Branching Laws for Some Unitary Representations of SL(4,R)

In this paper we consider the restriction of a unitary irreducible representation of type $A_{\mathfrak q}(\lambda)$ of $GL(4,{\mathbb R})$ to reductive subgroups $H$ which are the fixpoint sets of an involution. We obtain a formula for the restriction to the symplectic group and to $GL(2,{\mathbb C})$, and as an application we construct in the last section some representations in the cuspidal spectrum of the symplectic and the complex general linear group. In addition to working directly with the cohmologically induced module to obtain the branching law, we also introduce the useful concept of pseudo dual pairs of subgroups in a reductive Lie group.


Introduction
Understanding a unitary representation π of a Lie groups G often involves understanding its restriction to suitable subgroups H. This is in physics referred to as breaking the symmetry, and often means exhibiting a nice basis of the representation space of π. Similarly, decomposing a tensor product of two representations of G is also an important branching problem, namely the restriction to the diagonal in G × G. Generally speaking, the more branching laws we know for a given representation, the more we know the structure of this representation. For example, when G is semisimple and K a maximal compact subgroup, knowing the K-spectrum, i.e. the collection of K-types and their multiplicities, of π is an important invariant which serves to describe a good deal of its structure. It is also important to give good models of both π and its explicit K-types. There has been much progress in recent years (and of course a large number of more classical works, see for example [27,6,7,8]), both for abstract theory as in [11,12,14,13], and concrete examples of branching laws in [24,25,18,4,28].
In this paper, we shall study in a special case a generalization of the method applied in [19] and again in [8]; this is a method of Taylor expansion of sections of a vector bundle along directions normal to a submanifold. This works nicely when the original representation is a holomorphic discrete series for G, and the subgroup H also admits holomorphic discrete series and is embedded in a suitable way in G. The branching law is a discrete sum decomposition, even with finite multiplicities, so-called admissibility of the restriction to H; and the summands are themselves holomorphic discrete series representations for H. Since holomorphic discrete series representations are cohomologically induced representations in degree zero, it is natural to attempt a generalization to other unitary representations of similar type, namely cohomologically induced representations in higher degree. We shall focus on the line bundle case, i.e. the A q (λ) representations. In this case T. Kobayashi [13] obtained necessary and sufficient conditions that the restriction is discrete and that each representation appears with finite multiplicity, socalled admissibility of the representation relatively to the subgroup. Using explicit resolutions and filtrations associated with the imbedding of H in G, we analyze the derived functor modules and obtain an explicit decomposition into irreducible representations. It is perhaps not surprising, that with the appropriate conditions on the imbedding of the subgroup, the class of (in our case derived functor) modules is preserved in the restriction from H to G.
While the algebraic methods of derived functor modules, in particular the cohomologically induced representations, provide a very strong tool for the theoretical investigations of unitary representations of reductive Lie groups, it has been difficult to work with concrete models of these modules. It is however exactly these models that we use in this paper, as outlined above; the fact that one may consider these modules as Taylor expansions of appropriate differential forms, indicates that it should be natural to study these Taylor expansions along submanifolds -and when these submanifolds are natural for the subgroup for which one wants to do the branching law, there arises a useful link between the algebraic branching and the geometry of the imbedding of the subgroup. It is our hope, that this idea (that we carry out in some relatively small examples) will have a broader use in deciding the possible candidates for representations occuring in an admissible branching law.
Here is the general setting that we consider: Let G be a semisimple linear connected Lie group with maximal compact subgroup K and Cartan involution θ. Suppose that σ is another involution so that σ ·θ = θ ·σ and let H be the fixpoint set of σ in G. Suppose that L = L x is the centralizer of an elliptic element x ∈ i(g ∩ h) and let q = l ⊕ u, q H = q ∩ h be the corresponding θ-stable parabolic subgroups. Here we use as usual gothic letters for complex Lie algebras and subspaces thereof; a subscript will denote the real form, e.g. g o . We say that pairs of parabolic subalgebras q, q H which are constructed this way are well aligned. For a unitary character λ of L we define following Vogan/Zuckerman the unitary representations A q (λ).
In this paper we consider the example of the group G = SL(4, R). There are two G-conjugacy classes of skew symmetric matrices with representants . Let H 1 respectively H 2 be the symplectic subgroups defined by these matrices and H 1 , H 2 the centralizer of Q 1 , respectively Q 2 . All these subgroups are fix point sets of involutions σ i , i = 1, 2 and σ i , i = 1, 2 respectively. The matrix Q 2 has finite order, it is contained in all subgroups H i and iQ 2 ∈ g defines a θ-stable parabolic subalgebra q of sl(4, C) and also θ-stable parabolic subalgebras The parabolic subgroups q, q h 1 as well as q, q h 2 are well aligned.
We consider in this paper the unitary representation A q of G corresponding to trivial character λ. Its infinitesimal character is the same as that of the trivial representation. The representation A q was studied from an analytic point of view by S. Sahi [21]. Since the A q has nontrivial (g, K)-cohomology and is isomorphic to a representation in the residual spectrum, this representation is also interesting from the point of view of automorphic forms. See for example [23]. We show in this paper that the restriction of A q to H 1 and H 1 is a direct sum of irreducible unitary representations, where as the restriction to H 2 and H 2 has continuous spectrum. We also determine explicitly the restriction of A q to the subgroups H 1 and H 1 and conclude that for all unitary (h 1 , H 1 ∩ K)-modules V the dimension of Hom (h 1 ,H 1 ∩K) (A q , V ) is at most 1.
If we interpret SL(4, R) and Sp(2, R) as Spin (3,3) and Spin(2,3) then these branching laws can in some sense be considered as supporting the conjectures by B. Gross and D. Prasad [3] for the restriction of Vogan packets of representations of SO(n, n) to representations of SO(n−1, n).
The paper is organized as follows: After introducing all the notation in Section 2 we prove in Section 3 using a result of T. Kobayashi, that the restriction of A q to H 1 and H 1 is a direct sum of irreducible unitary representations, whereas the restriction to H 2 and H 2 does have a continuous spectrum. This discrete/continuous alternative, see [12], is one of the deep results that we invoke for symmetric subgroups.
We do not attempt in this paper to say anything about the continuous spectrum, and we mainly focus on the admissible situation, so the alternative is really admissible/non-admissible.
In Sections 4 and 5 we determine the representations of H 1 respectively of H 1 that appear in the restriction of A q to H 1 respectively H 1 and show that it is a direct sum of unitary representations of the form A q∩h 1 (µ) respectively A q∩h 1 (µ ), each appearing with multiplicity one. The main point is here, that we find a natural model in which to do the branching law, based on the existence results of T. Kobayashi; and also following experience from some of his examples, where indeed derived functor modules decompose as derived functor modules (for the smaller group).
In Section 6 we introduce pseudo dual pairs. This allows us to find another interpretation of the restrictions of A q to the pseudo dual pair H 1 , H 2 . This notion turns out to be extremely useful for analyzing the spectrum in the admissible situation, and combined with our idea of restricting a cohomologically induced module gives the complete branching law. In Section 7 we recall some more examples of branching laws.
In Section 8 we formulate a conjecture about the multiplicity of representations in the restriction of representations A q of semisimple Lie groups G to subgroups H, which are centralizers of involutions. If the restriction of A q to H is a direct sum of irreducible representation of H we expect that there is a θ-stable parabolic subalgebra q H of H so that all representations which appear in the restriction are of the form A q H (µ) and that a Blattner-type formula holds. See the precise conjecture at the end of Section 8, where we introduce a natural generalization of previously known Blattner-type formulas for the maximal compact subgroup.
In Section 9 we see how these results may be used to construct automorphic representations of Sp(2, R) and GL(2, C) which are in the discrete spectrum for some congruence subgroup. For Sp(2, R) these representations are in the residual spectrum, whereas for GL(2, C) these representations are in the cuspidal spectrum. We expect that our methods extend to other situations with similar applications to automorphic representations; and we hope the point of view introduced here also will help to understand in a more explicit way the branching laws for semisimple Lie groups with respect to reductive subgroups.

Notation and generalities
2.1. Let G be a connected linear semisimple Lie group. We fix a maximal compact subgroup K and Cartan involution θ. Let H be a θ-stable connected semisimple subgroup with maximal compact subgroup K H = K ∩ H. We pick a fundamental Cartan subgroup C H = T H · A H of H. It is contained in a fundamental Cartan subgroup C = T · A of G so that T H = T ∩ H and A H = A ∩ H. The complex Lie algebra of a Lie group (as before) is denoted by small letters and its real Lie algebra by a subscript o. We denote the Cartan decomposition by g o = k o ⊕ p.
Def inition. Let q and q H be θ-stable parabolic subalgebras of g, respectively h. We say that they are well aligned if q H = q ∩ h.
We fix x o in t H . Then i x o defines well aligned θ-stable parabolic subalgebras q = l ⊕ u and q H = l H ⊕ u H = q ∩ h of g respectively h; for details see page 274 in [10].
We write L and L H for the centralizer of x 0 in G and in H respectively. For a unitary character λ of L we write λ H for the restriction of λ to L H .

2.2.
For later reference we recall the construction of the representations A q (V ), V an irreducible (q, L ∩ K) module. We follow conventions of the book by Knapp and Vogan [10] (where much more detail on these derived functor modules is to be found -this is our standard reference) and will always consider representations of L and not of the metaplectic cover of L as some other authors. We consider U (g) as right U (q) module and write V = V ⊗ ∧ top u. Let p L be a L ∩ K-invariant complement of l ∩ k in l. We write r G = p L ⊕ u. Now we introduce the derived functor modules as on page 167 in [10], recalling that this formalizes Taylor expansions of certain differential forms. Since all the groups considered in the paper are connected we use the original definition of the Zuckerman functor [26] and do not use the the Hecke algebra R(g, K) to define the representations A q (V ). Consider the complex Here the subscript K denotes the subspace of K-finite vectors. We denote by T (x, U (·)) an element in Hom L∩K (U (g), Hom C (∧ n−1 r G , V )) K . The differential d is defined by where x ∈ U (g), X j ∈ r G and P r G is the projection onto r G along l ∩ k. Let s = dim(u ∩ k) and let χ be the infinitesimal character of V . If where , denotes the Killing form of g, then the cohomology is zero except in degree s and if V is irreducible this defines an irreducible ((U (g), K)-module A q (V ) in degree s (8.28 in [10]). By (5.24 in [10], see also the remark/example on page 344) the infinitesimal character of A q (V ) is χ + ρ(u) (usual shift of the half-sum of all positive roots in u).
If V is trivial the infinitesimal character of A q (V ) is ρ G and we write simply A q . Two representations A q and A q are equivalent if q and q are conjugate under the compact Weyl group W K . For

2.3.
Let H be the fix point set of an involutive automorphism σ of G which commutes with the Cartan involution θ. We write g o = h o ⊕ s o for the induced decomposition of the Lie algebra. T. Kobayashi proved [12] that the restriction of A q to H decomposes as direct sum of irreducible We have l = l H ⊕ l ∩ s. Put u H = u ∩ h. The representation of l H on u is reducible and as l Hmodule u = u H ⊕(u∩s). Let q = l⊕u be the opposite parabolic subgroup.

Now let G = SL(4, R). The skew symmetric matrices
represent the conjugacy classes of skew symmetric matrices under G. They define symplectic forms also denoted by Q 1 and Q 2 .
Let H 1 , respectively H 2 , be the θ-stable symplectic subgroups defined by Q 1 , respectively Q 2 . These subgroups are fix points of the involutions Since Q 1 and Q 2 are conjugate in GL(4, R), but not in SL(4, R), the symplectic groups H 1 and H 2 are not conjugate in SL(4, R). Let H 1 and H 2 be the fix points of the involutions Both groups H 1 and H 2 are isomorphic to but they are not conjugate in SL(4, R).

2.5.
We fix x 0 = Q 2 . It has finite order and is contained in 2 i=1 H i and in 2 i=1 H i . Now ix 0 ∈ ig defines a θ stable parabolic subalgebra q of sl(4, C) and also θ-stable well aligned parabolic subalgebras q H of the subalgebras h. Its centralizer L = L x 0 in SL(4, R), the Levi subgroup, is isomorphic to GL(2, C) = H 2 . For a precise description of the parabolic see page 586 in [10].
We have Let A q be the representation holomorphically induced from q which has a trivial infinitesimal character. This representations is a subrepresentation of a degenerate series representation induced from a one dimensional representation of the parabolic subgroup with Levi factor S(GL(2, R) × GL(2, R)) and thus all its K-types have multiplicity one. See [21] for details. The next proposition demonstrates how different imbeddings of the same subgroup (symplectic res. general linear complex) gives radically different branching laws.
1. The restriction of A q to H 1 and to H 1 is a direct sum of irreducible representation each appearing with finite multiplicity.
2. The restriction of A q to H 2 and to H 2 is not admissible and has continuous spectrum.
Proof . Since K H 1 = K H 1 and K H 2 = K H 2 it suffices by T. Kobayashi's Theorem 4.2 in [12] to show that A q is K H 1 admissible. We will prove this in the next section.
To prove (2) it suffices to show by Theorem 4.2 in [12] that A q is not K H 2 admissible. This is proved also in the next section.
Remark. We will prove later in the paper that the representations of H 1 have at most multiplicity one in the restriction of A q .

The restriction of
We use in this section the notation introduced on pages 586-588 in [10].
3.1. The Cartan algebra t o of so(4, R) consists of 2 by 2 blocks 0 θ j −θ j 0 down the diagonal.
We have a θ-stable Cartan subalgebra h o = t o ⊕ a o where a o consists of the 2 by 2 blocks Then the roots ∆(u) of (h, u) are and a compatible set of positive roots ∆ + (l) of (h, l) are The roots α 1 = e 1 + e 2 , α 2 = e 1 − e 2 are compatible positive roots of the Lie algebra k with respect to t. The highest weight of the minimal K-type of A q is Λ = 3(e 1 + e 2 ). See page 588 in [10]. All other K-types are of the form Λ + m 1 (e 1 + e 2 ) + 2m 2 e 1 , m 1 , m 2 ∈ N.
. We assume that the positive root of (t, k H 2 ) is α 2 whereas the α 1 can be identified with a character of the center of K H 2 . So the restriction of each K-type with highest weight (m + 3) Hence A q is not K H 2 -finite and thus by T. Kobayashi's theorem A q is not a discrete sum of irreducible representations of H 2 . This proves the second claim of Proposition 2.1.
The groups H 1 and H 2 are conjugate under the outer automorphism which changes the sign of e 2 . Hence the simple positive root of (t, k H 1 ) can be identified with α 1 and α 2 can be identified with a character of the center of K H 1 (see Fig. 1). A K-type with highest weight Λ + m 1 (e 1 + e 2 )+2m 2 e 1 is a tensor product of a representation with highest weight (3+m 1 +m 2 , 3+m 1 +m 2 ) and a representation with highest weight (m 2 , −m 2 ). Its restriction to K H 1 is a direct sum of representations with highest weights (3 + m 1 + m 2 + i, 3 + m 1 + m 2 − i), −m 2 ≤ i ≤ m 2 . Fig. 2 shows the highest weights of the K H 1 -types for the restriction of A q to K H 1 . Their multiplicities are indicated by a number.  Thus A q is K H 1 -finite. This completes the proof of the first assertion of Proposition 2.1.

Remark.
A second series of representations is obtained if we define another θ-stable parabolic subalgebra q using the matrix Q 1 ∈ g instead of Q 2 . We obtain a representation A q which is not equivalent to A q . The same arguments as in the previous case prove that restriction of A q to H 1 does not have a purely discrete spectrum whereas the restriction to H 2 is a direct sum of irreducible unitary representations.
The representation A q of SL +/− (4, R) (determinant ±1) obtained by inducing representation A q is irreducible and its restriction to SL(4, R) is equal to A q ⊕ A q . Hence the restriction of A q to H 1 does not have discrete spectrum. 4 The restriction of A q to the symplectic group H 1 In this section we determine H 1 -types of A q . Our techniques are based on homological algebra and the construction of an "enlarged complex" whose cohomology computes the restriction. We introduce it in 4.1 for semisimple connected Lie groups H and connected reductive subgroups H. Then we will compute the restriction to H 1 by restricting A q to a subgroup conjugate to H 1 . The motivation for this "enlarged complex" or "branching complex" is the same as when one is restricting holomorphic functions to a complex submanifold, and identifying the functions with their normal derivatives along the submanifold. In our case we are working with (formalizations of) differential forms satisfying a similar differential equation, so it is natural to try to identify them with their "normal derivatives"; this is what is formalized in our definition. As it turns out, with the appropriate conditions (well aligned parabolic subgroups, vanishing of the cohomology in many degrees, and the non-vanishing of explicit classes corresponding to small K-types) we can indeed make the calculation of the branching law effective, at least in the examples at hand. Here r H = r G ∩ h and d H is defined analogously to the differential d in 2.2. As a left U (l H )module where Q is the symmetric algebra S(s). (See [10, 2.56].) We have U (g) acts on the enlarged complex from the right and a quick check shows that d H also commutes with this action and therefore we have an action of U (g) on the cohomology of the complex. We also consider the "large complex" L * (Hom K∩L∩H (U (g), Hom(∧ i r G , C λ )) K∩H , d).
We have r G = r H ⊕ (u ∩ s) ⊕ (p L ∩ s), and so Using this we can define a "pull back" map of forms The pullback map commutes with the right action of U (h) and induces a map of complexes. This is the main observation we use in order to analyze the action of H on the cohomologically induced module.

4.2.
For the rest of this section we assume that G = SL(4, R). We will show that there exists a symplectic subgroup, which we denote by H w 1 conjugate to H 1 by an element w, so that the pullback induces a nontrivial map in cohomology. Since the restriction of A q depends only on the conjugacy class of H 1 this determines the restriction. Since by abuse of notation we will also write H 1 and H 1 for the groups defined by the skew symmetric form Thus Recall that q is defined by and that g = h 1 ⊕ s 1 . We need the fine structure of the parabolic relative to the symmetric subgroup, in order to compare the cohomology of these complexes during the branching. b) the representation of L ∩ H 1 acts by a nontrivial character µ 1 with differential (e 1 + e 2 ) on the one dimensional space u ∩ s 1 ; c) l ∩ s 1 is a direct sum of the trivial representation and the adjoint representation of l ∩ h 1 .
Proof . We have The nilradical of a parabolic subalgebra with this Levi subalgebra has dimension 3.
The dimension of l ∩ h 1 ∩ k is 2. Hence the dimension of u ∩ k ∩ h 1 is 1. On the other hand the dimension of l ∩ k is 4. So the dimension of u ∩ k is 1. Since u ∩ k ∩ h 1 ⊂ u ∩ k we have equality.
l ∩ h 1 acts on the 4 dimensional space l ∩ s 1 via the adjoint representation.
The representation of L ∩ H 1 on the symmetric algebra S((u ∩ s 1 ) ⊕ (u ∩ s 1 ) ⊕ (l ∩ s 1 )) is isomorphic to a direct sum of representations µ n 1 1 ⊗ µ −m 1 1 ⊗ ad r 1 with n 1 , m 1 , r 1 ∈ N. These powers of µ 1 will label the constituents in the branching law; it will also be sometimes convenient to think of their differentials in additive notation. Now it is important to note, that the parameter λ H 1 ⊗ µ n 1 1 , 0 ≤ n 1 is in the good range [10] and thus the representation on the cohomology in degree 1 = dim(u ∩ k ∩ h 1 ) of the complex L * H 1 has composition factors isomorphic to where 0 ≤ n 1 . In particular A q∩h 1 (λ H 1 ) is an (h 1 , K ∩ H 1 )-submodule module of the cohomology of L * H 1 .
is a composition factor of the restriction of A q to (h 1 , K ∩ H 1 ).
Proof . Note that dim(u ∩ k) = dim(u ∩ k ∩ h 1 ) and that 1 = dim(u ∩ k) is the degree in which the complexes defining the representations A q respectively A q∩h 1 (λ H 1 ) both have nontrivial cohomology [10]. Considering the complex defining A q as a subcomplex of the "large complex" L * the pullback pb i H 1 of forms defines a (h 1 , K ∩ H 1 )-equivariant map Recall the definition of the K-module R s K (λ) from V.5.70 in [10]. We have bottom layer maps of k o -modules.
where λ 0 is the trivial character of L ∩ K. These maps are defined by the inclusion of of complexes and hence of forms. See Theorem V.5.80 and its proof in [10]. The minimal K-types of A q , respectively K H 1 -type of A q∩h 1 (λ H 1 ) are in the bottom layer.
On the other hand we have an inclusion of complexes (the notation in analogy with the case in 4.1, now for the case where we take G = K) But C λ 0 = C λ H 1 and r G ∩ k ∩ h 1 = r G ∩ k and so using a forgetful functor we may consider as a subspace, respectively subcomplex, of where Q H is the symmetric algebra of the complement of h 1 ∩ k in k. Note that K ∩ H 1 and K is again a symmetric a pair and so we have a bottom-layer map for the representation Since the representation R 1 K (λ 0 ) is irreducible restricted to K ∩ H 1 this map is an isomorphism.
Def inition. We call A q∩h 1 (λ H 1 ) the minimal H 1 -type of A q . Theorem 4.3. The representation A q restricted to H 1 is the direct sum of the representations each occuring with multiplicity one, namely Proof . By the proof of the lemma A q∩h 1 (λ H 1 ) is a submodule of the restriction of A q to the symplectic group H 1 . Its minimal K H 1 -type is also a minimal K-type of A q and hence occurs with multiplicity one. Hence A q∩h 1 (λ H 1 ) is a H 1 -type of A q with multiplicity one. The minimal K H 1 -type of A q∩h 1 (λ) has highest weight λ + 3e 1 + 3e 2 . The roots of u ∩ h 1 ∩ p are 2e 1 , 2e 2 . Applying successively the root vectors to the highest weight vector of the minimal K H 1 -type of A q∩h 1 (λ) we deduce that A q∩h 1 (λ) contains the K H 1 -types with highest weight ((3 + 2r 1 )e 1 + (3 + 2r 2 )e 1 + λ), r 1 , r 2 ∈ N. Theorem 8.29 in [10] show that all these K H 1 -types have multiplicity one. Fig. 3 shows the K H 1 -type multiplicities of A q (λ H 1 ).
Note that we are here using quite a bit of a priori information about the derived functor modules for the smaller group; on the other hand, the branching problem has essentially been reduced to one for compact groups, K-type by K-type. The Borel subalgebra of k ∩ h 1 acts on the one dimensional space u ∩ s 1 by a character µ 1 with differential (e 1 + e 2 ). Let Y = 0 be in u ∩ s 1 and v = 0 a highest weight vector of the minimal K-types of A q . Then Y n · v = 0 is also the highest weight of an K H 1 -type of highest weight (3 + n)e 1 + (3 + n)e 2 of A q .
Let X k = 0 be in u ∩ k. The linear map which maps 1 to X k ⊗ C λ 0 is non-zero in cohomology and its class [T s ] is the highest weight vector of the minimal K-type. But The same argument shows that A q∩h 1 (λ H 1 ⊗ µ n 1 ), n ∈ N, is a H 1 -type of A q . Now every K-type with highest weight (n, n) has multiplicity n − 2 and is contained in exactly n − 2 composition factors. The multiplicity computations in Section 3 now show that every composition factor is equal to A q∩h 1 (λ H 1 ⊗ µ n 1 ) for some n. See Fig. 4.
Remark. Another proof of Theorem 4.3 can be obtained using Proposition 5.1 and the ideas of 6.1. By Proposition 8.11 in [10] for any (g, K)-module X we have Hom g,K (X, A q ) = Hom l,K∩L (H s (u, X), C ), where H s (u, X) is the Lie algebra homology as defined in [10] and s = dim(u ∩ k). Thus we have a "Blattner type formula" for the H 1 -types of A q .
Remark. The H 1 (·, ·) on the right refers to homology in degree one. The maximal Abelian split subalgebra a 1 in l ∩ h 1 are the diagonal matrices. So the parabolic subgroup of the Langlands parameter of the H 1 -types of A q is the so called "mirabolic", i.e. the maximal parabolic subgroup with Abelian nilradical. The other parts of Langlands parameter can de determined using the algorithm in [10]. We consider in Theorem 4.3 the restriction of a "small" representation A q of Spin(3,3) to Spin(2,3) similar to the restriction of "small" discrete series representations of SO(n + 1, n) to SO(n, n) considered by B. Gross and N. Wallach in [4]. It would be interesting to see if their techniques could be adapted to the problem discussed in the paper.

Restriction of A q to the group H 1
In this section we describe H 1 -types of A q using the same techniques as in the previous section.

5.1.
For H 1 we consider the complex and the map We write g = h 1 ⊕ s 1 . The intersection u ∩ s 1 is 2-dimensional and the representation of the group of L ∩ H 1 on u ∩ s 1 is reducible and thus a sum of 2 one dimensional representations χ 1 ⊕ χ 2 . The weights of these characters are 2e 1 and 2e 2 . So the symmetric algebra S(u ∩ s 1 ) is a direct sum of one dimensional representations of L ∩ H 1 with weights 2m 1 e 1 + 2m 2 e 2 .
In the cohomology in degree 1 of the complex L * H 1

we have composition factors
with 0 ≤ n 1 , n 2 . In particular A q∩h 1 (λ H 1 ) is an (h 1 , K ∩ H 1 )-submodule module of the cohomology in degree 1.
is a composition factor of the restriction of A q to (h 1 , H 1 ∩ K).
Proof . The maximal compact subgroups of H 1 and H 1 are identical. Thus dim u ∩ k ∩ h 1 = dim u ∩ k ∩ h 1 = 1 and the minimal K-type is irreducible under restriction to K H 1 . Thus the same argument as in Lemma 4.1 completes the proof.
Def inition. We call A q∩h 1 (λ H 1 ) the minimal H 1 type of A q .
Theorem 5.2. The representation A q restricted to H 1 is the direct sum of the representations each occurring with multiplicity one, namely Their minimal K∩H 1 -types have highest weights (3+m 1 +m 2 +i, 3+m Proof . The proof is the same in the previous section where we proved that the representations A q∩h 1 (λ H 1 ⊗ χ n 1 1 ⊗ χ n 2 2 ), n 1 , n 2 ∈ N appear in the restriction of the A q to H 1 The K ∩ H 1 -types of all unitary representations of GL(2, C) have multiplicity one. If the minimal K ∩ H 1 -type has highest weight l 1 e 1 + l 2 e 2 − 2, then the highest weights of the other K ∩ H 1 -types are (l 1 + j)e 1 + (l 2 + j)e 2 . Multiplicity considerations of K ∩ H 1 -types of A q conclude the proof.
Remark. Using Proposition 4.2, 4.1 and the ideas of 6.1 we can obtain another proof of this theorem. Fig. 5 shows the decomposition into irreducible representations. The highest weights of the K ∩ H 1 -types of a composition factors lie on the lines. For each highest weight there is exactly one composition factor which has a K ∩ H 1 -type with this weight as a minimal K ∩ H 1 -type.
We have again a "Blattner-type formula" for the H 1 types.
6 Pseudo dual pairs 6.1. Suppose now that G is a reductive connected Lie group with maximal compact subgroup K, Cartan involution θ of g. For an involution τ : G → G Figure 5. Figure 6.
commuting with θ we define Def inition. We call H and H a pseudo dual pair.
Suppose now that π be an irreducible unitary (g, K)-module. Then π is K ∩ H-admissible if and only if it is K ∩H -admissible and by Theorem 4.2 of [12] its restriction to h is a direct sum of irreducible (h, K ∩H)-modules if and only if π is a direct sum of irreducible (h , K ∩H )-modules.
Suppose that π is H ∩ K admissible and that V k ⊂ π is a minimal K-type of π. If π o is a (h, K ∩ H)-module, which occurs in the restriction of π to h then 6.2. These observations allow us to better understand the restriction of A q to the pseudo dual pair H 1 =Sp(2, R), H 1 = GL(2, C) in SL(4, R). The minimal K-type V K of A q has highest weight 3α 1 and is also irreducible under K ∩H 1 = K ∩H 1 . The (h 1 , K ∩H 1 )-submodule generated by V K is the minimal H 1 type and is isomorphic to a spherical principal series representation. We draw a diagram of its K ∩ H 1 -types using the same conventions as in the previous sections. The K ∩ H 1 -types are on the black line in Fig. 6.
Each K ∩H 1 = K ∩H 1 -type of this representation is the minimal K ∩H 1 -type of an irreducible (h 1 , K ∩ H 1 )-module as indicated in Fig. 7.  The K ∩ H 1 types of the minimal (h 1 , K ∩ H 1 )-module generated by the minimal K-type have multiplicity one and are indicated by the dots in Fig. 8.
Each of the K ∩ H 1 -types of this (h 1 , K ∩ H 1 )-module is the minimal K ∩ H 1 = K ∩ H 1 -type of a (h 1 , K ∩ H 1 )-module in the restriction of A q to H 1 , as illustrated in Fig. 9.

More branching
In this section we sketch the restriction to H 1 and H 1 of a representation A q (λ 1 ) with a parameter λ which is no longer in the weakly fair range and hence the representation is no longer irreducible and has a composition series of length 2. We indicate a procedure to use our previous techniques to compute the restriction of both composition factors of A q (λ 1 ) to H 1 and H 1 . Using the local isomorphisms we obtain a different proof of a result by T. Kobayashi and B. Ørsted of the branching of the minimal representation π of SO(3, 3) to SO(3, 1) × SO(2) and to SO(3, 2) [17].
7.1. We use the notation introduced in 3.1. Using the conventions on page 586 in [10] we denote the character of L by λ = m(e 1 + e 2 + e 3 + e 4 ). With this parametrization the representation A q (λ) is irreducible and unitary for m > −3 (see page 588 in [10]) and A q corresponds to the parameter m = 0. The representation A q (λ) for m = 3 is nonzero.
To simplify the notation we denote the representation A q (λ) for λ = m(e 1 + e 2 + e 3 + e 4 ) by A q (m).
We consider now the representation A q (−3). This representation is nonzero and has a trivial K-type. It is not irreducible, but has 2 composition factors. One composition factor is A q (−2). The other composition factor is a unitarily induced representation π 0 from one dimensional representation of a maximal parabolic subgroup. It is a ladder representation and the highest weights of its K-types are multiples of e 1 . See [17] for SO (3,3) or [1] for GL(4, R).
7.2. The same arguments as in Sections 4 and 5. allows us to compute the restriction of A q (−3) to H 1 and H 1 . In this case the minimal H 1 type is a spherical representation π H 1 0 and comparing the multiplicities of K ∩ H 1 -types of π H 1 0 and of π 0 we deduce that the restriction of π 0 to H 1 is irreducible and equal to π H 1 0 . A similar argument show that the restriction to H 1 is direct sum of principal series representations of GL(2, C).
Using the local isomorphism SL(4, R) and SO(3, 3) we obtain a new proof of the branching of π 0 determined by B. Ørsted and T. Kobayashi in [17].  [14] support the following conjecture: Let H be the connected fixpoint set of an involution σ. We write again g = h ⊕ s. Let A q (λ) be a representation, which is K ∩ H-admissible and thus decomposes discretely, when restricted to H. Suppose q, q h are defined by x o ∈ T H . Since A q = A p if q and p are conjugate under the compact Weyl group W K we use the following Def inition. Let y o ∈ T H and let p, p H be well aligned parabolic subalgebras defined by y o . We call the well aligned parabolic subalgebras p, p H related to q, q H , if x o and y o are conjugate by an element in the compact Weyl group W K of K with respect to T.
If x o and y o are not conjugate by an element in the Weyl group W K∩H of (K H , T H ) then the parabolic subalgebras q H , p H of H are not conjugate in H and thus we have up to conjugacy at most W K /W H∩K different pairs of well aligned pairs of θ-stable invariant parabolic subalgebras which are related to q, q H . If G = SL(4, R), H = H 1 and (q, q ∩ h 1 ) is the pair of well aligned parabolic subalgebras defined by x 0 = Q 2 , there there are at most 2 related pairs of well aligned parabolic subalgebras.
We expect the following Blattner-type formula to hold for the restriction to H: Conjecture. There exists a pair p, p H of well aligned θ-stable parabolic subgroups related to q, q H so that every H-type V of A q is of the form A p H (µ) for a character µ of L H and that dim Hom h,K H (V, A p ) = i j (−1) s−j dim Hom L∩H (H j (u ∩ h, V ), S i (u ∩ s) ⊗ C λ H ).
Remark. Some of the characters µ in this formula may be out of the fair range as defined in [10] and hence reducible.
If H is the maximal compact subgroup K, then |W K /W H∩K | = 1, all related pairs of well aligned parabolic subalgebra are conjugate to q, q H and hence we get the usual Blattner formula (5.108b on page 376 in [10]).
In the example discussed in this paper, G = SL(4, R), the representation A q and H the symplectic group considered in Section 4, all related pairs of well aligned parabolic subalgebras are conjugate to q, q h 1 and thus we obtain the Blattner type formula in Corollary 4.4.
Recall the definition of the (g, K)-module A q from 3.1. It is a submodule of L 2 (Z\G/Γ) for Γ small enough where Z the connected component of the center of GL(4, R). We will for the remainder of this sections consider it as an automorphic representation in the residual spectrum [23]. Then RES H 1 (A q ) and RES H 1 (A q ) are nonzero. Its discrete summands are contained in the space of automorphic forms. Theorem 9.3. The discrete summands of the two representations RES H 1 (A q ) respectively RES H 1 (A q ) are subrepresentations of the discrete spectrum of L 2 (H 1 /Γ H 1 ), respectively L 2 (H 1 /Γ H 1 ).
Proof . All the functions in A q decay rapidly at the cusps. Since the cusps of H 1 /Γ H 1 are contained in the cusps of G/Γ this is true for the functions in RES H 1 (A q ). Thus they are also contained in the discrete spectrum.
For Sp(2, R) the representations constructed in the previous theorem were first described by H. Kim (see [9] and [22]). For GL(2, C) we obtain the stronger result Theorem 9.4. The representations in the discrete spectrum of RES H 1 (A q ) are in the cuspidal spectrum of L 2 (H 1 /Γ H 1 ).
Proof . By 5.2 the representations in the discrete spectrum of RES H 1 (A q ) are unitarily induced principal series representations and so by a result of Wallach they are in fact cuspidal representations.
The embedding of H 1 = GL(2, C) into SL(4, R) is defined as follows: Write g = A + iB with real matrices A, B. Then Thus Γ H 1 is isomorphic to a congruence subgroup of GL(2, Z[i]). Since all the representations in the discrete spectrum of the restriction of A q do have nontrivial (h, K H )-cohomology with respect to some irreducible finite dimensional nontrivial representation F we obtained the well known result [5,20].