On-line version ISSN 1669-9637
Rev. Unión Mat. Argent. vol.49 no.2 Bahía Blanca July/Dec. 2008
To Mischa Cotlar with respect
Abstract. In this note, we give results about the restriction of a holomorphic discrete series of an exceptional simple Lie real group to a subgroup.
2000 Mathematics Subject Classification. Primary 22E46.
Key words and phrases. Holomorphic Discrete Series representations, branching laws.
Partially supported by FONCYT, CONICET, AgenciaCbaCiencia, SECYTUNC (Argentina), ICTP, TWAS (Italy)
A basic problem in representation theory of Lie groups is to derive "branching laws". By this we mean, for a given unitary irreducible representation of an ambient group , consider its restriction to a fixed subgroup and find the decomposition as a direct integral, and in particular compute the multiplicity of each irreducible factor of the restriction. There is a vast literature on this subject, and here we just direct the reader's attention to the extensive reviews of ,  and references therein. In this note, we consider a holomorphic discrete series of a connected simple exceptional Lie group, and determine whether or not it has an admissible restriction to a given closed connected reductive subgroup . Let us recall that a unitary representation of a topological group is admissible if it is a discrete Hilbert sum of irreducible unitary sub-representations and each irreducible summand occurs with finite multiplicity.
Holomorphic discrete series are associated to Hermitian symmetric spaces. We consider a Hermitian symmetric space , where is a simple connected real Lie group (which we shall assume for convenience, to minimize notations, with finite center), and a maximal compact subgroup. For a Lie group we denote its Lie algebra by the corresponding German lower case letter. We write the Cartan decomposition of as . Thus , the tangent space of at the origin, is provided with a complex structure corresponding to a choice of square root . To denote the complexification of a vector space, we add the subscript . We denote by and the eigenspaces of in with respective eigenvalues : a linear form is -linear if and only if its linear extension to is zero on the subspace . Moreover, , is the decomposition of as a direct sum of two irreducible -modules, dual to each other.
Recall (see ) that the center of is one dimensional, and that we can choose uniquely a basis (denoted by the same letter ) of whose adjoint action in is the complex structure of the tangent space at the origin of . We write . We have , and . Correspondingly, we have , where is isomorphic to , and is finite.
An irreducible unitary representation of is called holomorphic if its underlying Harish-Chandra module has a non zero vector which is annihilated by . An irreducible irreducible unitary representation of is called a discrete series representation if its coefficients are square integrable on with respect to a given Haar measure.
The exceptional connected simple Lie groups whose quotient by a maximal compact subgroup carries an invariant complex structure has been classified by E. Cartan. They are the connected groups with Lie algebras and . The respective complexified Cartan decompositions are :
In this paper, for and we give list closed connected reductive subgroups of such that an holomorphic discrete series of has an admissible restriction to . In , we gave several results concerning restrictions of more general discrete series for more general reductive groups, in particular, we introduced a sufficient condition —we call it condition (C)— which implies admissibility of restriction, and allows to compute multiplicities of restrictions by mean of a Blattner-Kostant type formula involving a partition function. However, there exist many cases of admissibility where condition (C) is not satisfied —many examples are given in , all of them for compact groups . One of our interests in studying precisely what happens for holomorphic discrete series of exceptional groups, besides our wish to understand the full picture, is to find other interesting examples. In particular, we give several non compact examples.
We would like to point out that in his Ph.D. thesis , S. Simondi has obtained the results on admissibility when rank of is equal to rank of they follow from Theorem 1. His technique is different from the one is used in this note.
The author would like to express his gratitude to Michel Duflo for the enlighten comments on the topics of this note.
We recall some results which we will use in our proofs. Let be a connected simple Lie group with finite center, choose a maximal compact group , and Cartan decomposition . We denote by the corresponding complex group. Let be a closed connected reductive subgroup. We assume that is a maximal compact subgroup of .
In , we prove a result which reduces the problem of admissibility of restriction of discrete series to the case of compact subgroups :
There are many criteria for admissibility of the restriction to a subgroup of an irreducible unitary representation of (see e. g. , ). When the subgroup is compact, we will use a criterium in term of the associated variety which we explain. We denote by the representation of in the space of -finite vectors of . Vogan  defined the associated variety , which is a Zariski-closed -invariant cone of dual of . Let us denote by the ring of regular functions on . The following criterium is known (see in particular Huang and Vogan , Kobayashi , Vergne ).
The most obvious example of proposition 3 is the group . The restriction to of an holomorphic discrete series is admissible (in fact it is true for any unitary irreducible representation of ), and we have also . Thus our problem of restriction is a particular case of a well known problem in invariant theory (see ,): Find pairs of connected reductive groups complex linear groups such that .
The subgroups and of deserve a special attention. For completeness, we recall the following well known result (which can serve as an illustration of proposition 3)
To study the restriction to , recall that Hermitian symmetric spaces are divided in two categories: the tube type, and the non tube type. One of the many equivalent definitions of tube type is (see ):
The Hermitian symmetric spaces is of non tube type if and only if .
We will also say that is of tube type. Hermitian symmetric spaces of tube type are related to simple Jordan algebras ; They are interesting because they have associated Zeta functions. However, from our point of view, non tube type is more interesting:
The list of Hermitian symmetric spaces of tube type is well known (see ). Among the two exceptional ones, is not of tube type, and is of tube type. Thus we have the following preliminary results, which explains why the case is richer.
We recall what is condition (C) of  in the particular case of a holomorphic discrete series. We choose a Cartan subgroup of , and denote by be set the roots of in . We choose a positive system such that the set of non compact roots is exactly the set of roots of in . We denote by the closed convex pointed cone generated by .
We assume that is a Cartan subgroup of . Let be the orthogonal of . Here is condition (C):
We rephrase condition (C). Let the cone dual to ; It is a closed convex cone whose interior contains . Then condition (C) is equivalent to condition (C'):
Proof. As a -module, is isomorphic to , where is an irreducible representation of . Thus, as a module, it is a finite direct sum of , where is a one dimensional representation of with weight . The weights of in are exactly the weights of contained in , occurring with finite multiplicity. The theorem follows. □
If condition (C) is satisfied for a torus , it is also satisfied for some one dimensional torus . Then satisfies condition (C) if and only if has a basis which belongs to . In particular, satisfies condition (C) (which is a way of proving proposition 4), and also all one dimensional torus not to far away from .
On the other hand, it is easy to see that condition (C) is never satisfied for . Thus, for , the group is an easy example where there is admissibility and condition (C) does not hold.
Let us explain more precisely what has to be done in general. We fix a compact connected semisimple group with a Cartan subgroup . Let be the connected component group of the centralizer of in . Then is a Cartan subgroup of .
Consider a connected closed groups such that . Up to conjugation, it will be of the form , where , the connected center of , is a closed connected subgroup of . Note that contains the center of . For clarity, we distinguish two cases.
First, assume there is admissible restriction of holomorphic discrete series of to — or equivalently, that is a -prehomogeneous space. Then there will be admissible restriction to any subgroup containing .
We assume now that the restriction of holomorphic discrete series of to is not admissible. Since contains , the restriction of a holomorphic discrete series of to the group is admissible. Let us choose a positive Weyl chamber for the group . Let the set of highest weights of the irreducible representations of which occur in , and the asymptotic cone of . It is known that is a closed convex polyhedral cone, independent of , contained in the projection on of the cone . We identify the orthogonal of in to . We consider the cone . The fact that the restriction of holomorphic discrete series of to is not admissible is equivalent to the fact that the cone is not reduced to . We consider its dual cone , and its interior . Note that belongs to .
Theorem 4. Let be a holomorphic discrete series of . Suppose that its restriction to is not admissible. Then the restriction of to is admissible if and only if one of the two following equivalent conditions hold:
Thus, discrete series of have admissible restriction to if and only if contains a closed subgroup , where is a one-dimensional subspace which intersects .
Theorem 4 suggests a method to find all closed connected reductive groups for which there is admissible restriction of holomorphic discrete series.
1. For each closed connected semisimple subgroup , determine whether there is admissibility of restriction of holomorphic discrete series. This step is not too difficult, for instance this is never the case when is of tube type, and we will give below the complete answer for .
2. When it is not the case, compute (with the notation as above) the algebra and the cone . This will give the list of closed connected subgroup such that for which there is admissibility of restriction of holomorphic discrete series.
For each particular , this is probably a feasible task, and we give several examples. However, we do not know an useful statement for all .
3. Given as in 2, list the closed connected reductive subgroups such that .
We use the notations of the previous subsection. We assume moreover that normalizes , or, equivalently, that . This means that is the sum of the root spaces for a certain subset of roots, and of the space generated by the corresponding coroots. We give some bounds on the cone .
For this we need to recall some important facts proven in . Let be the real rank of . There exists a set of pairwise strongly orthogonal roots such that the highest weights of the representations of occurring in are exactly those which belong to the cone generated by . We recall that is the highest weight of the -module , that is the maximal element (for a suitable order) among the roots orthogonal to , etc...
This means that is the cone generated by . Moreover, is a polyhedral cone such that .
For later use, we introduce some related notations. We will label the simple compact roots as ,…, , and the unique simple non compact root will be denoted by . Note that is the corresponding fundamental weight, and that , where is the longest element of the Weyl group of .
For this subsection denotes one of the groups We fix a holomorphic discrete series representation for Then,
In  is shown that a maximal connected subgroup of is either parabolic or reductive. For sake of completeness we list the maximal reductive subalgebras of The classification of maximal connected subgroups of was completed by . Some of the subalgebras has a compact abelian one dimensional factor which may not be the center of
Proof. The subgroups listed on the first seven lines corresponds to symmetric pairs The result follows from Kobayashi . Under his hypothesis, Kobayashi has shown that the multiplicity function is bounded. We do not know if this fact holds for other pairs
For we have that the center of the maximal compact subgroup of is contained in Hence, owing to Proposition 4 there is admissible restriction to the subgroup. In fact, center of K is contained in SU(2,1). For this, we consider the usual imbedding as a subgroup of Then, restricted to is equivalent to twice the spin representation of
Counting dimensions, we get restricted to is equivalent to twice Here, (resp. is the seven dimensional (resp. one dimensional) irreducible representation for It follows from a computation that the Cartan decomposition of is where is a subspace of the two copies of the trivial representation. From this we get that the center of is contained in
The maximal compact subgroup of is We show the center of is the center of In fact, in  is stated is a maximal subgroup of Hence, the projection of the center of on the direction of is trivial. Proposition 4 yields has admissible restriction to
Next, we dealt with in Let denote the noncompact simple root for the holomorphic system in and let be the compact simple root adjacent to We claim that of is spanned by The root system for the immersion of in is spanned by and the five compact simple roots different from Hence, belongs to the centralizer of Since, , is a subalgebra of and the centralizer of in is , the claim follows. Thus, a maximal compact subgroup of is Moreover, the factor of the immersion in is spanned by
Now, restricted to is equivalent to Also, takes on the values on each irreducible factor. Hence takes on the values Therefore, fix nonzero vectors in □
It may happen there is admissible restriction to a non compact subgroup which is not a maximal subgroup. In fact, we have,
In this section, . Following , we label the Dynkin diagram as follows.
The real rank is . We have , and .
We provide with the invariant scalar product for which for each root.This scalar product produces an isomorphism from to , and is the coroot corresponding to .
Let be the fundamental weight corresponding to . We have . We note that .
Since, by theorem 1, there is admissible restriction to , we consider proper maximal subgroups of . We show
i) restricted to is admissible.
ii) Let denote the usual imbedding and let denote the analytic subgroup of associated to then the restriction of to is admissible.
iii) For any other maximal subgroup of restricted to is not admissible.
iv) Let be a closed proper subgroup of Then restricted to is not an admissible representation.
Proof. To begin with, we recall the Cartan decomposition of where are the two spin representations of Let (resp. denote the simple factor of (resp. the center of
c) for a maximal subgroup of not locally isomorphic to
d) For subgroup and maximal subgroup or
In , we find a proof of Thus a) follows.
To continue, we fix an orthogonal basis of so that a system of positive compact roots is and the weights of the representation are with and odd number of The positive roots of are Let denotes an infinitesimal generator of chosen so that The module decomposes as
In order to show we list, up to conjugation, the maximal connected closed subgroups of These subgroups have been classified by Dynkin in . They are:
- for a connected, simple subgroup so that is an absolutely irreducible representation.
To continue, we assume for each maximal subgroup of not locally isomorphic to From this we derive a contradiction.
As before, denote the half spin representation. To begin with, we consider an irreducible, simple, maximal subgroup. decomposes as the sum irreducible modules
For of type if at least one is equivalent to then and hence, is one of has two irreducible representations of dimension ten whose highest weight are neither of these two representations are orthogonal . also has two ten dimensional representations of highest weight or  neither of them is orthogonal.
We are left to analyze the situation all are equivalent to Since is a subgroup of we have The case even was analyzed in the previous paragraph. The ten dimensional irreducible representations of have highest weight or they are not orthogonal. has no irreducible representation of dimension ten.
To conclude the proof of we show
- A half spin representations for restricted to
is equivalent to the spin representation
- The spin representation for restricted to is equivalent to the sum of the two half spin representations.
- An irreducible spin representation for is orthogonal.
- An irreducible spin representation for is symplectic.
For restricted to is equivalent to the spin representation of Since the spin representation of is orthogonal, we obtain
For besides acts on by Let denote a invariant quadratic form in Then is invariant under
For Hence, In  it is shown it is not an irreducible prehomogeneous vector space.
Here, and the restriction of to is equivalent to
Finally we examine Here, the restriction of to is equivalent to In  Apendix, it is shown that this representation is not a prehomogeneous vector space. Hence,
and we have verified c).
We now show d). For this we recall the work of  on the maximal subgroups of Up to conjugation, the maximal connect subgroups of are among the subgroups
Either the representation of or is orthogonal, , hence, an invariant for times one of these two groups, is given by an element of times the invariant quadratic form.
restricted to is equivalent to
The decomposition of under is
Finally, we analyze the admissibility of restricted to specific reductive subgroups of Let denote the fundamental weight of associated to The centralizer of in is equal to a semisimple Lie algebra plus the line spanned by
We fix real numbers, runs from
We define to be the subalgebra spanned by together with the vector We only consider such that the analytic subgroup associated to is compact. Either is isomorphic to are the usual two immersions of in From now on, we write for the analytic subgroup of associated to
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Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
Recibido: 17 de julio de 2008
Aceptado: 4 de septiembre de 2008