Dual pairs for matrix groups

In this paper we present two dual pairs that can be seen as the linear analogues of the following two dual pairs related to ﬂuids: the EPDiﬀ dual pair due to Holm and Marsden, and the ideal ﬂuid dual pair due to Marsden and Weinstein.


Motivation for the particular dual pairs considered in this paper
The original motivation for this paper was to find two dual pairs that can be seen as the linear analogues of the following two dual pairs related to fluids: the EPDiff dual pair introduced by Holm and Marsden in [5] and the ideal fluid dual pair introduced by Marsden and Weinstein in [14] (proven to be indeed dual pairs in [4], where the additional technicalities in defining infinite-dimensional dual pairs are discussed). EPDiff stands here for Euler-Poincaré equation on the diffeomorphism group.
The EPDiff dual pair involves the manifold of embeddings Emb(S, N ) of a compact manifold S into a manifold N , acted on by the diffeomorphism groups Diff(N ) from the left and by Diff(S) from the right. The momentum maps for the lifted cotangent actions, restricted to the open subset T * Emb(S, N ) × of T * Emb(S, N ) that consists of nowhere zero 1-form densities, define a dual pair 1 The linear analogues of these actions are the left GL(n, R)-action and the right GL(m, R)action on the manifold of rank m matrices M rk m n×m (R) (identified with linear injective maps ∶ R m → R n ). We show in Section 5 that the lifted cotangent momentum maps, restricted to an open subset of T * M rk m n×m (R), define a dual pair where gl(n, R) * J L denotes the image of J L in the dual Lie algebra gl(n, R) * , and similarly for gl(m, R) * J R . For the ideal fluid dual pair one notices that, given a compact manifold S endowed with a volume form µ, and a manifold M endowed with an exact symplectic form ω, the group It is not difficult to see that the maximal Lie subgroups of GL(2n, R) and GL(m, R) that preserve this symplectic form are the real symplectic group Sp(2n, R) and the orthogonal group O(m). We show in Section 4 that the momentum maps for these two actions define a dual pair In the special case m = 2n this dual pair appears in [17] (in the context of semiclassical quantum mechanics).

Outline of paper
In Section 2, we define the notion of mutually transitive actions, describe the coadjoint orbit and coadjoint orbit-reduced space correspondences, and discuss the relation between mutually transitivity and Lie-Weinstein dual pairs. In Section 3, we describe the (U(n), U(m)) dual pair, first considered by Balleier and Wurzbacher [2], and prove it satisfies mutual transitivity.
In Section 4, we construct the (Sp(2n, R), O(m)) dual pair, which is the analogue of the ideal fluid dual pair, prove it satisfies mutual transitivity, explicitly describe the (co)adjoint orbit correspondence, and point out some connections with the (U(n), U(m)) dual pair. In Section 5, we construct the (GL(n, R), GL(m, R)) dual pair, which is the analogue of the EPDiff dual pair, prove it satisfies mutual transitivity, explicitly describe the (co)adjoint orbit correspondence, and point out some connections with the (U(n), U(m)) dual pair.

Mutually transitive actions and dual pairs
In this section, we first introduce the notion of mutually transitive actions, and indicate the resulting coadjoint orbit correspondence. We then describe how mutual transitivity allows us to view reduced spaces of one action as coadjoint orbits of the other. We emphasise here that by contrast with other treatments in the literature, this correspondence invokes only smoothness of the actions, and does not require properness. Finally we outline the relationship between mutual transitivity and Lie-Weinstein dual pairs. Propositions 2.2, Lemma 2.7, and Proposition 2.8 are also proved in [2, Theorem 2.9]. We choose to include them here for completeness, since the first two are short, while our treatment of the last differs somewhat from that in [2].

Mutually transitive actions
Let (M, ω) be a symplectic manifold, and let Φ 1 ∶ G 1 × M → M and Φ 2 ∶ G 2 × M → M be symplectic actions. We assume M , G 1 , and G 2 are all finite-dimensional (dual pairs in infinite dimensions are discussed in [4]).
Definition 2.1. We say the actions Φ 1 , Φ 2 are mutually transitive if the following three properties hold: • Φ 1 and Φ 2 are Hamiltonian actions, with corresponding equivariant momentum maps • each level set of J 1 is a G 2 -orbit and vice versa, i.e., for any x ∈ M , Denoting the coadjoint orbit in g * i through µ i by O µ i , we then have

The relation between coadjoint orbits and reduced spaces
Let Φ be any Hamiltonian action of G on M with equivariant momentum map J, and let π ∶ M → M G denote the quotient map. For µ ∈ J(M ), let G µ denote the coadjoint stabiliser subgroup of G at µ, let M µ be the set J −1 (µ) G µ ≃ J −1 (O µ ) G, and let π µ ∶ J −1 (µ) → M µ ⊂ M G denote the restriction of π to J −1 (µ). In favourable situations (for example, if the group action Φ is free and proper), M µ can be given a differentiable structure with respect to which π µ is a submersion, and a symplectic structure ω Mµ satisfying (π µ ) * ω Mµ = (i µ ) * ω, where i µ ∶ J −1 (µ) ↪ M is the inclusion. The resulting symplectic manifold (M µ , ω Mµ ) is called the reduced space or the Marsden-Weinstein-Meyer quotient at µ ∈ g * . In this subsection, we demonstrate that as a consequence of mutual transitivity, such a differentiable and symplectic structure can always be defined on the reduced spaces corresponding to each action. Moreover, the reduced space M µ 1 of the G 1 -action is symplectomorphic to a coadjoint orbit O µ 2 ⊂ g * 2 (for some related µ 2 ). We first recall the concept of an initial submanifold, and its relationship to group orbits. Definition 2.4. [16, Section 1.1.8] Let M be a manifold, and N a subset of M endowed with its own manifold structure, such that the inclusion i ∶ N ↪ M is an immersion (i.e., N is an immersed submanifold of M ). We say N is an initial submanifold of M if for any manifold P , a map g ∶ P → N is smooth iff i ○ g ∶ P → M is smooth.  Before proving our main result Proposition 2.8, we first give a supplementary lemma. Lemma 2.7. Let Φ 1 , Φ 2 be mutually transitive actions, with equivariant momentum maps J 1 , J 2 . Choose x ∈ M , and let µ i = J i (x), i = 1, 2. Then the smooth map J 2 ∶ M → g * 2 restricts to a surjective submersion J µ 1 where ω + Oµ 2 is the positive Kostant-Kirillov-Souriau form on the coadjoint orbit O µ 2 ⊂ g * 2 .
Proposition 2.8. Let Φ 1 , Φ 2 be mutually transitive actions, with equivariant momentum maps J 1 , J 2 . Then any reduced space under the G 1 -action is symplectomorphic to a coadjoint orbit in J 2 (M ) ⊂ g * 2 , and similarly with 1 and 2 switched. Explicitly, for x ∈ M , via a resp. G 2 -and G 1 -equivariant symplectomorphism.
Pulling the smooth structure on O µ 2 back to M µ 1 via χ implies that π µ 1 1 is also a smooth submersion. The fibres (G 1 ) µ 1 ⋅ y of π µ 1 1 are integral manifolds of the degeneracy directions (g 1 ) µ 1 ⋅ y of the restriction (i µ 1 ) * ω. Then the usual Marsden-Weinstein-Meyer construction implies the existence of a reduced symplectic structure ω Mµ 1 on M µ 1 satisfying By (1) and commutativity of diagram (2), we have Then combining (3) and (4), and since π µ 1 1 is a surjective submersion, i.e., χ is a symplectomorphism. Since the G 1 -and G 2 -actions on M commute, the G 2 -action drops to M µ 1 . Then commutativity of the diagram (2) and G 2 -equivariance of J µ 1 2 implies that The map χ in the above proof has a natural interpretation: using the identity (i µ 1 ) * ω = (π µ 1 1 ) * ω Mµ 1 it is easily shown that χ is the momentum map of the induced G 2 -action on M µ 1 .

The relation to Lie-Weinstein dual pairs
In this subsection we make contact with the notion of dual pair in the Weinstein's original sense [18].
is called a Lie-Weinstein dual pair if J 1 , J 2 are surjective submersions satisfying Proposition 2.10. Let Φ 1 , Φ 2 be mutually transitive actions on M , and suppose the momentum maps J 1 , J 2 have constant rank. Then J 1 (M ), J 2 (M ) can be given smooth structures such Proof. Since the maps J i ∶ M → g * i are equivariant, they are Poisson ([13, Proposition 12.4.1]), and since J i (M ) is a union of symplectic leaves, this property still holds when the J i are corestricted to their images. Since the first equality being a consequence of the constant rank property (see for example the discussion on page 8 of [16]). Then where the second equality is a standard result. So the dual pair condition holds.
We define a smooth structure on J 1 (M ) as follows: let y ∈ J 1 (M ), and x ∈ J −1 1 (y). Since J 1 has constant rank, there exist local charts (U x , φ x ) about x and (V y , ψ y ) about y with respect to which J 1 takes the form of a projection, i.e., with k independent of x, y. The first k components of ψ y , restricted to W y = J 1 (U x ), defines a local coordinate chart η y ∶ W y → R k about y. To show any two such charts are compatible, ) if necessary, we can without loss of generality assume (5), the level sets of J 1 Ux∩U x ′ are expressed as (a 1 , . . . , a k ) = const. and (a ′ 1 , . . . , a ′ k ) = const. in respective local coordinates, and so the first k components of the x only depend on the coordinates (a 1 , . . . , a k ). From we deduce that for smooth functions f 1 , . . . , f k . From either of equations (6), Remark 2.11. In general, the image of a constant rank map can exhibit so-called multiple points, i.e., points where the tangent space to the image cannot be defined consistently-see [11, Appendix 1, Section 1.8] for a discussion. The latter part of the proof of Proposition 2.10 essentially shows that as a consequence of the fact that level sets of J 1 are G 2 -orbits, such multiple points do not exist for J 1 (M ). Remark 2.13. Careful examination of the proof of Proposition 2.10 shows that it is sufficient to know that one of the momentum maps has constant rank. From this, the Lie-Weinstein condition (ker T J 1 ) ω = ker T J 2 can be deduced, from which it follows that the other momentum map also has constant rank.
Example 2.14. The constant rank condition on J 1 , J 2 is necessary for proving that mutual transitivity implies the Lie-Weinstein condition. For an example where the Lie-Weinstein condition fails to hold, consider R 2 with its usual symplectic structure, and G 1 = G 2 = SO(2) with its usual action on R 2 . This action is Hamiltonian, with momentum map J(x, y) = 1 2 (x 2 + y 2 ) (on identifying so(2) with R). The action trivially commutes with itself, and the SO(2)-orbits agree with the level sets of the momentum map. However and so ker So the Lie-Weinstein condition fails to hold at the origin.
For pairs of group actions, the Lie-Weinstein condition is closely related with the notion of mutually completely orthogonal actions-see [11] for further details.
We conclude with a standard useful criterion for deducing that a momentum map has constant rank.
Lemma 2.15. [16,Corollary 4.5.13] If Φ is a (locally) free Hamiltonian G-action, then J is a submersion, and in particular has constant rank.
3 The (U(n), U(m)) actions on M n×m (C) Following [2], in this section we consider the natural Hamiltonian actions of U(n) and U(m) on M n×m (C). We show that these actions are mutually transitive, and consequently deduce the coadjoint orbit and reduced space correspondences (Corollary 2.3 and Proposition 2.8). We note that Balleier and Wurzbacher instead derive these properties as a consequence of the symplectic Howe condition In what follows, we view elements of M n×m (C) either as matrices or as linear maps from C m to C n , depending on context.

Commuting Hamiltonian actions
First, note that M n×m (C) is a complex inner product space, with Hermitian inner product The imaginary part of this inner product defines a linear symplectic form and M n×m (C) is a linear Kähler space, with obvious complex structure. It is straightforward to show that the natural left U(n)-and right U(m)-actions act symplectically on M n×m (C), considered as a symplectic manifold. In fact, these actions are Hamiltonian, and we can easily compute corresponding momentum maps.
Proof. Both results follow from the general expression for the momentum map of a linear symplectic action on a symplectic vector space-see for example [13, Section 12.4, Example (a)].
Remark 3.2. Both momentum maps are easily seen to be equivariant, hence Poisson with respect to the (+) Lie-Poisson structure on u(n) * , respectively (-) Lie-Poisson structure on u(m) * .
We can use the non-degeneracy of ⟪⋅, ⋅⟫ to translate the momentum maps J L and J R from Proposition 3.1 into Lie algebra-valued momentum maps. We note in particular that if g ⊂ gl(N, C) integrates to G ⊂ GL(n, C), then the identification g * ≃ g provided by the trace form is G-equivariant.

The mutually transitive property
Let E a denote the ath column of E, considered as a vector in C n . So we have the m 2 conditions The set {E 1 , E 2 , . . . , E m } ⊂ R n has a maximal linearly independent subset {E a 1 , . . . , E a k } for some k ≤ m, and such a subset constitutes a basis for the subspace im E ⊂ C n . We claim that Then for any c = 1, . . . , m, using the conditions (8).
(where ⊥ denotes orthogonality with respect to the usual inner product in C n ). Hence ∑ k i=1 α i E a i = 0, and so linear independence of the E a i guarantees that α i = 0 for all i = 1, . . . , k, proving linear independence of {E ′ a 1 , . . . , E ′ a k }. Also, for any c = 1, . . . , m, there exist (8) we see U is an isometry. It can be extended to the entire space C n by picking an arbitrary isometry (im E) ⊥ → (im E ′ ) ⊥ , giving U ∈ U(n).
From the discussion above, we see that if for all c = 1, . . . , m, and so E ′ = U E. Hence E and E ′ lie in the same U(n)-orbit.
(ii) Same method as part (i), except applied to rows of E instead of columns.
We have proved mutual transitivity of the (U(n), U(m)) actions on M n×m (C). Thus we get a (generalised) dual pair of momentum maps where u(n) j L and u(m) j R are the images of the left and right momentum maps respectively.
Remark 3.5. The momentum maps j L , j R in fact define a singular dual pair, in the sense of Ortega [15,16].

Adjoint orbit correspondence
We briefly recall the description of the adjoint orbit correspondence from [2]. Assuming for concreteness that n ≥ m, any E ∈ M n×m (C) has a unique singular-value decomposition E = U ΣV † , where U ∈ U(n), V ∈ U(m), and The expressions for the momentum maps (Proposition 3.3) imply that j L (E) is in the adjoint orbit of the diagonal matrix diag[ i 2 σ 2 1 , i 2 σ 2 2 , . . . , i 2 σ 2 m , 0, . . . , 0] ∈ u(n), The correspondence between such orbits, for all σ 1 ≥ σ 2 ≥ . . . ≥ σ m ≥ 0, is one-to-one (note our conventions for j R introduce a minus sign relative to [2]).

Restriction to a Lie-Weinstein dual pair
For completeness, we now characterise the subset of M n×m (C) where the generalised dual pair (9) becomes a Lie-Weinstein dual pair. As before, assume n ≥ m for concreteness.  Proof. Let E ∈ M n×m (C) have singular-value decomposition U ΣV † , where Σ is as described in the previous section. Suppose σ m−k is the last non-zero σ i (implying σ m−k+1 = . . . = σ m = 0). Note k = 0 is possible. From . This equals dim u(m) = m 2 iff all of the σ i are non-zero, which occurs iff E has full rank m.

Matrix analogue of ideal fluid dual pair
In this section, we describe a symplectic structure on M 2n×m (R) and demonstrate that the left (resp. right) action of Sp(2n, R) (resp. O(m)) is Hamiltonian. We then show that on a suitable subset of M 2n×m (R), the Sp(2n, R)-and O(m)-actions are mutually transitive, and deduce that they define a Lie-Weinstein dual pair. Finally, we describe explicitly the correspondence between adjoint orbits in the images of the respective momentum maps. This dual pair was originally discussed in [7, pp.502-506].

Commuting Hamiltonian actions
The vector space M 2n×m (R) has a symplectic form where J = 0 n I n −I n 0 n . As before, we think of the pair (M 2n×m (R), Ω) as a symplectic manifold by using the canonical isomorphism T E M 2n×m (R) ≃ M 2n×m (R).

The mutually transitive property on full rank matrices
Letting E a denote the ath column of E, considered as a vector in R 2n , this gives the m 2 conditions this is well-defined, since the columns E a are linearly independent). So the above condition becomes where ω(X, Y ) ∶= X ⊺ JY denotes the standard symplectic form on R 2n . By Witt's theorem, there exists a linear extension S ∶ R 2n → R 2n preserving ω. Then S ∈ Sp(2n, R), and E ′ = SE. So E ′ and E lie in the same Sp(2n, R)-orbit.

Adjoint orbit correspondence
By Corollary 2.3 there is a one-to-one correspondence between coadjoint orbits in the images sp(2n, R) * J L and o(m) * J R . Equivalently, since ⟪⋅, ⋅⟫ is Ad-invariant, we have a correspondence between adjoint orbits in sp(2n, R) j L and o(m) j R .
From [19] we know that every matrix E ∈ M 2n×m (R) of rank m has an singular-valuedecomposition-like representation as E = SDO with S ∈ Sp(2n, R), O ∈ O(m), and D given by where Σ is a diagonal block with positive entries σ 1 , . . . , σ p . Here q = m − 2p is imposed by the rank condition, and  We conclude that the adjoint orbit correspondence is between the two above mentioned orbits, characterized by the integer p and the positive values σ 1 , . . . , σ p .  , Ω R ) are isomorphic, where we now use obvious notation to distinguish between symplectic forms. In fact, under the identification we see that . We can realize u(n) as a Lie subalgebra of sp(2n, R) with the map . Denoting by i the inclusion of o(m) into u(m), we obtain: commutes, where here momentum maps are labelled by their corresponding groups.
and so The identity i * ○ J U(m) = J O(m) is proved similarly.

Matrix analogue of the EPDiff dual pair
In this section, upon identifying T * M n×m (R) with M 2n×m (R), we describe the lifted cotangent action of GL(n, R) and GL(m, R) on M 2n×m (R), and demonstrate that these actions are mutually transitive, and deduce that they define a Lie-Weinstein dual pair on a suitable subset of M 2n×m (R). We then give an explicit description of the adjoint orbit correspondence. Finally, we outline the relationship between (GL(n, R), GL(m, R)) momentum maps and the (Sp(2n, R), O(m)) momentum maps of the previous section.

Commuting Hamiltonian actions
We identify T * M n×m (R) with M n×m (R) × M n×m (R) ≃ M 2n×m (R) using the non-degenerate pairing on M n×m (R) given by (X, Y ) ↦ Tr(X ⊺ Y ). More precisely, Thus the canonical symplectic form on the cotangent bundle is the same as that induced by the constant symplectic form on the linear space M 2n×m (R): On M n×m (R) we consider the left GL(n, R)-action and the right GL(m, R)-action, together with their cotangent lifted action on T * M n×m (R): A ⋅ (Q, P ) = (AQ, (A ⊺ ) −1 P ), A ∈ GL(n, R) (12) and and ⟨J R (Q, P ), η⟩ = Tr(P ⊺ Qη), η ∈ gl(m, R).
When using the trace form ⟪X, Y ⟫ = Tr(XY ) to identify gl(n, R) * with gl(n, R), the momentum maps above take the concise expressions j L (Q, P ) = QP ⊺ ∈ gl(n, R), j R (Q, P ) = P ⊺ Q ∈ gl(m, R). Proof. The cotangent GL(m, R)-action (13) preserves the fibers of the momentum map j L in (14): Suppose now that j L (Q, P ) = j L (Q ′ , P ′ ). From QP ⊺ = Q ′ P ′⊺ we deduce that the linear injective mappings corresponding to Q and Q ′ have the same range, since both P ⊺ and P ′⊺ correspond to linear surjective maps R n → R m . Thus there exists B ∈ GL(m, R) with Q ′ = QB, and by inserting in the above identity we get QP ⊺ = QBP ′⊺ . By the injectivity of Q follows P ⊺ = BP ′⊺ . This ensures that (Q ′ , P ′ ) = (Q, P ) ⋅ B.
To prove the transitivity of the GL(n, R)-action on level sets of the right momentum map j R , we will need the fact that any two matrices in M rk m n×m (R) ⊂ M n×m (R) can be completed to invertible n × n matrices by using the same matrix. Lemma 5.3. Assume m < n. Given matrices Q 1 , Q 2 ∈ M rk m n×m (R), there exists X ∈ M n×(n−m) (R) such that the order n square matrices [Q 1 X] and [Q 2 X] are invertible.
Proof. An easy induction argument on m ensures that there exists a subspace V of R n that is simultaneously a complement to both m-dimensional subspaces im Q 1 and im Q 2 . Then we choose a basis of V and we build the matrix X whose columns are these basis vectors. (The induction argument is based on the fact that there exists v ∈ R n that doesn't belong to these two m-dimensional subspaces, so the subspaces Rv +im Q 1 and Rv +im Q 2 both have dimension m + 1.) Proposition 5.4. The group GL(n, R) acts transitively on level sets of the right momentum map j R restricted to M rk m n×m (R) × M rk m n×m (R).
Proof. The cotangent GL(n, R)-action (12) preserves the fibers of the momentum map j R in (14): Suppose now that j R (Q, P ) = j R (Q ′ , P ′ ), i.e., We are looking for A ∈ GL(n, R) with properties Q ′ = AQ and A ⊺ P ′ = P . The special case m = n is easy, because in this case all Q, Q ′ , P, P ′ ∈ GL(n, R), so we can put A = Q ′ Q −1 ∈ GL(n, R) which gives us A ⊺ P ′ = (Q −1 ) ⊺ (Q ′ ) ⊺ P ′ = P . Next we consider the general case m < n. Since both P and P ′ are injective, there exists a matrix C ∈ GL(n, R) such that P = C ⊺ P ′ . Since C −1 Q ′ ∈ M rk m n×m (R), by Lemma 5.3 there exists a matrix X ∈ M n×(n−m) (R) such that the two order n matrices D = [Q X] and [(C −1 Q ′ ) X] are both invertible. Putting X ′ = CX, the order n matrix we have that On the other hand thus getting the required transitivity conditions.
We have proved mutual transitivity of the (GL(n, R), GL(m, R)) actions. By injectivity of elements of M rk m n×m (R), it is straightforward to see that the right action (13) where gl(n, R) j L and gl(m, R) j R are the images of the left and right momentum maps respectively.

Adjoint orbit correspondence
As in the discussion of subsection 4.4, we have a one-to-one correspondence between GL(n, R)orbits in the image of j L and GL(m, R)-orbits in the image of j R . We now characterise the images of j L and j R , and the adjoint orbit correspondence between these images. Define the sets Proof. Thinking of (Q, P ) ∈ M rk m n×m (R) × M rk m n×m (R) as linear maps, we have that Q ∶ R m → R n is injective, while P ⊺ ∶ R n → R m is surjective. Then (i) rank j L (Q, P ) = dim im QP ⊺ = dim im P ⊺ = m, the second equality following from the injectivity of Q.
where we use that dim ker P ⊺ = n − m by the rank-nullity theorem.
In fact, the momentum maps j L , j R are surjective onto S L , S R : Proof. (i) Let ζ ∈ S L . We wish to show ζ is in the image of j L . By left GL(n, R)-equivariance of j L , we may assume ζ is in (real) Jordan canonical form where J c i (λ i ) ∈ M c i ×c i (R) denotes the (real) ith Jordan block corresponding to non-zero generalised eigenvalue λ i , and J d j (0) ∈ M d j ×d j (R) is the jth non-trivial (i.e., with d j ≥ 2) Jordan block corresponding to zero generalised eigenvalues. The dimension of the zero block follows from the condition ∈ M d×(d−1) (R), It is straightforward to check thať Take By (17), (Q, P ) ∈ M rk m n×m (R) × M rk m n×m (R). Using (19), (18), and (16), it may be checked that j L (Q, P ) = QP ⊺ = ζ.
(ii) Let ξ ∈ S R . Again, by right GL(m, R)-equivariance of j R we may assume that ξ is in (real) Jordan canonical form where c i , d j , and λ i obey the same conventions as in part (i) (in particular, d j ≥ 2), but where now we let J 1 (0) denote the 1 × 1 zero matrix instead of explicitly writing a zero block. DefiningǏ d−1 ,Î d−1 as before, we havê From the condition 2m − n ≤ rank ξ = m − q, implying n − m − q ≥ 0, we see that the matrices (19) are well-defined, with (Q, P ) ∈ M rk m n×m (R) × M rk m n×m (R). Using (19), (21), and (20), it may be checked that j R (Q, P ) = P ⊺ Q = ξ.
Examining the proof of Proposition 5.7 gives an explicit characterisation of the adjoint orbit correspondence for our dual pair: Corollary 5.8. Given • integers p, q with 0 ≤ p ≤ m and 0 ≤ q ≤ min{m, n − m}; • complex numbers λ 1 , . . . , λ p , with Im λ i ≥ 0; • integers c 1 , . . . , c p , d 1 , . . . , d q satisfying c i ≥ 1 and c i even if Im λ i > 0, d j ≥ 2, and ∑ Then there is a one-to-one correspondence between the adjoint orbits through elements (16) in gl(n, R) j L and (20) in gl(m, R) j R .