Branching laws, some results and new examples

For a connected, noncompact matrix simple Lie group $G$ so that a maximal compact subgroup $K$ has three dimensional simple ideal, in this note we analyze the admissibility of the restriction of irreducible square integrable representations for the ambient group when they are restricted to certain subgroups that contains the three dimensional ideal. In this setting we provide a formula for the multiplicity of the irreducible factors. Also, for general $G$ such that $G/K$ is an Hermitian $G$-manifold we give a necessary and sufficient condition so that a square integrable representations of the ambient group is admissible over the semisimple factor of $K.$


Introduction
Let G be a connected noncompact simple matrix Lie group. Henceforth, we fix a maximal compact subgroup K of G and we assume both groups have the same rank. We also fix T ⊂ K a maximal torus. Thus, T is a compact Cartan subgroup of G. Under these hypotheses, Harish-Chandra showed there exists irreducible unitary representations of G so that its matrix coefficients are square integrable with respect to a Haar measure on G. One aim of this note is to write down explicit branching laws for the restriction of some irreducible square integrable representation to specific subgroups H of G. A second objective is to show that when G is simple, the symmetric space G/K has G-invariant quaternionic structure, and H is a specific subgroup locally isomorphic to the group SU (2, 1), then an irreducible square integrable representations for G has an admissible restriction to H if and only if it is a quaternionic discrete series representation. The last objective is to present results on admissible restriction of square integrable representations to specific subgroups of G. To begin with, we recall a description of the irreducible square integrable representations for G. Harish-Chandra showed that the set of equivalence classes of irreducible square integrable representations is parameterized by a lattice contained in the dual of the Lie algebra of a compact Cartan subgroup. In order to state our results we need to explicit the parametrization and set up some notation. As usual, the Lie algebra of a Lie group is denoted by the corresponding lower case German letter. The complexification of a real vector space V is denoted by adding the subindex C. However, the root space for a root is denoted by the real Lie algebra followed by a subindex equal to the root. V ⋆ denotes the dual space to a vector space V. Let θ be the Cartan involution which corresponds to the subgroup K, the associated Cartan decomposition is denoted by g = k + p. Let Φ(g, t) denote the root system attached to the Cartan subalgebra t C . Hence, Φ(g, t) = Φ c ∪ Φ n = Φ(k, t) ∪ Φ n (g, t) splits up as the disjoint union of the set of compact roots and the set of noncompact roots. From now on, we fix a system of positive roots ∆ for Φ c . For this note, either the highest weight or the infinitesimal character of an irreducible representation of K is dominant with respect to ∆. The Killing form on the Lie algebra g gives rise to an inner product ( , ) in it ⋆ . As usual, let ρ = ρ g denote half of the sum of the roots for some system of positive roots for Φ(g, t).
A Harish-Chandra parameter for G is λ ∈ it ⋆ such that (λ, α) = 0 for every α ∈ Φ(g, t), and so that λ + ρ is the differential of a character of T.
To each Harish-Chandra parameter, λ, Harish-Chandra associated a unique irreducible square integrable representation (π G λ , V λ ) of G. Moreover, he showed the map λ → π G λ is a bijection from the set of Harish-Chandra parameters dominant with respect to ∆ onto the set of equivalence classes of irreducible square integrable representations for G. For a proof [W1]. In [GW], the authors have considered quaternionic real form G of a complex simple Lie group and constructed a specific subgroup H locally isomorphic to SU (2, 1), their setting is as follows: a system of positive roots Ψ so that ∆ ⊂ Ψ ⊂ Φ(g, t) is called small if the maximal root β for Ψ is compact, Ψ has at most two noncompact simple roots α 0 , α 1 and after we write β = n 0 α 0 + n 1 α 1 +a linear combination of compact simple roots, we have the inequality n 0 + n 1 ≤ 2. A noncompact connected simple Lie group G is a quaternionic real form if g is an inner form of a complex simple Lie algebra and if a compactly imbedded Cartan subalgebra t has the property that Φ(g, t) admits a small system of positive roots so that n 0 + n 1 = 2. In [GW], the list of the Lie algebras for the quaternionic real forms is presented, we reproduce the list in Section 3. It can be shown that the set of equivalence classes of the set of quaternionic real forms is equal to the set of equivalence classes of the set of noncompact simple Lie groups G so that the associated global symmetric space admits a G-invariant quaternionic structure.
In order to state Theorem 1, we fix a quaternionic real form G, a small system of positive roots Ψ ⊃ ∆ and a noncompact simple root α for Ψ. An irreducible square integrable representation (π G λ , V ) is called quaternionic discrete series representation if the Harish-Chandra parameter λ is dominant with respect to Ψ.
Thus, the Lie subalgebra h C of g C spanned by the root vectors corresponding to the roots {±α, ±β} is isomorphic to sl(3, C) and invariant under the conjugation of g C with respect to g. Hence, h := g ∩ h C is a real form for h C . This real form has a compactly embedded Cartan subalgebra, namely, u := t ∩ h. Thus, h is isomorphic to su(2, 1). Henceforth, we identify the set Φ(h, u) with the subset {±α, ±β, ±(β − α)} of Φ(g, t).
(1.0) Let H denote the analytic subgroup of G with Lie algebra h. Then, L := K ∩ H is a maximal compact subgroup for H. The system Φ(h, u) has three systems of positive roots to which the root β belongs to. The one of our interest is the non-holomorphic system The simple roots for Ψ q are β − α, α. For a root γ ∈ Φ(g, t), we denote its coroot byγ ∈ it. Let Λ 1 , Λ 2 denote the fundamental weights for Ψ q , labeled so that Λ 1 (α) = 0.
(1.1) Owing to results in [DV], [W2], [Kb2], which we will review in section 2, it follows that for a Harish-Chandra parameter λ dominant with respect to the small system Ψ the irreducible representation (π G λ , V λ ) restricted to H is an admissible representation. That is, there exists a sequence of Harish-Chandra parameters for H, dominant with respect to β, µ 1 , µ 2 , ..., µ j , ... in iu ⋆ and there exists positive integers n G,H (λ, µ j ), j = 1, 2, ... so that the restriction of (π G λ , V λ ) to H is unitarily equivalent to the discrete Hilbert sum In [GW] it is shown Ψ n := Ψ ∩ Φ n has 2d elements. Our hypothesis that G is a quaternionic real form, forces the root spaces for the roots ±β span a three dimensional simple ideal su 2 (β) in k. We denote by k 2 the complementary ideal to su 2 (β) in k. Hence, we have the decompositions For each λ ∈ t C , we write λ = λ 1 + λ 2 with λ 1 ∈ Cβ, λ 2 ∈ t 2 C := t C ∩ k 2 C . Let q u : t ⋆ → u ⋆ denote the restriction map.
(1.3) We will verify (in 2.9) that for a Harish-Chandra parameter λ dominant for the small system Ψ, we have λ 2 is a Harish-Chandra parameter for K 2 or perhaps for a two fold cover of K 2 . From now on, π K 2 λ 2 denotes the irreducible representation for k 2 of infinitesimal character λ 2 . As usual, ∆ T ∩K 2 π K 2 λ 2 denotes the set of T ∩ K 2 −weights for the representation π K 2 λ 2 and M (λ 2 , ν) stands for the multiplicity of the weight ν ∈ ∆ T ∩K 2 (π K 2 λ 2 ). In (2.9) we verify that for λ dominant with respect to small system Ψ the weight λ 1 + ν + aΛ 1 + bq u (λ 2 ) is dominant with respect to the system Ψ q for every a, b ∈ Z ≥0 , and for every U ∩ K 2 -weight ν of π K 2 λ 2 . One result of this note, is: A question that naturally arises is: What are the Harish-Chandra parameters for G, dominant with respect to ∆, so that π G λ has an admissible restriction to H? The answer to this question is given in Proposition 1.
A group G locally isomorphic to either SO(3, n) shares with the quaternionic real forms that a suitable copy of the algebra su 2 is an ideal in a maximal compactly embedded subalgebra for g. A group locally isomorphic to SO(3, 2p+1) has no square integrable representations. For a group locally isomorphic to SO(3, 2n) and n ≥ 2, in Proposition 2 we show that no irreducible square integrable representation of G has an admissible restriction to the usual copy of "SO(3)" contained in G. For the quaternionic group Sp(1, p) the usual factor "Sp(1)" of a maximal compact subgroup is contained in certain image H 0 of Sp(1, 1). In Proposition 3, for a quaternionic discrete series for Sp(1, p), we show it has admissible restriction to H 0 , we compute the Harish-Chandra parameters of the irreducible H 0 -factors and their respective multiplicities.
The group SU (2, 1) can be mapped into a simple Lie group G in perhaps several ways by maps φ : SU (2, 1) → G, a question is: What are the triple (G, π G λ , φ) such that π λ restricted to the image of φ is an admissible representation. In [Va1] we find that for the analytic subgroup H 1 that corresponds to the image of su(2, 1) in the rank one real form of a complex group type F 4 no square integrable representation of the ambient group has an admissible restriction to H 1 .
We would like to comment that this note grew up from results in the respective Ph. D. thesis of Sebastian Simondi and Oscar Marquez successfully defended at the Faculty of Mathematics Astronomy and Physics at the Universidad Nacional de Córdoba, Argentine, in 2007 and 2011 respectively.

Proof of Theorem 1
As in the hypothesis G is a connected, quaternionic simple Lie group and H is the subgroup locally isomorphic to SU (2, 1). To begin with, we sketch a proof for the statement: For λ dominant with respect to the small system Ψ, the representation (π G λ , V λ ) restricted to H is admissible. In fact, for a system of positive roots Σ ⊂ Φ(g, t) in [DV] is attached an ideal k 1 (Σ) for the Lie algebra k. The ideal is equal to the real form of the ideal of k (2.0) For the system Ψ, cf. [GW] Prop. 1.3, Table 2.5, we have that any root in Φ c ∩ Ψ not equal to β is a linear combination of compact simple roots for Ψ. Thus, for two noncompact roots in Ψ, its sum, is a root only when the sum is equal to β. Thus, k 1 (Ψ) is equal to su 2 (β) plus the contribution of the center. Now, from the list of the quaternionic real forms, we read that z k is nonzero only for G locally isomorphic to SU (2, p). For su(2, p), in [DV], it is shown that for Ψ the contribution of z k to k 1 (Ψ) is just the zero subspace. Hence, for a quaternionic system Ψ we have Because of the definition of H we have K 1 (Ψ) is contained in H, hence, Theorem 1 in [DV] yields that for λ dominant with respect to Ψ the representation (π G λ , V λ ) has an admissible restriction to H ∩ K as well as to the subgroup H. In [Kb3] we find a different proof of the admissibility. Therefore, there exists a sequence of Harish-Chandra parameters for H, µ 1 , . . . , µ n , . . . ∈ iu ⋆ , for which, we may assume for every j, (µ j , β) > 0, and positive integers n G,H (λ, µ j ) so that the restriction of (π G λ , V λ ) restricted to H is equivalent to the Hilbert sum We are left to compute µ j , to show each µ j is dominant for Ψ q and to compute the integers n G,H (λ, µ j ). For this we recall results in [DV], [He]. For γ ∈ it ⋆ . (resp in iu ⋆ .) we consider the Dirac distribution δ γ and the the discrete Heaviside distribution defined by the series For any strict multiset γ 1 , ..., γ r the convolution y γ 1 ⋆· · ·⋆y γr is a well defined distribution. In particular, we have We have q u : t ⋆ C → u ⋆ C denotes restriction map. Next, we recall the sub-root system Φ z := {γ ∈ Φ(k, t) : q u (γ) = 0}. Because of (2.1) and (2.3) The Weyl group for the system Φ z is denoted by W z . Because of (1.2) the Weyl group W for the pair (k, t) is equal to the product S β × W (k 2 , t). Thus, We set ρ z = 1 2 γ∈Ψ∩Φz γ and for σ ∈ it ⋆ , the Weyl polynomial is defined to be ̟(σ) := γ∈Ψ∩Φz (σ, γ) γ∈Ψ∩Φz (ρ z , γ) . As before, we write λ = λ 1 + λ 2 with λ 1 ∈ Riβ and λ 2 ∈ t 2 . Then, owing to (1.2) for γ ∈ Φ(k 2 , t) ∩ Ψ we have the equality λ(γ) = λ 2 (γ). Thus, λ 2 is a Harish-Chandra parameter for K 2 or perhaps for a two-fold cover of K 2 . Actually, it readily follows that λ 2 is a Harish-Chandra parameter for K 2 if and only if β 2 lifts to a character of T. Therefore, if necessary replacing G by a two-fold cover, we have λ 2 is a Harish-Chandra parameter for K 2 .
Another fact necessary for the proof is a formula in [DV] for the restriction of π G λ to the subgroup H. The hypothesis for the truth of the formula is K 1 (Ψ) being a subgroup of H which in our case holds because of our choice of Ψ and H. The hypothesis on G and on the system Ψ yields for each w ∈ W the multiset u). is strict. This, also follows from an explicit computation of S H w , which, we will carry out later on. The formula that encodes the parameters µ j and the multiplicities n G,H (λ, µ j ) is: To elaborate on (2.4) and on (2.6) we recall a few known results. It is convenient to think of (u ∩ su 2 (β)) ⋆ (resp. t ⋆ 2 ) as the linear functionals on t so that vanishes on t 2 (resp. on u ∩ su 2 (β)), hence, for λ 2 ∈ t 2 , we have the equality q u (λ 2 ) = q u∩k 2 (λ 2 ). For w ∈ W (k 2 , t 2 ) we have the equalities From table 2.5 in [GW] it follows that any root in Φ(k 2 , t) is linear combination of compact simple roots in Ψ. Thus, lemma 3.3 in [HS] yields wΨ n = Ψ n for w ∈ W (k 2 , t).
In fact, because of a Theorem of Kostant, every T 2 −weight of π K 2 λ 2 lies in the convex hull of {s(λ 2 ), s ∈ W (k 2 , t)}. Thus, there exists non negative real numbers c t so that σ = t∈W (k 2 ,t) c t q u (tλ 2 ) and t c t = 1. The hypothesis λ is regular and dominant with respect to Ψ yields, λ 1 (β) = λ(β) > 0.
and λ is regular dominant for Ψ. We have concluded the proof of Theorem 1, because we have shown that the left hand side of (2.6) is expressed as claims the statement of Theorem 1. This finishes the proof of Theorem 1. Note Wallach in [W2] considered the case the lowest K-type for π λ is equal to a representation of su 2 (β) times the trivial representation of K 2 .
For the corresponding groups, we show that a square integrable irreducible representation for G has an admissible restriction to H = "SU (2, 1)" if and only if the Harish-Chandra parameter is dominant with respect to the small system Ψ.
Proposition 1. Let G be a quaternionic real form, a small system of positive roots Ψ, su 2 (β), k 2 , H as in the previous section. Let Σ be a system of positive roots in Φ(g, t) so that ∆ ⊂ Σ. Then, a square integrable irreducible representation with Harish-Chandra parameter dominant with respect to Σ has an admissible restriction to H if and only if Σ = Ψ.
Proof: From the list of Vogan's diagram, we notice there exists a subgroup of H 1 of G so that (G, H 1 ) is a symmetric pair and H ⊂ H 1 and T ⊂ H 1 . Hence, if Σ ⊃ ∆ is a system of positive roots for Φ(g, t) so that some irreducible square integrable representation π G µ with µ dominant with respect to σ has admissible restriction to H, then, [Kb3] Theorem 2.8 implies π G µ has admissible restriction H 1 . Owing to [DV] Prop. 2, we have k 1 (Σ) is a subalgebra of h 1 , except for some G locally isomorphic to SO(4, 2n) the Lie algebra k is the sum of two simple ideals, hence k 1 (Σ) is equal to su 2 (β). A case by case computation forces Σ = Ψ. For a group G locally isomorphic to SO(4, 2n), n ≥ 2 we select two different choices of H 1 which forces once again k 1 (Σ) to be equal to a copy of su 2 (β) and Σ = Ψ.

Other groups
A group G locally isomorphic to either SO(3, n) or Sp(1, n) share with the quaternionic real forms that a copy su 2 is an ideal in any maximal compactly embedded subalgebra of g. Next, we analyze admissible restriction of square integrable representations to the subgroup corresponding to the copy of su 2 mentioned in the previous sentence.
We recall that from a criterium of Harish-Chandra it follows that a group locally isomorphic to SO(3, 2n + 1) has no irreducible square integrable representation, whereas, a group locally isomorphic to SO(3, 2n) does have a non empty discrete series. For a group locally G isomorphic to SO(3, p) a maximally compactly imbedded subalgebra is isomorphic to the direct sum of the ideals so(3), so(p). For the next statement we denote the analytic subgroup of G corresponding to so(3) by K 1 .
Proposition 2. For a group G locally isomorphic to SO(3, 2n) no irreducible square integrable representation has an admissible restriction to K 1 .
Proof: Because, n ≥ 1 we have that K 1 is contained in a subgroup H 1 of G locally isomorphic to SO(3, 1). Next, we recall Theorem 1.2 in [Kb1] which gives us: if a unitary representation of G has an admissible restriction to K 1 then it has an admissible restriction to H 1 . Hence, if irreducible square integrable representation of G had admissible restriction to K 1 we would have that H 1 has a nonempty discrete series, which is not true since H 1 is locally isomorphic to SO(3, 1). Another proof follows from [DV] and the fact that K 1 (Ψ) never is equal to K 1 .
For a group G locally isomorphic to Sp(1, q) we fix as maximal compact subgroup K and a compact Cartan subgroup T. Therefore, there exists an orthogonal basis {ǫ 1 , δ 1 , ..., δ q } for it ⋆ and a system of positive roots Σ so that Σ ∩ Φ c = {2ǫ 1 , δ i ± δ j , 1 ≤ i < j ≤ q, 2δ j , j = 1, ..., q} and Σ ∩ Φ n = {ǫ 1 ± δ j , j = 1, ..., q}. The simple roots are ǫ 1 − δ 1 , δ j − δ j+1 , j = 1, ..., q, 2δ q . The maximal root is β = 2ǫ 1 . It readily follows that k 1 (Σ) = su 2 (2ǫ 1 ). Let h 0 denote the real form of the Lie subalgebra spanned by the root vectors corresponding to the roots Then, h 0 is isomorphic to sp(1, 1). As for the quaternionic case, let H 0 denote the analytic subgroup of G associated to h 0 . Owing to [DV] Theorem 1, we have that for λ dominant with respect to Σ the representation π G λ restricted to H 0 is admissible when the Harish-Chandra. Let µ j , n G,H 0 (λ, µ j ) be as in (1.1). Let Σ q := Σ ∩ Φ(h 0 , u). Let HC L∩K 2 π K 2 λ 2 denotes the set of Harish-Chandra parameters for the L ∩ K 2 −irreducible factors of the restriction of π K 2 λ 2 to the subgroup L ∩ K 2 . We have, Proposition 3. Assume λ is dominant with respect to Σ. Then, for j = 1, ... the parameters µ j := λ 1 + σ + jǫ 1 are dominant with respect to Σ q . Besides, n G,H 0 (λ, µ) = 0 if and only if µ = µ j for some j. Moreover Proof: We begin writing the equalities (2.4) and (2.6) for the setting of the Proposition. For this particular case (2.4) reads The multiset is strict. This, follows from an explicit computation of S H 0 w , which, we will carry out after the next formula. The formula (2.6) becomes: In this case u ∩ k 1 = Riǫ 1 , u ∩ k 2 = Riδ 1 , l ∩ k 2 = su 2 (2δ 1 ). Furthermore For w ∈ W (k 2 , t 2 ), wΨ n = Ψ n , wS 2ǫ 1 Ψ n = −Ψ n . and Therefore, for w ∈ W (k 2 , t) we have, After replacing S H 0 w by the result obtained in the previous line, the right hand side of the formula similar to the one in (2.6) becomes By a reasoning similar to (2.9) we obtain that λ 1 + σ + pǫ 1 is dominant with respect to Ψ q and we have concluded the proof of Proposition 3.

Simondi's Thesis
The Ph. D. thesis of Sebastian Simondi was defended by March 2007. Most of his results were verified in a case by case checking. By now, some of his results are a consequence of work of Toshi Kobayashi [Kb2], [Kb3] and [DV]. This section does not include proofs of the stated results, we will point out those results that follows from the work of T. Kobayashi, Kobayashi-Oshima, DV and those results we believe are still not in the literature. A complete version of Simondi's thesis is on: www2.famaf.unc.edu.ar/publicaciones/documents/seried/DMat49.pdf.
(5.1) We now describe the results. For this, we fix a noncompact connected matrix simple Lie group G a maximal compact subgroup K for G. Henceforth, H is a closed reductive subgroup of G so that L := H ∩ K is a maximal compact subgroup for H and that (G, H) is a symmetric pair. Hence, (K, L) is Riemannian symmetric pair.
As in the previous sections we assume G admits irreducible square integrable representations, we would like to point out that in the course of the computation was made an extensive use of the description for the set of equivalence classes of square integrable irreducible representations given by Harish-Chandra in terms of Harish-Chandra parameters.
In [DV], [Kb1], [Kb2] and [Kb3] we find criteria for checking whether or not the restriction of an square integrable representation for G is an admissible representation for a subgroup H. By mean of these criteria, the classification of the symmetric pairs given by Berger and a case by case checking, we have, Theorem 2. Assume (G, H) is a symmetric pair and (π, V ) is an irreducible square integrable representation for G. If k is a simple Lie algebra, then the restriction of π to H is not an admissible representation.
Nowadays, this result follows from [DV] or from the work of [KO]. For the next result we fix a maximal compact connected subgroup L ′ for K so that the rank of K is equal to the rank of L ′ .
Theorem 3. Let (π, V ) be an irreducible square integrable representation for G. We assume k is a simple Lie algebra. Then, π restricted to L ′ is not an admissible representation.
When L ′ is a maximal compact subgroup of a reductive subgroup H of G so that (G, H) is symmetric pair Theorem 3 follows from Theorem 2 and results in [DV]. For the other subgroups L ′ the proof has been done in a case by case checking based on the classification of the equal rank maximal subgroups of K obtained by Borel-de Siebenthal and work of Toshi Kobayashi on criteria on admissibility of restriction of representations. Under the hypothesis k is not a simple Lie algebra, (G, H) a symmetric pair and the subgroups L, K are of the same rank, we obtain a complete list, in the language of Harish-Chandra parameters, of the square integrable representations for G which do not have an admissible restriction to H. Nowadays this results are included in [KO] [Va2].
For the last result of this note we further assume (G, K) is an Hermitian symmetric pair. Then, the center of K is a one dimensional torus. Let K ss denote the semisimple factor of K. We fix a maximal torus T for K. The hypothesis on (G, K) allows us to choose, once for all, a holomorphic system of positive roots Ψ h in Φ(g, t). In [Kb1] it is shown that either a holomorphic or a antiholomorphic discrete series for G has an admissible restriction to K ss if and only if G/K is not a tube domain. The next result gives a criteria which allows to determine when an arbitrary irreducible square integrable representation has admissible restriction to K ss . For this we recall set of equivalence classes for irreducible square integrable representations is parameterized by the set of Harish-Chandra parameters λ dominant with respect to Ψ h ∩ Φ c . The regularity of λ determines a system of positive roots Ψ λ := {α ∈ Φ(g, t) : λ(α) > 0} which satisfies Ψ λ ∩ Φ c = Ψ h ∩ Φ c . In Table  1 we list for each Hermitian symmetric pair (G, K) subsets I,Ĩ of Ψ h .
Theorem 4. Assume (G, K) is an Hermitian symmetric pair and fix an irreducible square integrable representation (π G λ , V λ ) for G of Harish-Chandra