Infinite Dimensional Multiplicity Free Spaces I: Limits of Compact Commutative Spaces

0.28 MB·English

by  Joseph A. Wolf

Infinite Dimensional Multiplicity Free Spaces I:
Limits of Compact Commutative Spaces
8
0
Joseph A. Wolf
0
2
Department of Mathematics

n University of California
a Berkeley, CA 94720–3840, USA
J
[email protected]
5
2
Summary. Westudydirectlimits(G,K)=l−i→m(Gn,Kn)ofcompactGelfandpairs.
First, we develop a criterion for a direct limit representation to be a multiplicity–
]
T freediscretedirectsumofirreduciblerepresentations. Thenwelookatdirectlimits
R G/K =l−i→m Gn/Knofcompactriemanniansymmetricspaces,wherewecombineour
criterionwiththeCartan–HelgasonTheoremtoshowingeneralthattheregularrep-
.
h resentationofG=l−i→mGn onacertainfunctionspace−li→mL2(Gn/Kn)ismultiplicity
at free.ThatmethodisnotapplicablefordirectlimitsofnonsymmetricGelfandpairs,
m soweintroducetwoothermethods.Thefirst,basedon“parabolicdirectlimits”and
“defining representations”, extends the method used in the symmetric space case.
[
Thesecond uses some(new) branchingrules from finitedimensional representation

1 theory.In both cases wedefinefunction spaces A(G/K), C(G/K) and L2(G/K) to
v which ourmultiplicity–free criterion applies.
9
6
8
1 Introduction
3
.
1
Gelfand pairs (G,K), and the corresponding “commutative” homogeneous
0
8 spaces G/K, form a natural extension of the class of riemannian symmetric
0 spaces. We recall some of their basic properties. Let G be a locally compact
v: topologicalgroup,K acompactsubgroup,andM =G/K.Thenthefollowing
i conditions are equivalent; see [W2007, Theorem 9.8.1].
X
r 1. (G,K) is a Gelfand pair, i.e. L1(K\G/K) is commutative under con-
a volution.
2. If g,g′ ∈ G then µ ∗µ = µ ∗µ (convolution of Dirac
KgK Kg′K Kg′K KgK
measures on K\G/K).
3. C (K\G/K) is commutative under convolution.
c
4. The measure algebra M(K\G/K) is commutative.
5. The representation of G on L2(M) is multiplicity free.
If G is a connected Lie group one can also add
2 Joseph A.Wolf
6.ThealgebraofG–invariantdifferentialoperatorsonM iscommutative.
When we drop the requirement that K be compact, conditions 1, 2, 3
and 4 lose their meaning because integration on M or K\G/K no longer
correspondsto integrationonG.Condition5stillmakessenseaslongasK is
unimodular in G. Condition 6 remains meaningful (and useful) whenever G
is a connected Lie group; there one speaks of “generalized Gelfand pairs”.
InthisnotewelookatsomecaseswhereGandK arenotlocallycompact,
infactareinfinitedimensional,andshowinthosecasesthatthe multiplicity–
free condition 5 is satisfied. We first discuss a multiplicity free criterion that
can be viewed as a variation on some of the combinatoric considerations of
[DPW2002]; it emerged from some discussions with Ivan Penkov in another
context.Wethenapplythe criterioninthesettingofsymmetricspaces,prov-
ing that direct limits of compact symmetric spaces are multiplicity free. This
applies in particular to infinite dimensional real, complex and quaternionic
Grassmann manifolds, and it uses some basic symmetric space structure the-
ory. In particular our argument for direct limits of compact riemannian sym-
metricspacesmakesessentialoftheCartan–HelgasonTheorem,andthusdoes
not extend to direct limits of nonsymmetric Gelfand pairs.
In order to extend the the multiplicity–free result to at least some direct
limits of nonsymmetric Gelfand pairs, we define the notion of “defining rep-
resentation” for a direct system {(G ,K )} where the G are compact Lie
n n n
groupsandtheK areclosedsubgroups.We show howa defining representa-
n
tion for {(G ,K )} leads to a direct system {A(G /K )} of C–valued poly-
n n n n
nomialfunctionalgebras,acontinuousfunctioncompletion{C(G /K )},and
n n
aLebesguespacecompletion{L2(G /K )}.ThedirectlimitspacesA(G/K),
n n
C(G/K) and L2(G/K) are the function spaces on G/K = lim G /K which
−→ n n
we study as G–modules.
Next, we prove the multiplicity free property, for the action of G on
A(G/K), C(G/K) and L2(G/K), when {(G ,K )} is any of six families of
n n
Gelfandpairsrelatedto spheresandGrassmannmanifolds.Thenwe goonto
prove multiplicity free for three other types of direct limits Gelfand pairs.
Finallywesummarizetheresults,extendingthemslightlybyincludingthe
possibilityofenlargingtheK withintheirG –normalizerswithoutlosingthe
n n
property that {K } is a direct system.
n
Our proofs of the multiplicity–free condition, for some direct limits of
nonsymmetric Gelfand pairs, use of a number of branching rules, new and
old, for finite dimensional representations. This lends a certain ad hoc flavor
which I hope can be avoided in the future.
Direct limits (G,K)=lim(G ,K ) of riemannian symmetric spaces were
−→ n n
studiedbyOl’shanskiifromaverydifferentviewpoint[Ol1990].Heviewedthe
G inside dualreductivepairsandexaminedtheiractiononHilbertspacesof
n
InfiniteDimensional Multiplicity Free Spaces, I 3
Hermite polynomials.Ol’shanskiimade extensive use offactor representation
theory and Gaussian measure, obtaining analytic results on limit–spherical
functions.SeeFaraut[Fa2006]foradiscussionofsphericalfunctionsintheset-
tingofdirectlimitpairs.IncontrasttotheworkofOl’shanskiiandFarautwe
usetherathersimplealgebraicmethodofrenormalizingformaldegreesofrep-
resentationstoobtainisometricembeddingsL2(G /K )֒→L2(G /K ).
n n n+1 n+1
That leads directly to our multiplicity–free results.
I am indebted to Ivan Penkov for discussions of multiplicities in direct
limit representationswhich areformalizedinTheorem2 below.I alsowish to
acknowledge hospitality from the Mathematisches Forschungsinstitut Ober-
wolfach and support from NSF Grant DMS 04 00420.
2 Direct Limit Groups and Representations
We consider direct limit groups G = limG and direct limit representations
−→ n
π =limπ of them. This means thatπ is a representationofG ona vector
−→ n n n
spaceV ,thattheV formadirectsystem,andthatπistherepresentationof
n n
G on V = limV given by π(g)v = π (g )v whenever n is sufficiently large
−→ n n n n
that V ֒→ V and G ֒→ G send v to v and g to g. The formal definition
n n n n
amounts to saying that π is well defined.
It is clear that a direct limit of irreducible representations is irreducible,
butthereareirreduciblerepresentationsofdirectlimitgroupsthatcannotbe
formulated as direct limits of irreducible finite dimensional representations.
Thisisacombinatoricmatterandisdiscussedextensivelyin[DPW2002].The
following definition is closely related to those combinatorics but applies to a
somewhat simpler situation.
Definition 1 We say that a representation π of G is limit–aligned if it
has form limπ in such a way that (i) each π is a direct sum of primary
−→ n n
representations and (ii) the corresponding representation spaces V = limV
−→ n
have the property every primary subspace of V is contained in a primary
n
subspace of V .
n+1
Theorem 2 A limit–aligned representation π = limπ of G = limG is a
−→ n −→ n
direct sum of primary representations. If the π are multiplicity free then π is
n
a multiplicity free direct sum of irreducible representations.
Proof. Let V = limV be the representation spaces. Decompose V =
−→ n n
V where the V are the subspaces for the primary summands of
Pα∈In n,α n,α
π . Write π for the representation of G on V , so π = π .
n n,α n n,α n Pα∈In n,α
Since π is limit–aligned, i.e. since each V ⊂ V for some β ∈ I ,
n,α n+1,β n+1
we may assume I ⊂ I in such a way that each V ⊂ V for every
n n+1 n,α n+1,α
α ∈ I . Now V = V , discrete sum, where I = I and V = V .
n Pα∈I α S n α S n,α
4 Joseph A.Wolf
The sum is direct, for if u +u +···+u = 0 where u ∈ V for distinct
1 2 r i αi
indices α ,...,α , then we take n sufficiently large so that each u ∈ V
1 r i n,αi
and conclude that u =u =···=u =0. Thus π is the discrete direct sum
1 2 r
of the representations π =limπ of G on V .
α −→ n,α α
Let C = {X : V → V linear | Xπ (g) = π (g)X for all g ∈ G}, the
α α α α α
commutingalgebraofπ .Ifπ failstobeprimarythenC containsnontrivial
α α α
commuting ideals C′ and C′′. Then for n large, the stabilizer N (V ) of
α α Cα n,α
Vn,α in Cα contains nontrivial commuting ideals NC′(Vn,α) and NC′′(Vn,α).
α α
That is impossible because π is primary. We have proved that π is the
n,α
discrete direct sum of primary representations π .
α
If the π are multiplicity free then the π are irreducible and it is im-
n n,α
mediate that the π = limπ are irreducible. That completes the proof of
α −→ n,α
Theorem 2. ⊓⊔
A direct limit of irreducible representations is irreducible, but it is not
immediate that every irreducible direct limit representation can be rewritten
as a direct limit of irreducible representations. With this and Theorem 2 in
mind, we extend Definition 1 as follows.
Definition 3 A representation π of G = limG is lim–irreducible if it
−→ n
has form π = limπ where each π is an irreducible representation of G .
−→ n n n
Similarly, π is lim–primary if it has form π = limπ where each π is a
−→ n n
primary representation of G .
n
Theorem 4 Consider a representation π = limπ of G = limG with rep-
−→ n −→ n
resentation space V = limV . Suppose that each π is a multiplicity free
−→ n n
direct sum of irreducible highest weight representations. Suppose for n >> 0
that the direct system map V ֒→V sends G –highest weight vectors to
n−1 n n−1
G –highest weight vectors. Then π is a multiplicity free direct sum of lim–
n
irreducible representations of G.
Proof. Byhypothesiseachπ isadirectsumofprimaryrepresentationswhich,
n
in fact, are irreducible highest weight representations. We recursively choose
highest weight vectors so that π = π where π has highest
n−1 λ,n−1 λ,n−1
P
weight vector v ∈ V that maps to a highest weight vector v ∈ V
λ,n−1 n−1 λ,n n
of anirreducible constituent π of π . This exhibits π as a limit–aligneddi-
λ,n n
rect sum because it embeds the summand V of V into the irreducible
λ,n−1 n−1
summand of V that contains v . Now Theorem 2 shows that π is a multi-
n λ,n
plicity free direct sum of lim–irreducible representations of G. ⊓⊔
3 Limit Theorem for Symmetric Spaces
We now apply Theorems 2 and 4 to direct limits of compact riemannian
symmetric spaces. Fix a direct system of compact connected Lie groups G
n
InfiniteDimensional Multiplicity Free Spaces, I 5
and subgroups K such that each (G ,K ) is an irreducible riemannian
n n n
symmetric pair. Suppose that the corresponding compact symmetric spaces
M =G /K are connected and simply connected. Up to re–numbering and
n n n
passage to a common cofinal subsequence the only possibilities are
Compact IrreducibleRiemannian Symmetric Mn =Gn/Kn
Gn Kn RankMn DimMn
1 SU(n)×SU(n) diagonal SU(n) n−1 n2−1
2 Spin(2n+1)×Spin(2n+1) diagonal Spin(2n+1) n 2n2+n
3 Spin(2n)×Spin(2n) diagonal Spin(2n) n 2n2−n
4 Sp(n)×Sp(n) diagonal Sp(n) n 2n2+n
5 SU(p+q), p=pn,q=qn S(U(p)×U(q)) min(p,q) 2pq (5)
6 SU(n) SO(n) n−1 (n−1)(n+2)
2
7 SU(2n) Sp(n) n−1 2n2−n−1
8 SO(p+q), p=pn,q=qn SO(p)×SO(q) min(p,q) pq
9 SO(2n) U(n) [n] n(n−1)
2
10 Sp(p+q), p=pn,q=qn Sp(p)×Sp(q) min(p,q) 4pq
11 Sp(n) U(n) n n(n+1)
Fix one of the direct systems {(G ,K )} of (5). Then we have involu-
n n
tive automorphisms θ of G such that the Lie algebras decompose into ±1
n n
eigenspaces of the θ ,
n
g =k +s in such a way that k =g ∩k and s =g ∩s .
n n n n n n+1 n n n+1
Then we recursively construct a system of maximal abelian subspaces
a : maximal abelian subspace of s such that a =g ∩a .
n n n n n+1
The restricted root systems
Σ =Σ (g ,a ): the system of a –roots on g
n n n n n n
formaninversesystemoflinearfunctionals:Σ =Σ(g,a)isthesystemlimΣ
←− n
oflinearfunctionalsona=lima .Inthis inversesystem,the multiplicities of
−→ n
the restricted roots will increase without bound, but we can make consistent
choices of positive subsystems
Σ+ =Σ+(g ,a ): system of positive a –roots on g
n n n n n n
so that Σ+ ⊂ Σ+| for m ≧ n ≧ n . Then the corresponding simple root
n m an 0
systems
Ψ =Ψ (g ,a )={ψ ,...,ψ }: simple a –roots on g
n n n n 1,n rn,n n n
satisfy Ψ ⊂Ψ | for m≧n≧n as well. Here r =dima , rank of M .
n m an 0 n n n
6 Joseph A.Wolf
Recursivelydefineθ –stableCartansubalgebrasofh =t +a ofg with
n n n n n
h = g ∩h . Here t is a Cartan subalgebra of the centralizer m of a
n n n+1 n n n
in k . Now recursively construct positive root systems Σ+(m ,t ) such that
n n n
if α ∈ Σ+(m ,t ) then either α| = 0 or α| ∈ Σ+(m ,t ). Then we
n+1 n+1 tn tn n n
have positive root systems
Σ+(g ,h )={α∈ih∗ |α| =0 or α| ∈Σ+(g ,a )},
n n n an an n n n
the corresponding simple root systems, and the resulting systems of funda-
mental highest weights.
The Cartan–Helgason Theorem says that the irreducible representation
π of g of highest weight λ gives a summand of the representation of G on
λ n n
L2(M )ifandonlyif(i)λ| =0,sowemayviewλasanelementofia∗,and
n tn n
(ii)ifψ ∈Ψ (g ,a )then hλ,ψi isaninteger≧0.Condition(i)persistsunder
n n n hψ,ψi
restriction λ 7→ λ| because t ⊂ t . Given (i), condition (ii) says that
hn−1 n−1 n
1λbelongstothe weightlatticeofg ,soits restrictiontoh exponentiates
2 n n−1
to a well defined function on the corresponding maximal torus of G and
n−1
thus belongs to the weight lattice of g . Given condition (i) now condition
n−1
(ii) persists under restriction λ7→λ| . Define
hn−1
Λ =Λ(g ,k ,a )= λ∈ia∗ hλ,ψi integer ≧0 for all ψ ∈Ψ (g ,a ) .
n n n n n n(cid:12)hψ,ψi n n n o
(cid:12)
(cid:12)
ThisisthesetofhighestweightsforrepresentationsofG onL2(M ),andwe
n n
have just verified that Λ | ⊂ Λ . Now define the fundamental highest
n an−1 n−1
weights of Λ :
n
ξ ∈ia∗ defined by hξℓ,n,ψm,ni =δ for 1≦ℓ,m≦r ,
ℓ,n n hψm,n,ψm,ni ℓ,m n
and Ξ =Ξ(g ,k ,a )={ξ ,...,ξ }.
n n n n 1,n rn,n
Lemma 6 For n sufficiently large, and passing to a cofinal subsequence, if
ξ ∈Ξ there is a unique ξ′ ∈Ξ such that ξ′| =ξ.
n−1 n an−1
Proof. In the group manifold cases, lines 1, 2, 3 and 4 of Table 5, express
Gn =Ln×Ln and note that the complexification (Ln−1)C is the semisimple
component of a parabolic subgroup of (Ln)C. The restricted root and weight
systemsof(G ,K )arethesameastheunrestrictedrootandweightsystems
n n
of L , and the assertion follows.
n
In the Grassmann manifold cases, lines 5, 8 and 10 of Table 5, we first
consider the case where {p } is bounded. Then we may assume p = p con-
n n
stant and q increasing for n >> 0. Thus a = a , Ψ = Ψ (though
n n−1 n n−1 n
the multiplicities of the restricted roots will increase) and Ξ = Ξ . The
n−1 n
assertion now is immediate.
InfiniteDimensional Multiplicity Free Spaces, I 7
IntheGrassmannmanifoldcaseswemaynowassumethatbothp andq
n n
are unbounded. If p =q on a cofinal sequence of indices n we may assume
n n
p =q for all n, so Ψ is always of type C . Then we interpolate pairs and
n n n rn
renumber sothat p =q =p +1=q +1 for alln andnotice thatthe
n n n−1 n−1
DynkindiagraminclusionsC ⊂C areuniquelydeterminedbytheinteger
r−1 r
r. If p = q for only finitely many n and p < q on a cofinal sequence of
n n n n
indices n we may assume that r =p <q for all n, so Ψ is always of type
n n n n
B .Thenweinterpolate(p ,qn−1),(p ,q ),(p +1,q ),...,(p ,q )
rn n−1 n−1 n n−1 n n n
and renumber so that we always have r =r or r =r +1 and notice
n n−1 n n−1
that the Dynkin diagram inclusions B ⊂ B are uniquely determined by
r−1 r
the integer r. If p = q for only finitely many n and also p = q for only
n n n n
finitelymanynthenp >q onacofinalsequenceofindicesn,wemayassume
n n
p > q = r for all n, we interpolate as before exchanging the rˆoles of p
n n n ℓ
and q , and we note again that the Dynkin diagram inclusions B ⊂ B
ℓ r−1 r
are uniquely determined by the integer r. Thus in all cases the fundamental
highest weights restrict as asserted.
In the lower–rank cases, lines 6 and 7 of Table 5, Ψ is of type A , so
n n−1
again restriction to a has the required property. In the hermitian sym-
n−1
metric case, line 11 of Table 5, a is a Cartan subalgebra of g and g
n n n−1
complexifies to the semisimple partof a parabolic subalgebraof (gn)C, so the
assertionfollows as in the groupmanifold cases.In the remaining case,line 9
of Table 5, Ψ is of type C for n even, type B for n odd. Passing to
n n/2 (n−1)/2
a cofinal subsequence we may assume n always even or always odd, and we
may interpolate as necessary by pairs so that n increases in steps of 2. Then,
again, there is no choice about the restriction, and the assertion follows. ⊓⊔
In view of Lemma 6, after passage to a cofinal subsequence and re-
numbering, we may assume the sets Ξ ordered so that
n
Ξ =Ξ(g ,k ,a )={ξ ,...,ξ } with
n n n n 1,n rn,n (7)
ξ =ξ | for 1≦ℓ≦r .
ℓ,n−1 ℓ,n an−1 n−1
Now define
I : all r –tuples I =(i ,...,i ) of non–negative integers,
n n 1 rn
I =limI where I ֒→I by (i ,...,i )7→(i ,...,i ,0,...,0),
−→ n n m 1 rn 1 rn (8)
π : rep of G with highest weight ξ =i ξ +···+i ξ ,
I,n n I 1 1 p rn
π =limπ for I ∈I.
I −→ I,n
According to the Cartan–Helgason Theorem, the π exhaust the represen-
I,n
tations of G on L2(M ). Denote
n n
V : representation space for the abstract representation π . (9)
I,n I,n
ThenV occurswithmultiplicity1intherepresentationofG onL2(M ).In
I,n n n
effect,therepresentationofG onL2(M )ismultiplicity free,andL2(M )∼=
n n n
8 Joseph A.Wolf
V as a G –module. However, in the following we must distinguish
LI∈I I,n n
between V as a G –module and L2(M ) as a space of functions.
LI∈I I,n n n
Let U(g ) denote the (complex) universal enveloping algebra of g . Let
n n
v be a highest weight unit vector in V for the action of G . Then
n+1 I,n+1 n+1
we have the G –submodule U(g )(v )⊂V ⊂L2(M ).
n n n+1 I,n+1 n+1
If u,v ∈V we write f for g 7→hu,π (g)vi, the matrix coefficient
I,n u,v;I,n I,n
function on G . These matrix coefficient functions span a space E that is
n I,n
invariantunderleftandrighttranslationsbyelementsofG .Asa(G ×G )–
n n n
module E ∼= V ⊠V∗ . If u∗ is the (unique up to scalar multiplication)
I,n I,n I,n n
K –fixed unit vector in V∗ then the right K –fixed functions in E form
n I,n n I,n
the left G –module EKn ∼=V ⊗u∗C∼=V .
n I,n I,n n I,n
Inthe following,itis crucialto distinguish betweenthe abstractrepresen-
tation space V and the space EKn of functions on G /K .
I,n I,n n n
We normalize Haar measure on G (and the resulting measure in M )
n n
to total mass 1. If u,v,u′,v′ ∈ V then we have the Schur Orthogonality
I,n
relation hfu,v;I,n,fu′,v′;I,ni|L2(Gn) =(degπI,n)−1hu,u′ihv,v′i.
Theorem 10 The space EKn of functions on G /K is G –module equiva-
I,n n n n
lent to U(g )(v ⊗u∗ ) ⊂ EKn+1. We map EKn into EKn+1 as follows.
n n+1 n+1 I,n+1 I,n I,n+1
Let {w } be a basis of V and define
j I,n
ψn′+1,n(cid:16)Xcjfwj,u∗n;I,n(cid:17) (11)
=(degπI,n+1/degπI,n)1/2Xcjfwj,u∗n+1;I,n+1 ∈EIK,nn++11.
Then ψ′ : EKn → EKn+1 is is G –equivariant and is isometric for L2
n+1,n I,n I,n+1 n
norms on G /K and G /K . In particular, as I varies with n fixed,
n n n+1 n+1
ψ′ :L2(G /K )→L2(G /K ) is a G –equivariant isometry.
n+1,n n n n+1 n+1 n
Proof. We have a(v )=ξ (a)v for all a∈a. The inclusionG ֒→G
n+1 I n+1 n n+1
is G –equivariant, so restriction of functions is G –equivariant and thus is
n n
A–equivariant, and (v ⊗u∗ )| is a ξ –weight vector in L2(M ). If α
n+1 n+1 Mn I n
is a positive restricted root for G and e ∈g is an α root vector then
n+1 α n+1
e (v ) = 0. If α is already a root for G and if e ∈ g then we have
α n+1 n α n
e ((v ⊗u∗ )| )=0. Thus either the restriction (v ⊗u∗ )| =0
α n+1 n+1 Mn n+1 n+1 Mn
or (v ⊗u∗ )| is a highest weight vector in EKn.
n+1 n+1 Mn I,n
Suppose (v ⊗ u∗ )| = 0 as a function on M = G /K . De-
n+1 n+1 Mn n n n
note V′ = U(g )(v ). It is a cyclic highest weight module for G with
n n n+1 n
highest weight ξ , and (V′ ⊗ u∗ C)| = 0, and it contains a unique
I n n+1 Mn
(up to scalar multiple) K –invariant unit vector u′. The coefficient function
n n
ϕ(g) := hu′n,πI,n+1(g)u′niVn′ = RGn(u′n ⊗ u∗n)(x)(u′n⊗u∗n)(x−1g)dx is iden-
tically zero because the u′(x) factor in the integrand vanishes for x ∈ G .
n n
InfiniteDimensional Multiplicity Free Spaces, I 9
But ϕ| is the positive definite (G ,K )–spherical function on G for the
Gn n n n
representation π , and in particular ϕ(1) = 1. That is a contradiction. We
I,n
conclude that (v ⊗u∗ )| 6=0, so (v ⊗u∗ )| is a highestweight
n+1 n+1 Mn n+1 n+1 Mn
vector in EKn. In particular EKn ∼= (V′ ⊗u∗ C)| ⊂ EKn+1| . That is
I,n I,n n n+1 Mn I,n+1 Mn
the equivariant map assertion. The unitary map assertion follows by Schur
Orthogonality. ⊓⊔
Theorem 10 gives isometric embeddings ψ′ : L2(M ) → L2(M ) for
m,n n m
n≦m. By construction ψ′ is G –equivariant. Define
m,n n
L2(G/K)=lim{L2(G /K ),ψ′ }: direct limit in the
−→ n n m,n
(12)
category of Hilbert spaces and unitary injections.
We emphasize the renormalizations of Theorem 10. Without those renormal-
izations we lose the Hilbert space structure of L2(G/K).
Theorem 13 The left regular representation of G on L2(G/K) is a multi-
plicity free discrete direct sum of lim–irreducible representations. Specifically,
that left regular representation is π where π = limπ is the irre-
PI∈I I I −→ I,n
ducible representation of G with highest weight ξ := i ξ . This applies to
I r r
P
all the thedirect systems of Table5. In particular wehave thethirteen infinite
dimensional multiplicity free spaces
1. SU(∞)×SU(∞)/diagSU(∞), group manifold SU(∞),
2. Spin(∞)×Spin(∞)/diagSpin(∞), group manifold Spin(∞),
3. Sp(∞)×Sp(∞)/diagSp(∞), group manifold Sp(∞),
4. SU(p+∞)/S(U(p)×U(∞)), Cp subspaces of C∞,
5. SU(2∞)/[S(U(∞)×U(∞))], C∞ subspaces of infinite codim in C∞,
6. SU(∞)/SO(∞),
7. SU(2∞)/Sp(∞),
8. SO(p+∞)/[SO(p)×SO(∞)], oriented Rp subspaces of R∞,
9. SO(2∞)/[SO(∞)×SO(∞)], R∞ subspaces of infinite codim in R∞,
10. SO(2∞)/U(∞),
11. Sp(p+∞)/[Sp(p)×Sp(∞)], Hp subspaces of H∞,
12. Sp(2∞)/[Sp(∞)×Sp(∞)], H∞ subspaces of infinite codim in H∞,
13. Sp(∞)/U(∞).
Proof. λ is limit–aligned by Theorem 10. Denote V = V = limV .
I S I,n −→ I,n
Then G acts irreducibly on it by π = limπ , and the various π are mu-
I −→ I,n I
tually inequivalent because they have different highest weights ξ := i ξ ,
I r r
P
and are lim–irreducible by construction. Now let V = V . Then V =
PI∈I I
limL2(G /K )=L2(G/K). ⊓⊔
−→ n n
10 Joseph A.Wolf
4 Gelfand Pairs and Defining Representations
In this section we set the stage for extension of Theorem 13 to a number of
direct systems {(G ,K )} of compact nonsymmetric Gelfand pairs.A glance
n n
at [Ya2004] or [W2007] exhibits many such pairs, but here we will only con-
sider those for which the compact groups G are simple. Here is the Kr¨amer
n
classification of Gelfand pairs corresponding to compact simple Lie groups
(see [Kr1979] or [Ya2004] or [W2007, Table 12.7.1]).
Mn =Gn/Hn weakly symmetric Gn/Kn symmetric
Gn Hn conditions Kn with Hn⊂Kn ⊂Gn
1 SU(m+n) SU(m)×SU(n)] n>m≧1 S[U(m)×U(n)]
2 SO(2n) SU(n) n odd, n≧3U(n)
3 E Spin(10) Spin(10)·Spin(2)
6
4 SU(2n+1) Sp(n) n≧1 U(2n)=S[U(2n)×U(1)]
5 SU(2n+1) Sp(n)×U(1) n≧1 U(2n)=S[U(2n)×U(1)]
(14)
6 Spin(7) G (thereis none)
2
7 G SU(3) (thereis none)
2
8 SO(10) Spin(7)×SO(2) SO(8)×SO(2)
9 SO(9) Spin(7) SO(8)
10Spin(8) G Spin(7)
2
11SO(2n+1) U(n) n≧2 SO(2n)
12Sp(n) Sp(n−1)×U(1) n≧1 Sp(n−1)×Sp(1)
That gives us the nonsymmetric direct systems {(G ,K )} where
n n
(a) G =SU(p +q ) and K =SU(p )×SU(q ), p <q
n n n n n n n n
(b) G =SO(2n) and K =SU(n), n odd, n≧3
n n
(c) G =SU(2n+1) and K =Sp(n), n≧1
n n
(15)
(d) G =SU(2n+1) and K =U(1)×Sp(n), n≧1
n n
(e) G =SO(2n+1) and K =U(n), n≧2
n n
(f) G =Sp(n) and K =U(1)×Sp(n−1), n≧2
n n
Definition 16 Let {(G ,K )} be a direct system of Lie groups and closed
n n
subgroups. Suppose that π = limπ is a lim–irreducible representation of
−→ n
G = limG , with representation space V = limV , such that (i) π (K ) is
−→ n −→ n n n
theπ (G )–stabilizerofavectorv ∈V and(ii)eachv =v +w where
n n n n n+1 n n+1
π (G )leavesw fixed.(Thusthev giveacoherentsystemofembeddings
n n n+1 n
oftheG /K .) Supposefurtherthatforn>>0theπ havethesamehighest
n n n
weight vector. Then we say that π = limπ is a defining representation
−→ n
for {(G ,K )}.
n n
Now let’s consider some important examples of defining representations.
We will use those examples later.