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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.49 n.2 Bahía Blanca jul./dic. 2008
Admissible restriction of holomorphic discrete series for exceptional groups
Jorge Vargas
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 [13], [14] 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 [7]) 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 :
![𝔢6 = 𝔢6(-14)ℂ = 𝔰𝔬(10,ℂ ) + ℂJ + (𝔰+ ⊕ 𝔰- ).](/img/revistas/ruma/v49n2/2a0848x.png)
![𝔰±](/img/revistas/ruma/v49n2/2a0849x.png)
![16](/img/revistas/ruma/v49n2/2a0850x.png)
![𝔢7 = 𝔢7(- 25)ℂ = 𝔢6 + ℂJ + (ϖ1 ⊕ ϖ6 ).](/img/revistas/ruma/v49n2/2a0851x.png)
![ϖ ⋆](/img/revistas/ruma/v49n2/2a0852x.png)
![𝔢6](/img/revistas/ruma/v49n2/2a0853x.png)
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 [6], 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 [6], 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 [21], 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.
2.1. A criterium for admissibility of restriction.
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 [6], we prove a result which reduces the problem of admissibility of restriction of discrete series to the case of compact subgroups :
Proposition 1. Let be a discrete series for
. Then its restriction to
is admissible if and only if its restriction to
is admissible.
There are many criteria for admissibility of the restriction to a subgroup of an irreducible unitary representation of
(see e. g. [13], [14]). 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 [26] 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 [9], Kobayashi [12], Vergne [25]).
Proposition 2. Let be an irreducible unitary representation of
. Then its restriction to
is admissible if and only if
, that is the only
invariant regular functions on
are the constant.
Assume now that is hermitian symmetric. The criterium is particularly pleasant for holomorphic discrete series (see [12], [6], [25]) :
Proposition 3. Let be a holomorphic discrete series of
. Then
is the orthogonal of
in
. Thus its restriction to
is admissible if and only if
.
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 [22],[23]): Find pairs of connected reductive groups complex linear groups
such that
.
Remark 1. If is semi-simple, the condition
holds if and only if
has an open orbit in
.
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)
Proposition 4. Let be a holomorphic discrete series of
. Its restriction to
(and also to any closed subgroup
which contains
) is admissible.
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 [7]):
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 [7]; They are interesting because they have associated Zeta functions. However, from our point of view, non tube type is more interesting:
Proposition 5. Let be a holomorphic discrete series of
. Its restriction to
is admissible if and only if
is not of tube type.
The list of Hermitian symmetric spaces of tube type is well known (see [7]). 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.
Theorem 1. Let be a holomorphic discrete series of
with Lie algebra
. Its restriction to
is admissible.
Theorem 2. Let be a holomorphic discrete series of
with Lie algebra
. Its restriction to
(and to any of its closed subgroups
) is not admissible.
We recall what is condition (C) of [6] 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'):
![]() |
Condition (C) depends only on the maximal torus
of
. We have:
Theorem 3. Let be a holomorphic discrete series of
. Let
be a compact connected torus. Then the restriction of
to
is admissible if and only if condition (C) holds.
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.
2.3. Formulation of the problem.
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:
![]() |
or
![]() |
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 [20]. 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,
Theorem 5. For a maximal connected reductive subgroup of
restricted to
is admissible if and only if the center of
is a subgroup of
When is so that
a symmetric space, the theorem is a result of Kobayashi, [12], [15].
In [3] 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 [16]. 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 [12]. 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 [16] 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, [16],
is a subalgebra of
and the centralizer of
in
is
[1], 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,
Theorem 6. A holomorphic discrete series for has admissible restriction to any of the subgroups
![SO ⋆(10), SO (2,8), SU (4,1) × SU (2).](/img/revistas/ruma/v49n2/2a08390x.png)
We notice that none of the subgroups listed above contain the center of A consequence of Theorem 2 and Theorem 8 is
Theorem 7. If a holomorphic Discrete Series of an exceptional group has an admissible restriction to then center of
is a torus.
Theorem 7 does not hold for classical groups because Proposition 3 yields that holomorphic Discrete series for has an admissible restriction to
In this section, . Following [8], 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
Theorem 8. Let be a holomorphic discrete series representation for
Then,
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
Owing to Proposition 3, Theorem 8 follows from:
a) b)
c) for a maximal subgroup
of
not locally isomorphic to
d) For subgroup and maximal subgroup
or
and
Then
In [2], 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
![V 1 ⊕ V 1 ⊕ V 1 . 2(ε1+ε2+ε3+ ε4+ε5) ε1- 2(ε1+ε2+ ε3+ε4+ε5) ε1+ε2+ ε3- 2(ε1+ε2+ ε3+ε4+ε5)](/img/revistas/ruma/v49n2/2a08464x.png)
![(SL (2m + 1 ),Λ1 (ℂ2m+1 )⋆ ⊕ Λ2 (ℂ2m+1 ))](/img/revistas/ruma/v49n2/2a08465x.png)
![SU (5)ℂ](/img/revistas/ruma/v49n2/2a08466x.png)
![V ε1 ⊕ Vε1+ε2+ ε3](/img/revistas/ruma/v49n2/2a08467x.png)
![U (5) ℂ](/img/revistas/ruma/v49n2/2a08468x.png)
![𝔰+.](/img/revistas/ruma/v49n2/2a08469x.png)
![S [𝔰+ ]U (5) = ℂ](/img/revistas/ruma/v49n2/2a08470x.png)
![b)](/img/revistas/ruma/v49n2/2a08471x.png)
![SU (5)](/img/revistas/ruma/v49n2/2a08472x.png)
![1 V 2(ε1+ε2+ε3+ ε4+ε5),](/img/revistas/ruma/v49n2/2a08473x.png)
![d )](/img/revistas/ruma/v49n2/2a08474x.png)
![L1 = {e},L2 = SU (5).](/img/revistas/ruma/v49n2/2a08475x.png)
![V1(ε+ ε+ε +ε +ε) 2 1 2 3 4 5](/img/revistas/ruma/v49n2/2a08476x.png)
![Z5](/img/revistas/ruma/v49n2/2a08477x.png)
In order to show we list, up to conjugation, the maximal connected closed subgroups of
These subgroups have been classified by Dynkin in [5]. They are:
for
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
![V1 ⊕ ⋅⋅⋅ ⊕ Vr.](/img/revistas/ruma/v49n2/2a08496x.png)
![ρj](/img/revistas/ruma/v49n2/2a08497x.png)
![Vj](/img/revistas/ruma/v49n2/2a08498x.png)
![𝔰+ ∘ ρ.](/img/revistas/ruma/v49n2/2a08499x.png)
![S [V ]ρj(L ) = ℂ j](/img/revistas/ruma/v49n2/2a08500x.png)
![j = 1,⋅⋅⋅ ,r.](/img/revistas/ruma/v49n2/2a08501x.png)
![(L,ρj,Vj)](/img/revistas/ruma/v49n2/2a08502x.png)
![L](/img/revistas/ruma/v49n2/2a08503x.png)
![ρj](/img/revistas/ruma/v49n2/2a08504x.png)
![ρ (L) S [Vj] j = ℂ.](/img/revistas/ruma/v49n2/2a08505x.png)
![n+1 n(2n+1) 2n (An,Λ1, ℂ ), (A2n,Λ2, ℂ ), (Cn, Λ1, ℂ ).](/img/revistas/ruma/v49n2/2a08506x.png)
![Vj](/img/revistas/ruma/v49n2/2a08507x.png)
![(Cn, Λ1).](/img/revistas/ruma/v49n2/2a08508x.png)
![SL2](/img/revistas/ruma/v49n2/2a08509x.png)
![n ≥ 2.](/img/revistas/ruma/v49n2/2a08510x.png)
![L](/img/revistas/ruma/v49n2/2a08511x.png)
![C ,n ≥ 2 n](/img/revistas/ruma/v49n2/2a08512x.png)
![r = 1](/img/revistas/ruma/v49n2/2a08513x.png)
![n = 8,](/img/revistas/ruma/v49n2/2a08514x.png)
![L](/img/revistas/ruma/v49n2/2a08515x.png)
![Cn,n ≥ 2](/img/revistas/ruma/v49n2/2a08516x.png)
![r ≥ 2](/img/revistas/ruma/v49n2/2a08517x.png)
![+ L S[𝔰 ] ⁄= ℂ,](/img/revistas/ruma/v49n2/2a08518x.png)
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 [3].
also has two ten dimensional representations of highest weight
or
[3] 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
![+ 𝔰𝔬(p)⊕ 𝔰𝔬(q) S[𝔰 ] ⁄= ℂ for p ≥ 1, q ≥ 1,p + q = 10.](/img/revistas/ruma/v49n2/2a08543x.png)
We recall the following facts, for a proof, see [1], [3] Table 1.
- 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 [11] it is shown it is not an irreducible prehomogeneous vector space.
For
Here, and the restriction of
to
is equivalent to
![ℂ4 ⊠ ℂ2 ⊠ ℂ ⊕ (ℂ4 )⋆ ⊠ ℂ ⊠ ℂ2](/img/revistas/ruma/v49n2/2a08580x.png)
![+ 𝔰](/img/revistas/ruma/v49n2/2a08581x.png)
![L](/img/revistas/ruma/v49n2/2a08582x.png)
![ℂ4 ×2 ⊕ ℂ4×2](/img/revistas/ruma/v49n2/2a08583x.png)
![]() |
![t p(X, Y ) = det(Y X ).](/img/revistas/ruma/v49n2/2a08585x.png)
![S [𝔰+ ]SO (4)×SO(6)](/img/revistas/ruma/v49n2/2a08586x.png)
![⁄= ℂ.](/img/revistas/ruma/v49n2/2a08587x.png)
![p.](/img/revistas/ruma/v49n2/2a08588x.png)
Finally we examine Here, the restriction of
to
is equivalent to
In [11] 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 [5] on the maximal subgroups of Up to conjugation, the maximal connect subgroups of
are among the subgroups
![SO (5), S (U (k ) × U (5 - k))k = 1,2,3,4, (SU (2),ρ)](/img/revistas/ruma/v49n2/2a08596x.png)
![ρ](/img/revistas/ruma/v49n2/2a08597x.png)
![SU (2 ).](/img/revistas/ruma/v49n2/2a08598x.png)
Either the representation of or
is orthogonal, [3], 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
![4 3 4 2 4 ℂ ⊕ (ℂ ⊕ ℂe5 ) ⊕ (Λ (ℂ ) ⊕ Λ (ℂ ) ∧ ℂe5 ).](/img/revistas/ruma/v49n2/2a08605x.png)
![S(U (4) × U(1))](/img/revistas/ruma/v49n2/2a08606x.png)
![ℂ4 ⊕ Λ3 (ℂ4)](/img/revistas/ruma/v49n2/2a08607x.png)
![J5](/img/revistas/ruma/v49n2/2a08608x.png)
![V12(ε1+ ε2+ε3+ε4+ ε5) ⊕ Vε1- 12(ε1+ε2+ε3+ ε4+ε5) ⊕ V ε1+ε2+ε3- 12(ε1+ε2+ε3+ ε4+ε5)](/img/revistas/ruma/v49n2/2a08609x.png)
![5 3 1 --i, - -i, --i. 6 6 6](/img/revistas/ruma/v49n2/2a08610x.png)
![V1(ε1+ ε2+ε3+ε4+ε5) 2](/img/revistas/ruma/v49n2/2a08611x.png)
![Z5S (U (4) × U(1)).](/img/revistas/ruma/v49n2/2a08612x.png)
The decomposition of under
is
![]() |
![2 1 2 S [ℂ ⊕ Λ (ℂ )]](/img/revistas/ruma/v49n2/2a08616x.png)
![∑ r XrYr,](/img/revistas/ruma/v49n2/2a08617x.png)
![U (2 )-](/img/revistas/ruma/v49n2/2a08618x.png)
![S [ℂ3 ⊕ Λ2(ℂ3 )]](/img/revistas/ruma/v49n2/2a08619x.png)
![U (3),](/img/revistas/ruma/v49n2/2a08620x.png)
![∑ ZjWj. j](/img/revistas/ruma/v49n2/2a08621x.png)
![∑ ∑ (1 ⊗ Xr) (Zj ⊗ 1)(Wj ⊗ Yr) r j](/img/revistas/ruma/v49n2/2a08622x.png)
![U (3) × U (2).](/img/revistas/ruma/v49n2/2a08623x.png)
![Z5S (U (3 ) × U (2)).](/img/revistas/ruma/v49n2/2a08624x.png)
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
Proposition 6. A holomorphic discrete series for has an admissible restriction to the subgroups:
[1] Adams, J.F., Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, (1996). University of Chicago Press. [ Links ]
[2] Benson, C., Ratcliff, G., A classification of multiplicity free actions, J. Algebra, (181), 152-186, (1996). [ Links ]
[3] Bourbaki, N., Groupes et Algèbres de Lie, Chap 8, Masson et Cie, (1982). [ Links ]
[4] Brion, M., Sur l'image de l'application moment. Springer Lecture Notes in Math., 1296, 177-192 (1987) [ Links ]
[5] Dynkin, E.B., Maximal subgroups of classical groups, Transl. AMS (2)6, 245-378, (1957). [ Links ]
[6] Duflo, M., Vargas, J., Proper maps and multiplicities, preprint. An announcement is in [24]. [ Links ]
[7] Faraut, J., Koranyi, A., Analysis on symmetric connes, Oxford University Press (1995). [ Links ]
[8] Freundenthal, H., de Vries, H., Linear Lie Groups, Academic Press,(1969). [ Links ]
[9] Huang, J., Vogan, D., The moment map and the geometry of admissible representations, preprint. [ Links ]
[10] Kimura, T., A classification of prehomogeneous vector spaces of simple algebraic groups with scalar multiplication, J. Algebra, 83, 72-100, (1983). [ Links ]
[11] Kimura, T., Introduction to prehomogeneous vector spaces, Translation of Mathematical Monographs, Vol. 215, AMS, (2002). [ Links ]
[12] Kobayashi, T., Discrete decomposability of the restriction of with respect to reductive subgroups, Invent. Math. 131, 229-256 (1998). [ Links ]
[13] Kobayashi, T., Branching Problems of Unitary representations, Proceedings of the International Congress of Mathematicians, Vol II, 615-627, (2002). [ Links ]
[14] Kobayashi, T. Restrictions of unitary representations of real reductive groups. In: Lie theory: Unitary representations and compactifications of symmetric spaces, 139-207, Progr. Math., 229, Birkhäuser Boston, Boston, MA, 2005. [ Links ]
[15] Kobayashi, T., Multiplicity free Representations and Visible actions on complex manifolds, preprint. [ Links ]
[16] Komrakov, B., Maximal subalgebras of real Lie algebras and a problem of Sophus Lie, Soviet Math. Dokl. Vol. 41, 269-273, (1990). [ Links ]
[17] Littelmann, P. Koreguläre und äquidimensionale Darstellungen. J. Algebra 123 (1989), no. 1, 193-222. [ Links ]
[18] Ness, L., with an appendix by Mumford, D., A stratification of the nullcone via moment map, Amer. J. Math. 106, 1281-1330 (1984) [ Links ]
[19] Paradan, E., Multiplicities of holomorphic representations relatively to compact subgroups, Harmonische Analysis und Darstellungstheorie Topologischer Gruppen, Oberwolfach Report, 49/2007. [ Links ]
[20] Schmid, W., Die Randerwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Inventiones Math.. vol 9, 61-80, (1969). [ Links ]
[21] Simondi, S., Restricción de representaciones de cuadrado integrable, Ph. D. Thesis, FAMAF, Universidad Nacional de Córdoba, Argentina, (2006). [ Links ]
[22] Solomon, S., Irreducible linear group-subgroup pairs with the same invariants, J. Lie Theory 15, 105-123, (2005). [ Links ]
[23] Solomon, S., Orthogonal linear group-subgroup pairs with the same invariants, J. Algebra 299 623-647, (2006). [ Links ]
[24] Vargas, J., Restriction of square integrable representations, a review, Symposium in representation theory, edited by Yamashita, J., Hokadate, Japon, 62-79, (2003). [ Links ]
[25] Vergne, M., Quantization of algebraic cones and Vogan's conjecture, Pacific Journal of Math. Vol. 182, 114-135, (1998). [ Links ]
[26] Vogan, D., Associated varieties and unipotent representations, Harmonic Analysis on Reductive Lie groups, Progress in Math. Vol. 101, Birkhäuser, 315-388, (1991). [ Links ]
Jorge Vargas
CIEM-FaMAF,
Universidad Nacional de Córdoba,
Córdoba 5000, Argentina
vargas@famaf.unc.edu.ar
Recibido: 17 de julio de 2008
Aceptado: 4 de septiembre de 2008