0.Moreover, the minimality of c 0 guarantees then that every vertex c − with c < c 0 is free in G(ω ′′ ), and G(ω ′′ ) contains no mark at a + for all a ∈ {1, . . ., p} (because (a, 0) cannot belong to E(ω ′′ ), due to the minimality of c 0 ).
If c 0 = 0, then a + 0 is a marked vertex in G(ω ′′ ).There may be more than one marked vertex of this type in G(ω ′′ ), and we choose a 0 minimal for this property.Thus, in any situation, we have r a 0 ,c 0 (ω ′′ ) = 1.
• If a 0 = 0, then G(ω ′′ ) has a mark at c − 0 , and G(ω ′ ) is obtained by replacing this mark by a mark at c − 1 , whereas c − 0 becomes a free vertex.This corresponds to Case 5 -lower subcase in Figure 1.
The fact that a 0 = 0 implies that a + 0 is a vertex in G(ω).Either a + 0 is incident with an edge/marked in G(ω), in which case E(ω) contains an element of the form (a 0 , d 0 ) with c 0 < d 0 ≤ q, or a + 0 is a free vertex in G(ω), in which case we set d 0 = q + 1.We choose a 1 ∈ {a ′ 0 , . . ., a 0 − 1} maximal such that (a 1 , c 1 ) ∈ E(ω) for some c 1 with c 0 ≤ c 1 < d 0 .There is an element ω ′ ∈ T such that In each situation, the graph G(ω ′ ) yields G(ω) by one of the moves of Figure 1: • In the case where d 0 ≤ q, the graph G(ω) has an edge (a is also an edge in G(ω), and the relation between G(ω ′ ) and G(ω) is as depicted in Case 1 of Figure 1.If c 1 = 0 (resp., a 1 = 0), then G(ω) has a mark at a + 1 (resp., c − 1 ) and the relation with G(ω ′ ) is as in Case 2 -upper subcase (resp., lower subcase) of Figure 1.• In the case where d 0 = q + 1, the vertex a + 0 is a free vertex in G(ω).The relation between G(ω ′ ) and G(ω) is as described in Case 3 -upper subcase, Case 5 -upper subcase, or Case 4 -lower subcase of Figure 1, depending on whether a 1 , c 1 = 0, c 1 = 0 (and a 1 = 0), or a 1 = 0 (and c 1 = 0).In particular we have ω ≺ ω ′ (by Lemma 3.6).
The conclusion of the lemma is implied by the following general fact, taking also Lemma 3.1 into account.Fact: Given a connected solvable algebraic group acting on an algebraic variety, the boundary A proof of this fact can be found in [14, Lemmas 2.12-2.13].Now we are in position to proceed with the proof of Theorems 2.2 (4) and 2.3.

Proof of Theorem 2.2 (4).
The "only if" part is shown in Lemma 3.7.For the inverse implication, let ω, ω ′ ∈ T be such that ω ω ′ .Repeated applications of Lemma 3.8 yield a sequence of elements fits in one of the cases of Figure 1 for all k.Then, Lemma 3.6 shows that the following sequence of inclusions holds: In particular, we get the desired inclusion Proof of Theorem 2.3.Assume that condition (1) of Theorem 2.3 holds.First, the equality dim O ω ′ = dim O ω + 1 follows from Lemma 3.9.Next, we have in particular ω ≺ ω ′ in view of Theorem 2.2 (4).Lemma 3.8 yields an element ω 0 ∈ T with ω ω 0 ω ′ and such that (ω, ω 0 ) or (ω 0 , ω ′ ) fits in one of the cases of Figure 1.By Theorem 2.2 (4) again, we get O ω ⊂ O ω 0 ⊂ O ω ′ , and therefore ω = ω 0 or ω 0 = ω ′ , due to the assumption that O ω ′ covers O ω .In both cases this implies that the pair (ω, ω ′ ) fits in one of the cases of Figure 1, that is, the graph G(ω) is obtained from G(ω ′ ) by one of the moves listed in Figure 1.This yields condition (2) of Theorem 2.3.
Conversely, assume (2).By Lemma 3.6, the inclusion O ω ⊂ O ω ′ holds.This inclusion, combined with the fact that dim 3.4.Further remarks.(a) In view of Lemma 3.1, the results shown in Section 3 establish the properties of the K-orbits on X as well as of the B K -orbits on Gr(V, r).Specifically, we obtain the decomposition Note that Gr(V, r) is a fortiori a union of finitely many orbits for the action of K.The description of these K-orbits is well known, and it can be related to the decomposition into B K -orbits in the following way.
Note that the number s := r p,0 (ω) (resp., t := r 0,q (ω)) is the number of marks among the positive (resp., negative) vertices of the graph G(ω).In view of Theorem 2.2 (2), the and only if the degree of vertices is nondecreasing from left to right along the row of positive (resp., negative) vertices of G(ω) (i.e., marked vertices are located on the right and free vertices are located on the left), and each pair of edges has a crossing.Thus there are k 2 crossings, where k := r − (s + t) is the number of edges, and we have For example, if p = q = r = 2, the variety Gr(V, r) is the union of six K-orbits.In Figure 5 we indicate the graphs G(ω) corresponding to the B K -orbits B K • [ω] which are dense in their K-saturation K •[ω], the dimensions of the K-orbits, and the cover relations; this diagram is deduced from Figure 2 given in Example 2.4.
(b) In [9], Matsuki and Oshima classify the orbit set K\G/B, where B ⊂ G = GL p+q (C) is a Borel subgroup.It appears that our parametrization of X/K ∼ = B K \G/P ressembles to theirs; here P ⊂ G denotes the maximal parabolic subgroup obtained as the stabilizer of an r-dimensional subspace, and which satisfies B ⊂ P .In particular, by gluing orbits, Matsuki and Oshima's classification yields a parametrization of K\G/P which coincides with the one described in part (a) above.We can speculate on a deeper relation between the two orbit sets K\G/B and B K \G/P .We also refer to [15], where the image of the moment map for the action of K on G/B is considered.
. The parameters of the K-orbits of Gr(V, r) and the cover relations for p = q = r = 2.
for a matrix ω = τ 1 τ 2 ∈ T = T (p,q),r /S r , where [ω] ∈ Gr(V, r) stands for the image of where n + k ⊂ gl k (C) denotes the subalgebra of strictly upper-triangular matrices.Hence, the conormal bundle to the orbit O ω is obtained as This immediately implies: For the computation of the maps Φ k and Φ s , we need a more detailed description of the conormal direction D ω .We use the following notation: if a is a k ×ℓ matrix, then for every subsets R ⊂ {1, . . ., k} and S ⊂ {1, . . ., ℓ}, we denote by (a) R,S the submatrix of a formed by the coefficients a i,j with i ∈ R, j ∈ S, and we view it as a linear map from C S to C R .Recall from Section 2.1 that ω gives rise to decompositions p := {1, . . ., p} = I ∪ L ∪ L ′ and q := {1, . . ., q} = J ∪ M ∪ M ′ and to a bijection σ : J → I which we view as a linear map from C J to C I .(a) p,L = 0, (c) q,L = 0, (b) p,M = 0, (d) q,M = 0, (b) p,J = −(a) p,I σ, (d) q,J = −(c) q,I σ, (4.2) (a) L ′ ,p = 0, (b) L ′ ,q = 0, (c) M ′ ,p = 0, (d) M ′ ,q = 0, (a) I,p = σ(c) J,p , (b) I,q = σ(d) J,q .(4.3) Proof.Let (e + 1 , . . ., e + p ) be the standard basis of V + = C p × {0} q and let (e − 1 , . . ., e − q ) be the standard basis of V − = {0} p × C q .Then (4.4) [ω] = e + i : i ∈ L; e − j : j ∈ M; e + σ(j) + e − j : j ∈ J .For every matrix x such that a ∈ n + p and d ∈ n + q , we have x ∈ D ω if and only if Im x ⊂ [ω] ⊂ Ker x, and in view of (4.4) the second inclusion is equivalent to (4.2) while the first inclusion is equivalent to (4.3).
Figure 6 illustrates the form of the elements in the conormal direction D ω .The matrix is represented blockwise with blocks indicating the submatrices (X) R,S relative to the subsets R, S ∈ {I, J, L, L ′ , M, M ′ }.Note that in the decompositions I ∪ L ∪ L ′ = p and J ∪M ∪M ′ = q the subsets are not consecutive, and the matrix is thus represented modulo permutation within the rows and the columns.In particular, it is required in addition that the blocks a and d be strictly upper triangular.

4.2.
A review of the orbit sets, and an involution.As explained in Section 2.3, the orbits of K in the nilpotent cone are parametrized by pairs of partitions (λ, µ) ∈ P(p) × P(q), and we denote by O λ,µ the orbit corresponding to the pair (λ, µ).Note that though here the notation O µ,λ refers to an orbit of K * := GL q (C) × GL p (C) on gl q (C) × gl p (C).
Recall also from Section 2.3 that the orbits of K in the nilpotent cone are parametrized by signed Young diagrams of signature (p, q), and we denote by O Λ the orbit corresponding to Λ.We have where Λ * denotes the signed Young diagram of signature (q, p) obtained from Λ by switching the +'s and the −'s, and O Λ * is an orbit of the group K * .As shown in Theorem 2.2, the orbits of K in are parametrized by the elements of T, and O ω is the orbit corresponding to ω.If ω = τ 1 τ 2 is an element of T = T (p,q),r /S r , then ω * := τ 2 τ 1 is an element of T * := T (q,p),r /S r which thus yields an orbit O ω * of K * in a suitable multiple flag variety X * .The graphic representation G(ω * ) of ω * is obtained from G(ω) by switching the two rows of vertices, i.e., by relabeling every vertex i + (resp., j − ) as i − (resp., j + ).This implies that, if (I, J, L, L ′ , M, M ′ , σ) are the data corresponding to ω in the sense of Section 2.1, then the relevant data for ω * are (4.7) Finally, by the description of the conormal direction in (4.1), we have ( There is an abuse of notation in that statement, since we use the notation Φ k and Φ s to designate also the symmetrized and exotic Steinberg maps relative to the symmetric pair (G, K * ) = (GL p+q (C), GL q (C) × GL p (C)).
Proof of Theorem 2.5 (1).(where the symbol * stands for some matrix in M p,q (C), whose precise description is not needed).
Proof.Every element x ∈ D ω is such that x 2 = 0. On the other hand, we can see that (x 2 ) k = (x k ) 2 + (x s ) 2 .The formula for (x s ) 2m ensues.
It readily follows that For every ℓ ≥ 1, using the notation of Figure 6, we can see that (−1) ℓ d ℓ c is the matrix (written blockwise) (4.11) , and this coincides with (cτ ) ℓ c.Letting ℓ = 2m, this yields the formula claimed for (x s ) 2m+1 in the case of m ≥ 1.The claimed formula is immediate if m = 0.
We consider the following extension-by-zero mappings: M J∪M,I∪L ′ (C) M q,p (C) = M q,p (C) M S,R (C).
We have R = R ′ ∪ I ∪ L ′ and S = J ∪ M ∪ S ′ where R ′ := {−i} t i=1 and S ′ := {q + j} s ′ j=1 .Let z be an element in M S,R (C), and let us write it (blockwise) as Note that we have i < j whenever (i, j) ∈ R ′ × (I ∪ L ′ ).We have also i < j whenever (i, j) ∈ (J ∪ M) × S ′ .We obtain that (wz, zw) ∈ n + R × n + S if and only if the following conditions are satisfied: • the blocks z 1 , z 4 , z 7 , z 8 , z 9 are zero, • the submatrices Combining these observations with the description of the space Γ given above, we conclude that the elements in the space Z := {z ∈ M S,R (C) : wz ∈ n + R , zw ∈ n + S } are exactly of the form z = η 2 (γ) for γ ∈ Γ (see (4.12)-(4.13))and, therefore, η 2 restricts to an isomorphism (η 2 )| Γ : Γ → Z as asserted.

Example 2 .
10. (a) The 5-tuple corresponding to the element ω of Example 2.1 is gRS

Lemma 3 . 4 .
Every p × r matrix can be written in the form b 1 τ b 2 for some b 1

4 .
Calculation of symmetrized and exotic Steinberg maps 4.1.Conormal direction.As shown in Theorem 2.2, every K-orbit in X takes the form

Lemma 4 . 2 .
A matrix x = a b c d (with a ∈ n + p , d ∈ n + q) belongs to D ω if and only if it satisfies the following conditions: The observations made in (4.5), (4.6), (4.8) combined with Lemma 4.1 yield the following statement: Lemma 4.3.

4. 3 .
Symmetrized Steinberg map Φ k .For a permutation w ∈ S k we consider the space n

Lemma 4 . 5 .( 1 )
Let a ∈ n + p .The following conditions are equivalent: There are matrices b, c, d such that x := a b c d ∈ D ω ;

9 .
The bijection w = w s,+ : S → R of (2.4) can be written in the following matrix form σ 0 ∈ M R ′ ,M (C) is yielded by the unique decreasing bijection M → R ′ ; this corresponds to a block with 1's on the antidiagonal and 0's elsewhere.The block σ ′ 0 ∈ M L ′ ,S ′ (C) is defined in the same way.We get wz =
and (the graph of ) ω is obtained from (the graph of ) ω ′ by modifying the pattern of at most four vertices a + , b + , c − , d − (a < b, c < d), Elementary moves yielding cover relations in the poset ({O ω }, ⊂).follows from Theorem 2.3 that the boundary ∂O ω := O ω \O ω of every non-closed orbit is equidimensional of codimension one in O ω .This boundary is in general not irreducible as it already appears in Example 2.4 (a).