Branching laws, some results and new examples
aa r X i v : . [ m a t h . R T ] J un BRANCHING LAWS, SOME RESULTS AND NEWEXAMPLES
OSCAR MARQUEZ, SEBASTIAN SIMONDI, JORGE A. VARGAS
Abstract.
For a connected, noncompact matrix simple Lie group G sothat a maximal compact subgroup K has three dimensional simple ideal,in this note we analyze the admissibility of the restriction of irreduciblesquare integrable representations for the ambient group when they arerestricted to certain subgroups that contains the three dimensional ideal.In this setting we provide a formula for the multiplicity of the irreduciblefactors. Also, for general G such that G/K is an Hermitian G -manifoldwe give a necessary and sufficient condition so that a square integrablerepresentations of the ambient group is admissible over the semisimplefactor 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 havethe same rank. We also fix T ⊂ K a maximal torus. Thus, T is a compactCartan subgroup of G . Under these hypotheses, Harish-Chandra showedthere exists irreducible unitary representations of G so that its matrix co-efficients are square integrable with respect to a Haar measure on G . Oneaim of this note is to write down explicit branching laws for the restrictionof some irreducible square integrable representation to specific subgroups H of G . A second objective is to show that when G is simple, the symmetricspace G/K has G -invariant quaternionic structure, and H is a specific sub-group locally isomorphic to the group SU (2 , G has an admissible restriction to H if andonly if it is a quaternionic discrete series representation. The last objectiveis to present results on admissible restriction of square integrable represen-tations to specific subgroups of G. To begin with, we recall a descriptionof the irreducible square integrable representations for G . Harish-Chandrashowed that the set of equivalence classes of irreducible square integrablerepresentations is parameterized by a lattice contained in the dual of the Liealgebra of a compact Cartan subgroup. In order to state our results we need Date : October 7, 2018.2010
Mathematics Subject Classification.
Primary 22E46; Secondary 17B10.
Key words and phrases.
Square integrable representation, admissible restriction, mul-tiplicity formulae .Partially supported by CONICET, SECYTUNC (Argentina). to explicit the parametrization and set up some notation. As usual, the Liealgebra of a Lie group is denoted by the corresponding lower case Germanletter. The complexification of a real vector space V is denoted by addingthe subindex C . However, the root space for a root is denoted by the real Liealgebra followed by a subindex equal to the root. V ⋆ denotes the dual spaceto a vector space V. Let θ be the Cartan involution which corresponds to thesubgroup 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 unionof 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 highestweight or the infinitesimal character of an irreducible representation of K is dominant with respect to ∆ . The Killing form on the Lie algebra g givesrise to an inner product ( , ) in i t ⋆ . As usual, let ρ = ρ g denote half of thesum of the roots for some system of positive roots for Φ( g , t ) . A Harish-Chandra parameter for G is λ ∈ i t ⋆ such that ( λ, α ) = 0 forevery 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 uniqueirreducible square integrable representation ( π Gλ , V λ ) of G . Moreover, heshowed the map λ π Gλ is a bijection from the set of Harish-Chandraparameters dominant with respect to ∆ onto the set of equivalence classesof irreducible square integrable representations for G . For a proof [W1].In [GW], the authors have considered quaternionic real form G of a complexsimple Lie group and constructed a specific subgroup H locally isomorphicto SU (2 , , 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 , α and after we write β = n α + n α +a linear combination of compact simple roots, we have theinequality n + n ≤ . A noncompact connected simple Lie group G is a quaternionic real form if g is an inner form of a complex simple Lie algebraand if a compactly imbedded Cartan subalgebra t has the property thatΦ( g , t ) admits a small system of positive roots so that n + n = 2 . In [GW],the list of the Lie algebras for the quaternionic real forms is presented, wereproduce the list in Section 3. It can be shown that the set of equivalenceclasses of the set of quaternionic real forms is equal to the set of equivalenceclasses of the set of noncompact simple Lie groups G so that the associatedglobal symmetric space admits a G -invariant quaternionic structure.In order to state Theorem 1, we fix a quaternionic real form G, a smallsystem of positive roots Ψ ⊃ ∆ and a noncompact simple root α for Ψ . Anirreducible square integrable representation ( π Gλ , V ) is called quaternionicdiscrete series representation if the Harish-Chandra parameter λ is dominantwith respect to Ψ . -S-V 3 For the quaternionic real form G , a particular copy h of su (2 ,
1) containedin g is constructed in [GW]. For this, they verify the equality2( β, α )( α, α ) = 1 . Thus, the Lie subalgebra h C of g C spanned by the root vectors correspondingto the roots {± α, ± β } is isomorphic to sl (3 , C ) and invariant under theconjugation 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 , . Henceforth, we identify the setΦ( 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 ) hasthree systems of positive roots to which the root β belongs to. The one ofour interest is the non-holomorphic systemΨ q := Ψ ∩ Φ( h , u ) = { β − α, α, β } . The simple roots for Ψ q are β − α, α. For a root γ ∈ Φ( g , t ), we denote itscoroot by ˇ γ ∈ i t . Let Λ , Λ denote the fundamental weights for Ψ q , labeledso that Λ ( ˇ α ) = 0 . (1.1) Owing to results in [DV], [W2], [Kb2], which we will review in section2, it follows that for a Harish-Chandra parameter λ dominant with respectto the small system Ψ the irreducible representation ( π Gλ , 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 , µ , ..., µ j , ... in i u ⋆ and there exists positive integers n G,H ( λ, µ j ) , j = 1 , , ... so that the restriction of ( π Gλ , V λ ) to H is unitarily equivalent to the discreteHilbert sum ∞ X j =1 n G,H ( λ, µ j ) ( π Hµ j , V µ j ) . In [GW] it is shown Ψ n := Ψ ∩ Φ n has 2 d elements. Our hypothesis that G isa quaternionic real form, forces the root spaces for the roots ± β span a threedimensional simple ideal su ( β ) in k . We denote by k the complementaryideal to su ( β ) in k . Hence, we have the decompositions t = R i ˇ β + ( t ∩ k ) and ∆ = { β } ∪ Φ( k , t ∩ k ) ∩ Ψ . For each λ ∈ t C , we write λ = λ + λ with λ ∈ C ˇ β, λ ∈ t C := t C ∩ k 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 dominantfor the small system Ψ, we have λ is a Harish-Chandra parameter for K orperhaps for a two fold cover of K . From now on, π K λ denotes the irreducible Branching laws representation for k of infinitesimal character λ . As usual, ∆ T ∩ K (cid:16) π K λ (cid:17) denotes the set of T ∩ K − weights for the representation π K λ and M ( λ , ν )stands for the multiplicity of the weight ν ∈ ∆ T ∩ K ( π K λ ) . In (2.9) we verify that for λ dominant with respect to small system Ψ theweight λ + ν + a Λ + bq u ( λ ) is dominant with respect to the system Ψ q forevery a, b ∈ Z ≥ , and for every U ∩ K -weight ν of π K λ . One result of thisnote, is:
Theorem 1.
Let G be a quaternionic real form, H as in (1.0) and ( π Gλ , V λ ) a quaternionic discrete series representation for G . Then, n G,H ( λ, µ ) = 0 if and only if µ = ( n + d )Λ + ( m + d )Λ + q u ( λ ) + q u ( ν ) with arbitrary m, n ∈ Z ≥ and T ∩ K -weight ν for π K λ . Moreover n G,H ( λ, µ ) = X ν ∈ ∆ T ∩ K (cid:16) π K λ (cid:17) ,m,n ∈ Z ≥ ,µ =( n + d )Λ +( m + d )Λ + q u ( λ )+ q u ( ν ) M ( λ , ν ) (cid:18) m + d − d − (cid:19)(cid:18) n + d − d − (cid:19) . A question that naturally arises is: What are the Harish-Chandra pa-rameters for G, dominant with respect to ∆ , so that π Gλ has an admissiblerestriction 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 quater-nionic real forms that a suitable copy of the algebra su is an ideal in amaximal compactly embedded subalgebra for g . A group locally isomorphicto SO (3 , p +1) has no square integrable representations. For a group locallyisomorphic to SO (3 , n ) and n ≥
2, in Proposition 2 we show that no irre-ducible square integrable representation of G has an admissible restrictionto 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 con-tained in certain image H of Sp (1 , Sp (1 , p ), we show it has admissible restriction to H , wecompute the Harish-Chandra parameters of the irreducible H -factors andtheir respective multiplicities.The group SU (2 ,
1) can be mapped into a simple Lie group G in perhapsseveral ways by maps φ : SU (2 , → G, a question is: What are the triple( G, π Gλ , φ ) such that π λ restricted to the image of φ is an admissible represen-tation. In [Va1] we find that for the analytic subgroup H that correspondsto the image of su (2 ,
1) in the rank one real form of a complex group type F no square integrable representation of the ambient group has an admissiblerestriction to H . -S-V 5 We would like to comment that this note grew up from results in therespective Ph. D. thesis of Sebastian Simondi and Oscar Marquez success-fully defended at the Faculty of Mathematics Astronomy and Physics at theUniversidad Nacional de C´ordoba, Argentine, in 2007 and 2011 respectively.2.
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 , . To begin with, we sketcha proof for the statement:
For λ dominant with respect to the small system Ψ , the representation ( π Gλ , V λ ) restricted to H is admissible. In fact, for asystem of positive roots Σ ⊂ Φ( g , t ) in [DV] is attached an ideal k (Σ) for theLie algebra k . The ideal is equal to the real form of the ideal of k C spannedby { [ Y γ , Y φ ] : γ, φ ∈ Σ ∩ Φ n , Y γ ∈ g γ } together with a subspace of the center z k of k .(2.0) For the system Ψ , cf. [GW] Prop. 1.3, Table 2.5, we have that anyroot in Φ c ∩ Ψ not equal to β is a linear combination of compact simple rootsfor Ψ . Thus, for two noncompact roots in Ψ , its sum, is a root only whenthe sum is equal to β. Thus, k (Ψ) is equal to su ( β ) plus the contributionof the center. Now, from the list of the quaternionic real forms, we readthat 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 (Ψ) is just the zerosubspace. Hence, for a quaternionic system Ψ we have k (Ψ) = su ( β ) . Because of the definition of H we have K (Ψ) is contained in H, hence,Theorem 1 in [DV] yields that for λ dominant with respect to Ψ the repre-sentation ( π Gλ , V λ ) has an admissible restriction to H ∩ K as well as to thesubgroup H. In [Kb3] we find a different proof of the admissibility.Therefore, there exists a sequence of Harish-Chandra parameters for H , µ , . . . , µ n , . . . ∈ i u ⋆ , for which, we may assume for every j, ( µ j , β ) > , andpositive integers n G,H ( λ, µ j ) so that the restriction of ( π Gλ , V λ ) restricted to H is equivalent to the Hilbert sum X j n G,H ( λ, µ j ) π Hµ j . We are left to compute µ j , to show each µ j is dominant for Ψ q and tocompute the integers n G,H ( λ, µ j ) . For this we recall results in [DV], [He]. For γ ∈ i t ⋆ . (resp in i u ⋆ . ) weconsider the Dirac distribution δ γ and the the discrete Heaviside distributiondefined by the series y γ := X n ≥ δ γ + nγ = δ γ + δ γ + γ + δ γ +2 γ + ... Branching laws
For any strict multiset γ , ..., γ r the convolution y γ ⋆ · · · ⋆y γ r is a well defineddistribution. In particular, we have y γ ⋆ · · · ⋆ y γ | {z } r := y rγ = X n ≥ (cid:18) n + r − r − (cid:19) δ ( r + n ) γ We have u C = C ˇ α + C ˇ β, u C ∩ k C = u C ∩ t C = h C ∩ k C = C ( ˇ β − α ) = C Λ .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)Φ z = { γ : ( γ, α ) = ( γ, β ) = 0 } = { γ ∈ Φ( k , t ) : q u ∩ t ( γ ) = 0 } . The Weyl group for the system Φ z is denoted by W z . Because of (1.2) theWeyl group W for the pair ( k , t ) is equal to the product h S β i × W ( k , t ) . Thus, W z \ W = h S β i × W z \ W ( k , t ) . Let∆( k / l ) := q u [Ψ ∩ Φ( k , t ) \ Φ z ] \ Φ( l , u ) = q u [ { β } ∪ Ψ( k , t ) \ Φ z ] \{ β } = q u ∩ k (Ψ ∩ Φ( k , t ∩ k ) \ Φ z ) =: ∆( k / u ∩ k ) . We set ρ z = 12 X γ ∈ Ψ ∩ Φ z γ and for σ ∈ i t ⋆ , the Weyl polynomial is defined tobe ̟ ( σ ) := Q γ ∈ Ψ ∩ Φ z ( σ, γ ) Q γ ∈ Ψ ∩ Φ z ( ρ z , γ ) . As before, we write λ = λ + λ with λ ∈ R i ˇ β and λ ∈ t . Then, owing to (1.2) for γ ∈ Φ( k , t ) ∩ Ψ we have the equality λ (ˇ γ ) = λ (ˇ γ ) . Thus, λ is a Harish- Chandra parameter for K or perhapsfor a two-fold cover of K . Actually, it readily follows that λ is a Harish-Chandra parameter for K if and only if β lifts to a character of T. Therefore,if necessary replacing G by a two-fold cover, we have λ is a Harish-Chandraparameter for K .We now state according to [He] the branching law for the restriction ofthe irreducible representation π K λ of infinitesimal character λ to the onedimensional torus H ∩ K = U ∩ K . The restriction of π K γ to H ∩ K is thesum of one-dimensional representations σ , ..., σ r with multiplicity M ( λ , σ j )for j = 1 , ..., r. The formula of Heckman for this particular case reads X µ ∈ ∆ U ∩ K (cid:16) π K λ (cid:17) M ( λ , µ ) δ µ = X s ∈ W z \ W ( k , t ) ǫ ( s ) ̟ ( sλ ) δ q u ∩ k ( sλ ) ⋆ y ∆( k / u ∩ k ) . -S-V 7 Another fact necessary for the proof is a formula in [DV] for the restriction of π Gλ to the subgroup H. The hypothesis for the truth of the formula is K (Ψ)being a subgroup of H which in our case holds because of our choice of Ψand H. The hypothesis on G and on the system Ψ yields for each w ∈ W the multiset S Hw := [∆( k / l ) ∪ q u ( w Ψ n )] \ Φ( h , u ) . is strict. This, also follows from an explicit computation of S Hw , which, wewill carry out later on. The formula that encodes the parameters µ j andthe multiplicities n G,H ( λ, µ j ) is: X µ ∈ i u ⋆ :( µ,β ) > n G,H ( λ, µ )( δ µ − δ S β µ ) = X w ∈ W z \ W ǫ ( w ) ̟ ( wλ ) δ q u ( wλ ) ⋆ y S Hw . To elaborate on (2.4) and on (2.6) we recall a few known results. It isconvenient to think of ( u ∩ su ( β )) ⋆ (resp. t ⋆ ) as the linear functionals on t so that vanishes on t (resp. on u ∩ su ( β )) , hence, for λ ∈ t , we have theequality q u ( λ ) = q u ∩ k ( λ ) . For w ∈ W ( k , t ) we have the equalities q u ( wλ ) = λ + q u ∩ k ( wλ ) q u ( wS β λ ) = S β ( λ ) + q u ∩ k ( wλ ) ̟ ( wλ ) = ̟ ( wλ ) ̟ ( wS β λ ) = ̟ ( wλ ) . From table 2.5 in [GW] it follows that any root in Φ( k , t ) is linear com-bination of compact simple roots in Ψ . Thus, lemma 3.3 in [HS] yields w Ψ n = Ψ n for w ∈ W ( k , t ) . In [GW] Proposition 1.3 it is shown that Ψ n = n γ ∈ Ψ : β,γ )( β,β ) = 1 o , andthat the map γ β − γ is an involution in Ψ n . Thus, the number of elementsof Ψ n is an even number 2 d and we may writeΨ n = { γ , ..., γ d , β − γ , ..., β − γ d , α, β − α } . Hence, we have S β (Ψ n ) = − Ψ n . Also in [GW] Proposition 2 it is shown that q u ( γ j ) = Λ for j = 2 , . . . , d. The equality Λ + Λ = β yields q u ( β − γ j ) = Λ . From these and (2.8) weconclude for w ∈ W ( k , t ) q u ( w Ψ n ) = { Λ , . . . , Λ | {z } d − , Λ , . . . , Λ | {z } d − , α, β − α } .q u ( wS β Ψ n ) = S β ( q u ( w Ψ n )) = { Λ , ..., Λ | {z } d − , Λ , . . . , Λ | {z } d − , α, α − β } . The previous calculations let us conclude.
Branching laws
For w ∈ W ( k , t ), S Hw = { Λ , . . . , Λ | {z } d − , Λ , . . . , Λ | {z } d − } ∪ ∆( k / u ∩ k ) .S HS β w = {− Λ , ..., − Λ | {z } d − , − Λ , . . . , − Λ | {z } d − } ∪ ∆( k / u ∩ k ) . The right hand side of (2.6), after we apply the previous calculations, be-comes equal to X s ∈ W z \ W ( k ) ǫ ( s ) ̟ ( sλ ) δ λ ⋆ δ q u ∩ k ( sλ ) ⋆ y d − ⋆ y d − ⋆ y ∆( k / u ∩ k ) − X s ∈ W z \ W ( k ) ǫ ( s ) ̟ ( sλ ) δ S β λ ⋆ δ q u ∩ k ( sλ ) ⋆ y d − S β Λ ⋆ y d − S β Λ ⋆ y ∆( k / u ∩ k ) = X σ ∈ ∆ U ∩ K ( π K λ ) M ( λ , σ ) δ σ ⋆ [ δ λ ⋆ y d − ⋆ y d − ⋆ δ Sβλ ⋆ y d − S β Λ ⋆ y d − S β Λ ]= X σ,p,q ∈ Z ≥ M ( λ , σ ) (cid:18) p + d − d − (cid:19)(cid:18) q + d − d − (cid:19) δ λ + σ ⋆ δ p Λ ⋆ δ q Λ + X σ,p,q ∈ Z ≥ M ( λ , σ ) (cid:18) p + d − d − (cid:19)(cid:18) q + d − d − (cid:19) δ S β ( λ + σ ) ⋆ δ pS β Λ ⋆ δ qS β Λ . (2.9) We now show: For every p, q ∈ Z ≥ , and for every U ∩ K − weight σ of π K λ the weight λ + σ + p Λ + qq u ( λ ) is dominant with respect to thesystem Ψ q = { α, β − α, β } . In fact, because of a Theorem of Kostant, every T − weight of π K λ lies inthe convex hull of { s ( λ ) , s ∈ W ( k , t ) } . Thus, there exists non negative realnumbers c t so that σ = P t ∈ W ( k , t ) c t q u ( tλ ) and X t c t = 1 . The hypothesis λ is regular and dominant with respect to Ψ yields, λ ( ˇ β ) = λ ( ˇ β ) > . We write ( λ + σ + p Λ + qλ , α ) = λ ( ˇ β ) α ( ˇ β ) + X t ( q u ( tλ ) , α ) + p (Λ , α )and q u ( tλ ) = ( tλ , β − α )( β − α ) . Now, since α ∈ i u ⋆ , we have,( q u ( tλ ) , α ) = ( tλ , α ) = ( λ , t − α ) = ( λ, t − α ) > t is a product of reflections about compact simple roots for Ψ , α ∈ Ψ n and (2.7). -S-V 9 For( λ + σ + p Λ + qλ , β − α )= X t c t ( tλ , β − α ) + X t c t ( tλ , β − α ) + q ( λ , β − α )= X t c t ( q u ( tλ + tλ ) , β − α ) + q ( λ , β − α )= X t c t ( tλ, β − α ) > β − α ∈ Ψ n , t ∈ W ( k , t ) and λ is regular dominant for Ψ . Wehave concluded the proof of Theorem 1, because we have shown that the lefthand side of (2.6) is expressed as claims the statement of Theorem 1. Thisfinishes the proof of Theorem 1.
Note
Wallach in [W2] considered the case the lowest K -type for π λ isequal to a representation of su ( β ) times the trivial representation of K . Admissible restrictions to ” SU (2 , of discrete series forquaternionic real forms To begin with we list the the Lie algebra of the Lie groups where Theorem1 applies. Up to equivalence, the list of the Lie algebras for quaternionicreal forms is: su (2 , n ), so (4 , n ), EII = e , EV I = e − , EIX = e − , F I = f and G = g . For the corresponding groups, we show that a square integrable irreduciblerepresentation for G has an admissible restriction to H = ” SU (2 , . Proposition 1.
Let G be a quaternionic real form, a small system of pos-itive roots Ψ , su ( β ) , k , H as in the previous section. Let Σ be a system ofpositive roots in Φ( g , t ) so that ∆ ⊂ Σ . Then, a square integrable irreduciblerepresentation 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 subgroupof H of G so that ( G, H ) is a symmetric pair and H ⊂ H and T ⊂ H .Hence, if Σ ⊃ ∆ is a system of positive roots for Φ( g , t ) so that some irre-ducible square integrable representation π Gµ with µ dominant with respectto σ has admissible restriction to H, then, [Kb3] Theorem 2.8 implies π Gµ has admissible restriction H . Owing to [DV] Prop. 2, we have k (Σ) is asubalgebra of h , except for some G locally isomorphic to SO (4 , n ) the Liealgebra k is the sum of two simple ideals, hence k (Σ) is equal to su ( β ) . Acase by case computation forces Σ = Ψ . For a group G locally isomorphicto SO (4 , n ) , n ≥ H which forces onceagain k (Σ) to be equal to a copy of su ( β ) and Σ = Ψ . (cid:3) Other groups
A group G locally isomorphic to either SO (3 , n ) or Sp (1 , n ) share with thequaternionic real forms that a copy su is an ideal in any maximal compactlyembedded subalgebra of g . Next, we analyze admissible restriction of squareintegrable representations to the subgroup corresponding to the copy of su mentioned in the previous sentence.We recall that from a criterium of Harish-Chandra it follows that a grouplocally isomorphic to SO (3 , n + 1) has no irreducible square integrablerepresentation, whereas, a group locally isomorphic to SO (3 , n ) does havea non empty discrete series. For a group locally G isomorphic to SO (3 , p ) amaximally compactly imbedded subalgebra is isomorphic to the direct sumof the ideals so (3) , so ( p ) . For the next statement we denote the analyticsubgroup of G corresponding to so (3) by K . Proposition 2.
For a group G locally isomorphic to SO (3 , n ) no irre-ducible square integrable representation has an admissible restriction to K . Proof: Because, n ≥ K is contained in a subgroup H of G locally isomorphic to SO (3 , . Next, we recall Theorem 1.2 in [Kb1] whichgives us: if a unitary representation of G has an admissible restriction to K then it has an admissible restriction to H . Hence, if irreducible squareintegrable representation of G had admissible restriction to K we wouldhave that H has a nonempty discrete series, which is not true since H islocally isomorphic to SO (3 , . Another proof follows from [DV] and the factthat K (Ψ) never is equal to K . (cid:3) For a group G locally isomorphic to Sp (1 , q ) we fix as maximal compactsubgroup K and a compact Cartan subgroup T. Therefore, there exists anorthogonal basis { ǫ , δ , ..., δ q } for i t ⋆ and a system of positive roots Σ sothat Σ ∩ Φ c = { ǫ , δ i ± δ j , ≤ i < j ≤ q, δ j , j = 1 , ..., q } and Σ ∩ Φ n = { ǫ ± δ j , j = 1 , ..., q } . The simple roots are ǫ − δ , δ j − δ j +1 , j = 1 , ..., q, δ q . The maximal root is β = 2 ǫ . It readily follows that k (Σ) = su (2 ǫ ) . Let h denote the real form of the Lie subalgebra spannedby the root vectors corresponding to the rootsΦ( h , u ) := {± ǫ , ± δ , ± ( ǫ ± δ ) } . Then, h is isomorphic to sp (1 , . As for the quaternionic case, let H denotethe analytic subgroup of G associated to h . Owing to [DV] Theorem 1, wehave that for λ dominant with respect to Σ the representation π Gλ restrictedto H is admissible when the Harish-Chandra. Let µ j , n G,H ( λ, µ j ) be as in(1.1). Let Σ q := Σ ∩ Φ( h , u ) . Let HC L ∩ K (cid:16) π K λ (cid:17) denotes the set of Harish-Chandra parameters for the L ∩ K − irreducible factors of the restriction of π K λ to the subgroup L ∩ K . We have,
Proposition 3.
Assume λ is dominant with respect to Σ . Then, for j = 1 , ... the parameters µ j := λ + σ + jǫ are dominant with respect to Σ q . Besides, -S-V 11 n G,H ( λ, µ ) = 0 if and only if µ = µ j for some j. Moreover n G,H ( λ, µ ) = X σ ∈ HC L ∩ K ( π K λ ) ,p ∈ Z ≥ µ = λ + σ + pǫ M ( λ , σ ) (cid:18) p + 2 q − q − (cid:19) . Proof: We begin writing the equalities (2.4) and (2.6) for the setting ofthe Proposition. For this particular case (2.4) reads X µ ∈ HC L ∩ K ( π K λ ) M ( λ , µ ) X r ∈ W ( L ∩ K ,U ∩ K ) ǫ ( r ) δ rµ = X s ∈ W z \ W ( k , t ) ǫ ( s ) ̟ ( sλ ) δ q u ∩ k ( sλ ) ⋆ y ∆( k / l ∩ k ) . The multiset S H w := [∆( k / l ) ∪ q u ( w Ψ n )] \ Φ( h , u )is strict. This, follows from an explicit computation of S H w , which, we willcarry out after the next formula. The formula (2.6) becomes: X µ ∈ i u ⋆ ( µ,ǫ ) > , ( µ,δ ) > n G,H ( λ, µ ) X t ∈ W ( L,U ) ǫ ( t ) δ tµ = X w ∈ W z \ W ǫ ( w ) ̟ ( wλ ) δ q u ( wλ ) ⋆ y S Hw . In this case u ∩ k = R i ˇ ǫ , u ∩ k = R i ˇ δ , l ∩ k = su (2 δ ). FurthermoreΨ ∩ Φ z = { δ i ± δ j , ≤ i < j ≤ q } ,W = h S ǫ i × W ( k , t ) and W z \ W = h S ǫ i × W z \ W ( k , t ) . For w ∈ W ( k , t ), w Ψ n = Ψ n , wS ǫ Ψ n = − Ψ n . and q u ( w Ψ n ) \ Φ( h , u ) = { ǫ , ..., ǫ } | {z } q − .q u ( wS ǫ Ψ n ) \ Φ( h , u ) = {− ǫ , ..., − ǫ } | {z } q − . ∆( k , l ) = q u (Ψ c \ Φ z ) \ Φ( h , u ) = { ǫ , δ , δ , ..., δ | {z } q − }\ Φ( h , u ) = ∆( k , l ∩ k ) . Therefore, for w ∈ W ( k , t ) we have, S H w = { ǫ , ..., ǫ } | {z } q − ∪ ∆( k , l ∩ k ) .S H S ǫ w = { ǫ , ..., ǫ } | {z } q − ∪ ∆( k , l ∩ k ) . After replacing S H w by the result obtained in the previous line, the righthand side of the formula similar to the one in (2.6) becomes X t ∈{ ,S ǫ } s ∈ W z \ W ( k , t ) ǫ ( t ) ǫ ( s ) ̟ ( sλ ) δ tλ ⋆ δ q u ( sλ ) ⋆ y ∆( k / l ∩ k ) ⋆ y q − tǫ = X t ∈{ ,S ǫ } ǫ ( t ) X r ∈ W ( k , t ) σ ∈ HC ǫ ( r ) M ( λ , σ ) δ tλ + rσ ⋆ y q − tǫ = X t,r,σp ∈ Z ≥ ǫ ( t ) ǫ ( s ) M ( λ , σ ) (cid:18) p + 2( q − − q − − (cid:19) δ tr ( λ + σ + pǫ ) = X p ≥ X σ M ( λ , σ ) (cid:18) p + 2 q − q − (cid:19) X w ∈ W ( L,U ) ǫ ( w ) δ w ( λ + σ + pǫ ) . By a reasoning similar to (2.9) we obtain that λ + σ + pǫ is dominant withrespect to Ψ q and we have concluded the proof of Proposition 3. (cid:3) Simondi’s Thesis G a maximal compact subgroup K for G. Henceforth, H is a closed reductive subgroup of G so that L := H ∩ K isa 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 in-tegrable representations, we would like to point out that in the course ofthe computation was made an extensive use of the description for the set of -S-V 13 equivalence classes of square integrable irreducible representations given byHarish-Chandra in terms of Harish-Chandra parameters.In [DV], [Kb1], [Kb2] and [Kb3] we find criteria for checking whetheror not the restriction of an square integrable representation for G is anadmissible representation for a subgroup H. By mean of these criteria, theclassification of the symmetric pairs given by Berger and a case by casechecking, we have,
Theorem 2.
Assume ( G, H ) is a symmetric pair and ( π, V ) is an irreduciblesquare integrable representation for G . If k is a simple Lie algebra, then therestriction 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 representationfor G . We assume k is a simple Lie algebra. Then, π restricted to L ′ is notan 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 2and results in [DV]. For the other subgroups L ′ the proof has been donein a case by case checking based on the classification of the equal rankmaximal subgroups of K obtained by Borel-de Siebenthal and work of ToshiKobayashi on criteria on admissibility of restriction of representations.Under the hypothesis k is not a simple Lie algebra, ( G, H ) a symmetricpair and the subgroups
L, K are of the same rank, we obtain a completelist, in the language of Harish-Chandra parameters, of the square integrablerepresentations 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 Hermitiansymmetric 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. Thehypothesis on (
G, K ) allows us to choose, once for all, a holomorphic systemof positive roots Ψ h in Φ( g , t ). In [Kb1] it is shown that either a holomor-phic or a antiholomorphic discrete series for G has an admissible restrictionto K ss if and only if G/K is not a tube domain. The next result gives acriteria which allows to determine when an arbitrary irreducible square in-tegrable representation has admissible restriction to K ss . For this we recallset of equivalence classes for irreducible square integrable representations isparameterized by the set of Harish-Chandra parameters λ dominant withrespect to Ψ h ∩ Φ c . The regularity of λ determines a system of positive rootsΨ λ := { α ∈ Φ( g , t ) : λ ( ˇ α ) > } which satisfies Ψ λ ∩ Φ c = Ψ h ∩ Φ c . In Table1 we list for each Hermitian symmetric pair (
G, K ) subsets I, ˜ I of Ψ h . Theorem 4.
Assume ( G, K ) is an Hermitian symmetric pair and fix an ir-reducible square integrable representation ( π Gλ , V λ ) for G of Harish-Chandra parameter λ dominant with respect Ψ h ∩ Φ c . Then, π Gλ restricted to K ss isan admissible representation if and only if either I or ˜ I is a subset of Ψ λ . The proof of the last Theorem is carried out in basis of classification ofHermitian symmetric pairs and criteria due to Kobayashi [Kb2], [Kb3]. g ksp ( n, R ) u ( n ) If n = 2 lI = { ( e k + e n − k +1 ) } lk =1 , ˜ I = − I If n = 2 l + 1 I = { e l +1 } ∪ { ( e k + e n − k +1 ) } lk =1 , ˜ I = − I so ∗ (2 n ) u ( n ) If n = 2 lI = { ( e k + e n − k +1 ) } lk =1 , ˜ I = − I If n = 2 l + 1 I = { ( e k + e n − k +1 ) } lk =1 ∪ { e l +1 + e l +2 } , ˜ I = { ( − e k − e n − k +1 ) } lk =1 ∪ {− e l − e l +1 } su ( p, q ) su ( p ) ⊕ u ( q ) I = { e i − e γ i } pi =1 ˜ I = {− e i + e b i } pi =1 e − so (10) ⊕ so (2) I = { ε , ε , e + e , e + e } ˜ I = − I e − e ⊕ so (2) I = { η , η , e + e } ˜ I = − I Table 1Here, in notation of Bourbaki: γ i = i + h ( i − q − p ) p i + p, for 1 ≤ i ≤ p,b i = ( i + 1 + [ i ( q − p ) p ] + p if i ( q − p ) p / ∈ Z ,i + i ( q − p ) p + p if i ( q − p ) p ∈ Z , for 1 ≤ i ≤ p,ε = ( − e − e − e − e + e − e − e + e ), ε = ( − e − e + e + e + e − e − e + e ), η = ( − e + e − e − e + e + e − e + e ) η = ( − e − e + e + e − e + e − e + e ) References [DV] Duflo, M., Vargas, J., Branching laws for square integrable representations, Proc.Japan Acad. Ser. A Math. Sci. , n 3, 49-54 (2010). -S-V 15 [GW] Gross, B., Wallach, N., On quaternionic discrete series representations, and theircontinuation, J. reine angew. Math., , 73-123 (1996).[HS] Hecht, H., Schmid, W., A Proof of Blattner’s Conjecture, Inventiones math. , 129-154 (1975).[He] Heckman, G. J., Projections of orbits and asymptotic behavior of multiplicities forcompact connected Lie groups, Invent. Math., , 333-356 (1982).[Kb1] Kobayashi, T., Discrete decomposability of the restriction of A q ( λ ) with respect toreductive subgroups and its applications, Inv. Math. , 181-205 (1994).[Kb2] Kobayashi, T., Discrete decomposability of the restriction of A q ( λ ) with respect toreductive subgroups III, Inv. Math. , 229-256 (1997).[Kb3] Kobayashi, T., Discrete decomposability of the restriction of A q ( λ ) with respectto reductive subgroups II-micro local analysis and asymptotic K -support, Annals ofMath. , 709-729 (1998).[KO] Kobayashi, T., Oshima, Y., Classification of discretely decomposble A q ( λ ) with re-spect to reductive symmetric pairs. Advances in Mathematics, , 2013-2047 (2012).[Va1] Vargas, J., Harish Chandra Modules of Rank One Lie Groups with AdmissibleRestriction to Some Reductive Subgroup. , Journal of Lie Theory , 643-663 (2010).[Va2] Vargas, J., Associated symmetric pair and multiplicities of admissible restriction ofdiscrete series, International Journal of Mathematics No. 12, 1650100 (2016)[W1] Wallach, N., Real reductive groups I, Academic Press, (1988).[W2] Wallach, N., Generalized Whittaker vectors for holomorphic and quaternionic rep-resentations, Comm. Math. Helv. , 266-307 (2003). Universidad Federal de Santa Mara, Santa Maria, RS Brasil; Facultad deCiencias Exactas y Natutales, Universidad Nacional de Cuyo, 5500 Mendoza,Argentine; FAMAF-CIEM, Ciudad Universitaria, 5000 C´ordoba, Argentine
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