Symmetry breaking and the Gross Prasad conjectures for...

Symmetry breaking and the Gross Prasadconjectures for orthogonal groups

Puerto Rico, January 2014

Birgit Speh

Cornell University

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I would like to discuss some work on the Gross Prasad conjec-

tures for tempered representations of real orthogonal groups and

also examples extensions of these conjectures to Arthur-Vogan

packets of orthogonal groups.

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G = SO(p, q), UG an irreducible unitary representation of G on

Hilbert space

H = S0(p − 1, q) ⊂ G, UH an irreducible unitary representations

of H on a Hilbert space

2





G = SO(p, q), UG an irreducible unitary representation of G on

Hilbert space

H = S0(p − 1, q) ⊂ G, UH an irreducible unitary representations

of H on a Hilbert space

Gross Prasad conjectures concern

dim HomH(UG ⊗ UH ,C)

for tempered representations UG and UH.

2

HomH(UG, U∨H) = HomH(UG⊗UH ,C) is of interest for automor-

phic representations

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Periods of automorphic representations

If πA is an automorphic representation of GA and f ∈ πA then∫HA

f(HA)dhA is defines a period of πA if it converges and in

particular an element in HomH(πA,C).

If πA =∏πν then it defines an H- invariant linear functional on

all representations πν i.e an element in HomH(πν, UH) for a trivial

representation UH.

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Periods of automorphic representations

If πA is an automorphic representation of GA and f ∈ πA then∫HA

f(HA)dhA is defines a period of πA if it converges and in

particular an element in HomH(πA,C).

If πA =∏πν then it defines an H- invariant linear functional on

all representations πν i.e an element in HomH(πν, UH) for a trivial

representation UH.

The integral is related to special values of L-functions and thelinear invariant linear functionals are important in the relativetrace formula of Jacquet.

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HomH(UG, UH) is also of interest in other areas of mathe-

matics

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HomH(UG, UH) is also of interest in other areas of mathe-

matics

Analysis on G/H

We get a function on G/H by

v → T (U(g)v).

This function is an eigenfunction of the invariant differential op-

erators. Classical problem is determine the spectrum of the in-

variant differential operators on G/K. More generally determine

the spectrum of invariant operators on G/H where H is fix points

of an involution. Considered and solved by H.Schlichtkrull, E.

van den Ban, T. Oshima and many others ...

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For real orthogonal groups:

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For real orthogonal groups:

Sun-Zhou and Aizenbud -Gurevich proved that for irreduciblerepresentations

HomH(UG, UH) has at most dimension 1.

B.Gross and D.Prasad have a precise conjecture for the dimen-sion of HomH(UG, UH) if both UG and UH are tempered involvingarithmetic invariants (epsilon factors)

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A special case :

DISCRETE Branching laws at the real place : We consider

irreducible unitary representations U of G, whose restriction U|Hto H is a direct sum of irreducible representations of H and

determine explicitly the irreducible representations of H in U|H.

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A special case :




For tempered representations (discrete series) discrete branching

laws due to Gross-Wallach, T. Kobayashi confirm part of the

conjectures of Gross-Prasad,

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A special case :




For tempered representations (discrete series) discrete branching

laws due to Gross-Wallach, T. Kobayashi confirm part of the

conjectures of Gross-Prasad,

Complete description of all discrete branching laws for unitary

representations with nontrivial cohomology for pairs G=SO(n+1,1),

H=SO(n,1) which are usually not tempered is due to T.Kobayashi,

Y.Oshima, but no arithmetic conjectures about them yet.

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Symmetry breaking Integral operators and Differential op-

erators for real groups

They are explicit operators in HomH(UG, UH). For principal se-

ries representations this is an analysis problem related to the

construction of intertwining operators.

In this case the restriction of UG to H is not a discrete sum.

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Symmetry breaking Integral operators and Differential op-

erators for real groups

They are explicit operators in HomH(UG, UH). For principal se-

ries representations this is an analysis problem related to the

construction of intertwining operators.

In this case the restriction of UG to H is not a discrete sum.

So we proceed as follows:

We consider the space of C∞-vectors of UG and UH. It is a

Frechet space and UG and UH act continuously on this Frechet

space and we obtain the smooth representations U∞G and U∞H .

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Then

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Then

HomH(UG, UH)

denotes the space of continuous (with respect to the Frechet

topology) intertwining operators

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Then

HomH(UG, UH)

denotes the space of continuous (with respect to the Frechet

topology) intertwining operators

We have

HomH(UG ⊗ U∨H ,C) = HomH(UG, UH)

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Results about symmetry breaking operators:

Let P = MAN and P ′ = M ′A′N ′ be Langlands decompositions

of parabolic subgroups G = SO(n + 1,1) and G′ = SO(n,1),

respectively, satisfying

M ′ = M ∩G′, A′ = A ∩G′, N ′ = N ∩G′, P ′ = P ∩G′.

Fix finite-dimensional representations

σ : M → GLC(V )

and

τ : M ′ → GLC(W ).

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For λ ∈ a∗C and ν ∈ (a′C)∗, we define Vλ of P and Wν of P ′ by

P = MAN 3 meHn 7→ σ(m)eλ(H) ∈ GLC(V )

P ′ = M ′A′N ′ 3 m′eH′n′ 7→ τ(m′)eν(H ′) ∈ GLC(W ),

Consider the induced representations

I(P, Vλ) on the C∞-sections of Vλ := G×P Vλ.

I(P ′,Wν) on the C∞-sections of Wν := G×P Wν.

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For λ ∈ a∗C and ν ∈ (a′C)∗, we define Vλ of P and Wν of P ′ by

P = MAN 3 meHn 7→ σ(m)eλ(H) ∈ GLC(V )

P ′ = M ′A′N ′ 3 m′eH′n′ 7→ τ(m′)eν(H ′) ∈ GLC(W ),

Consider the induced representations

I(P, Vλ) on the C∞-sections of Vλ := G×P Vλ.

I(P ′,Wν) on the C∞-sections of Wν := G×P Wν.

A example of a symmetry breaking operator in

HomH(I(P, Vλ)), I(P ′, (Vλ)|P ′) is the restriction of a section f in

I(P, Vλ) to the subvariety G′/P ′.

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Consider now the case V = W trivial and the spherical principal

series representations I(λ) and J(ν) of O(n+1,1) and O(n,1)

using normalized induction.

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In joint work with T. Kobayashi

dim HomH(I(λ), J(ν)) ≤ 2

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In joint work with T. Kobayashi

dim HomH(I(λ), J(ν)) ≤ 2

If I(λ) and I(ν) are tempered then dim HomH(I(λ), J(ν)) = 1.

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Remarks about the (symmetry breaking) intertwining operators

This G′ intertwining operator has similar form to the usual in-

tertwining operator. Consider the spherical principal series rep-

resentations as functions on the large Bruhat cell N ∼ Rn.

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Remarks about the (symmetry breaking) intertwining operators

This G′ intertwining operator has similar form to the usual in-

tertwining operator. Consider the spherical principal series rep-

resentations as functions on the large Bruhat cell N ∼ Rn.

Theorem 1. Joint with T.KobayashiThere exists an analytic family

Aλ,ν ∈ HomG′(I(λ), J(ν)

of symmetry breaking operators with the distribution kernel

π∗

L(1/2, eλ1+ν1)L(1/2, eλ1−ν1)|xn|λ+ν−1/2(|x|2 + x2

n)−ν−(n−1)/2.

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Remark:

If the representations are tempered then the normalization factor

is closely related to the

L-function of the representation I(λ)× J(ν).

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Remark:

If the representations are tempered then the normalization factor

is closely related to the

L-function of the representation I(λ)× J(ν).

In this case λ and ν are are purely imaginary and I(λ) × J(ν)

and I(−λ) × J(−ν) are isomorphic and the L-function of the

representation I(λ)× J(ν) contains the factor

L(1/2, eλ1+ν1)L(1/2, eλ1−ν1)L(1/2, e−λ1+ν1)L(1/2, e−λ1−ν1)

The L-factor doesn’t have a pole and so the intertwining operator

is nonzero for tempered representations.

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Outline of ”a naive version” of the Gross Prasad conjectures for

orthogonal groups

Idea

Consider the complex dual group L(G)C of G. The tempered

representations of orthogonal groups are parametrized by homo-

morphisms τ of the Weil group WR into L(G)C.

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Outline of ”a naive version” of the Gross Prasad conjectures for

orthogonal groups

Idea

Consider the complex dual group L(G)C of G. The tempered

representations of orthogonal groups are parametrized by homo-

morphisms τ of the Weil group WR into L(G)C.

Unfortunately if we look at a homomorphism

τ : WR →L (G)Cwe cannot easily read of the inner form for which this definesa representation. It defines packet AG(τ) of representations ofseveral forms of orthogonal groups.

Similarly we can define a packet AH(τH).

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We now to define a subset A of compatible representations of

AG(τ)×AH(τH).

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AG(τ)×AH(τH).

Gross Prasad Conjecture (simple version)

There exists exactly one pair πG × πH ∈ A so that

HomH(πG, πH) = 1

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AG(τ)×AH(τH).

Gross Prasad Conjecture (simple version)

There exists exactly one pair πG × πH ∈ A so that

HomH(πG, πH) = 1

Furthermore an ε-factors determines the pair πG × πH.

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A simple first example:

G = SO(2,1) = GL(2,R)/D+, H = SO(2). Every representation

UH of H is tempered and are parametrized by the even integers.

The tempered representations of G are spherical principal series

representations and discrete series representations.

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If UG is a spherical principal series representation and UH is any

representations of H, then the set A = {UG × UH} satisfies the

G-P conjecture.

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If UG is a spherical principal series representation and UH is any

representations of H, then the set A = {UG × UH} satisfies the

G-P conjecture.

If π is a discrete series representation and Fπ the finite dimen-sional representation with the same infinitesimal character as πthen for every representation UH the set

A = {π × UH , Fπ × UH}satisfies the G-P conjecture.

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Problem: Fπ is not tempered

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Problem: Fπ is not tempered

Solution: Consider tempered representations of SO(2,1)×SO(2)

and of SO(3)×SO(2) instead and consider the final dimensional

representation Fπ as a representation of SO(3). . The set Fnow contains 2 irreducible tempered representations of SO(2,1)×SO(2) and one representation of SO(3)× SO(2) and

⊕U∈F dim HomH(U,C) = 1

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Second Example

Assume that both SO(p,q) and SO(p-1,q) has discrete series

representations

In this case A = AG(τ)×AH(τH)

Gross-Wallach proved about 20 years ago that under some ad-

ditional assumption on the function there exists a pair πG, πHAso that

HomH(πG, πH) = 1

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Second Example

Assume that both SO(p,q) and SO(p-1,q) has discrete series

representations

In this case A = AG(τ)×AH(τH)

Gross-Wallach proved about 20 years ago that under some ad-

ditional assumption on the function there exists a pair πG, πHAso that

HomH(πG, πH) = 1

Open Problem: There are no other such pairs in A.

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I am interested here in the case of tempered representations

which are not discrete series representations and also to

see if there is an extension of this conjecture to cohomological

relevant representations.

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I am interested here in the case of tempered representations

which are not discrete series representations and also to

see if there is an extension of this conjecture to cohomological

relevant representations.

To state the conjecture I have to introduce Vogan ”packets” ofrepresentations of a collection of orthogonal groups . We needto discuss

• Inner forms of orthogonal groups

• Relevant inner forms (and relevant parabolic subgroups)

• L-groups and Vogan packets of representations

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Inner forms:

We consider the families of real forms of SO(p+q,C). We are

interested in families which contain an orthogonal group of real

rank 1. ,

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Inner forms:

We consider the families of real forms of SO(p+q,C). We are

interested in families which contain an orthogonal group of real

rank 1. ,

First case

p+q =2n+1, Consider G0 = SO(n,n+1), i.e that the form defin-

ing G0 has signature n,n+1

Consider the family SO(n-2r, n+1+2r) for an integer r with |r| ≤n Note this family contains the pure inner form G0, a compact

form is always in this family and either SO(2n,1) or S0(1,2n) is

in this family.

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Second case:

p+q =2n ∼= 0 mod 4. We may take G0 = SO(n,n), i.e the

form defining GO has signature n,n. consider the family S0(n-2r,

n+2r) |r| ≤ n. Note the family contains G0, the compact form

is always in this family and that SO(2n-1,1) or S0(1,2n-1) is not

in this family,

Since we are interested in families which contain rank one or-

thogonal groups we can ignore this family.

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Second case:

p+q =2n ∼= 0 mod 4. We may take G0 = SO(n,n), i.e the

form defining GO has signature n,n. consider the family S0(n-2r,

n+2r) |r| ≤ n. Note the family contains G0, the compact form

is always in this family and that SO(2n-1,1) or S0(1,2n-1) is not

in this family,

Since we are interested in families which contain rank one or-

thogonal groups we can ignore this family.

Instead we consider G0 =SO(n-1,n+1) and the family S0(n-2-1r,

n+2r+1) |r| ≤ n which contains SO(2n-1,1) or S0(1,2n-1).

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Third case:

p+q =2n 6∼= 0 mod 4. We may take G0= SO(n,n). We consider

the family S0(n-2r, n+2r) |r| ≤ n. Note the compact form is

not in this family and either SO(2n-1,1) or S0(1,2n-1) is in this

family.

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Third case:




family.

Relevant families of pairs of inner forms

Consider a family G of real forms (G,H) of SO(n+1,C)×SO(n,C)

where both groups are in one of the families defined above.

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Third case:




family.

Relevant families of pairs of inner forms

Consider a family G of real forms (G,H) of SO(n+1,C)×SO(n,C)

where both groups are in one of the families defined above.

We are interested in the situation H ⊂ G. So we say that a pairin G is relevant if this is satisfied.We denote this subset of G by Gr.

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Examples of relevant families of pairs of inner forms;

First example:

Gr={ S0(n-2r,n+1+2r), Hr=S0(n-2r,n+2r) }

Remark: If n is odd then this family does contains a pair of rank

one orthogonal groups

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Examples of relevant families of pairs of inner forms;

First example:

Gr={ S0(n-2r,n+1+2r), Hr=S0(n-2r,n+2r) }

Remark: If n is odd then this family does contains a pair of rank


Second example :

Gr={ S0(n-2r,n+1+2r), Hr=S0(n-2r-1,n+2r +1) }

Remark: n is even then this family does contain a pair of rank


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Third example:

Gr={ SO(n+2r,n-2r), Hr = S0(n-1+2r, n-2r) }

Remark: If n is odd then this family contains a pair of rank one

orthogonal subgroups.

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Third example:

Gr={ SO(n+2r,n-2r), Hr = S0(n-1+2r, n-2r) }

Remark: If n is odd then this family contains a pair of rank one

orthogonal subgroups.

A choice of compatible maximal parabolic subgroups PG, PH with

PH ⊂ PG in the rank one pairs in Gr defines pairs of relevant max-

imal parabolic subgroups for all members of a family of relevant

inner forms.

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An example:

Consider the case SO(4,1) ,SO(3,1)

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An example:


Relevant pairs of inner forms:

G ={ ( SO(4,1) , SO(3,1)) and (SO(2,3), SO(1,3))}

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An example:


Relevant pairs of inner forms:

G ={ ( SO(4,1) , SO(3,1)) and (SO(2,3), SO(1,3))}

Relevant parabolic subgroups have Levi subgroups

(SO(3)R∗, S0(2,1)R∗) and (SO(1,2)R∗,SO(2)R∗).

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The Langlands dual group of SO(4,1) is Sp(4,C)

The Langlands dual group of SO(3,1) is SO(4,C)

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The Weil group WR is a non split extension of the Galois group

{1, τ} by C∗.

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{1, τ} by C∗.

Consider 4 dimensional symplectic and orthogonal representa-

tions of WR defining a tempered representations of SO(4,1)×SO(3,1)

induced from relevant parabolic subgroups. These representa-

tions of WR define also tempered representations of SO(2,3)×SO(1,3).

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{1, τ} by C∗.

Consider 4 dimensional symplectic and orthogonal representa-

tions of WR defining a tempered representations of SO(4,1)×SO(3,1)

induced from relevant parabolic subgroups. These representa-

tions of WR define also tempered representations of SO(2,3)×SO(1,3).

We obtain a set A = with 2 representations of groups G× H

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Gross Prasad conjectures in this case:

If L(1/2, τsp× τorth) 6= 0 exactly one of these representations has

an H-invariant linear functional

Theorem 2. Consider a family G which contains the pair G=SO(4,1),

H=SO(3,1) and a family A of representations of the groups in

G so that the tempered spherical principal series representations

of G× H are in A. This family satisfies the G-P conjecture

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Theorem 3. Consider a family G which contains the pair G

=SO(5,1) , H=SO(4,1) and and a family A of representations

of the groups in G so that the tempered spherical principal series

representation on G × H are in A. This family A satisfies the

G-P conjectures.

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Theorem 3. Consider a family G which contains the pair G

=SO(5,1) , H=SO(4,1) and and a family A of representations

of the groups in G so that the tempered spherical principal series

representation on G × H are in A. This family A satisfies the

G-P conjectures.

A consequence of the work with T. Kobayashi

Theorem 4. Consider a family G which contains the pairs G

=SO(n+1,1) , H=SO(n,1) and a family A of the groups in Gso that the tempered spherical principal series representations of

G×H are in A. This family contains at least one representation

with an H invariant linear functional.

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Is is possible to generalize this idea to representations which are

not tempered?

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not tempered?

Possible interesting families :

Find families of representations of cohomologically induced rep-

resentations Aq(λ) (generalization of Adams/Johnson packets)

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not tempered?

Possible interesting families :

Find families of representations of cohomologically induced rep-

resentations Aq(λ) (generalization of Adams/Johnson packets)

Families of representations which contain representations of arith-

metic interest, i.e. reps in Arthur packets

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Some experimental results:

Consider the complementary series representation I(3) of SO(5,1)

which is isomorphic to a subrepresentation of L2(SO(5,1)/SO(4,1)).

The complementary series representation I(2) of SO(4,1) is iso-

morphic to a subrepresentation of L2(SO(4,1)/SO(3,1)). For

these orthogonal groups the Ramanujan conjecture doesn’t hold

and so these representation are infinity components of automor-

phic representations.

The infinitesimal character of these representations is singular

and there are only 2 irreducible unitary representations in the

family A which contains I(3)×I(2).

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Some experimental results:

Consider the complementary series representation I(3) of SO(5,1)

which is isomorphic to a subrepresentation of L2(SO(5,1)/SO(4,1)).

The complementary series representation I(2) of SO(4,1) is iso-

morphic to a subrepresentation of L2(SO(4,1)/SO(3,1)). For

these orthogonal groups the Ramanujan conjecture doesn’t hold

and so these representation are infinity components of automor-

phic representations.

The infinitesimal character of these representations is singular

and there are only 2 irreducible unitary representations in the

family A which contains I(3)×I(2).

Joint work with Kobayashi implies In this packet G-P is true.

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We consider the unitary infinite dimensional subquotients TG and

TH of the spherical principal series representations of SO(5,1)

respectively SO(4,1) with infinitesimal character ρ. These rep-

resentations are automorphic representations with nontrivial co-

homology, are holomorphically induced from a θ stable parabolics

with L = T1SO(3,1) respectively L = T1SO(2,1).

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We consider the unitary infinite dimensional subquotients TG and

TH of the spherical principal series representations of SO(5,1)

respectively SO(4,1) with infinitesimal character ρ. These rep-

resentations are automorphic representations with nontrivial co-

homology, are holomorphically induced from a θ stable parabolics

with L = T1SO(3,1) respectively L = T1SO(2,1).

Joint with Kobayashi

dim HomSO(4,1)(TG, TH) = 1

TG and TH are both members of an enlarged Arthur packet Acohoof representations in which G-P holds.

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