Invariant linear forms on tensor products of ... - Cornell Math
Invariant linear forms on tensor products of representations
Simons Symposium April. 2016
1
Outline of the talk
2
Outline of the talk
• Review of the restriction problem of infinite dimensional representations.
2
Outline of the talk
• Review of the restriction problem of infinite dimensional representations.
• Example 1. Representations of rank one orthogonal groups. (joint work with T. Kobayashi)
2
Outline of the talk
• Review of the restriction problem of infinite dimensional representations.
• Example 1. Representations of rank one orthogonal groups. (joint work with T. Kobayashi)
• Example 2: Invariant trilinear forms on principal series representations of PGL(2, R) (joint work R. Gomez)
2
Outline of the talk
• Review of the restriction problem of infinite dimensional representations.
• Example 1. Representations of rank one orthogonal groups. (joint work with T. Kobayashi)
• Example 2: Invariant trilinear forms on principal series representations of PGL(2, R) (joint work R. Gomez) • Open problems: 2
Short review of restrictions of representations Π to a subgroup H ⊂ G: What do mean by multiplicity m(Π, π)?
3
Short review of restrictions of representations Π to a subgroup H ⊂ G: What do mean by multiplicity m(Π, π)?
Assume that V is a topological vector space, G a reductive noncompact Lie group and H a noncompact subgroup . A representation Π of G is a continuous homomorphism of G into End(V ). I do not assume that the representation is unitary but I assume that it is admissible.
3
Fact 1 There are many different realization of the same irreducible representation (i.e the same representation appears in different disguises) since Two representations are equivalent if their underlying Harish Chandra modules are equivalent
4
Fact 1 There are many different realization of the same irreducible representation (i.e the same representation appears in different disguises) since Two representations are equivalent if their underlying Harish Chandra modules are equivalent Example: P a parabolic subgroup of G, Then G/P is compact. G acts on G/P and hence on functions by (Π(g0)F )(gP ) = F (go−1gP ) so we have equivalent representations on VS = S(G/P ) Schwartz space V ∞ = C ∞(G/P ) Vc = C(G/P ) . 4
Which realization of representations are we considering?
5
Which realization of representations are we considering? Consider Π : G → End(HΠ) continuous irreducible representation on a Banach space HΠ
5
Which realization of representations are we considering? Consider Π : G → End(HΠ) continuous irreducible representation on a Banach space HΠ Extend the action of G to the action of the convolution algebra of S(G) Schwartz functions on G.
5
Which realization of representations are we considering? Consider Π : G → End(HΠ) continuous irreducible representation on a Banach space HΠ Extend the action of G to the action of the convolution algebra of S(G) Schwartz functions on G. Endow ∞ := Π(S(G))H HΠ Π
with a Fr´ echet topology. By a result of Casselman Wallach, this gives rise to a continuous representation Π∞ of G on a nuclear Fr´ echet space, which also naturally a module for the Lie algebra. 5
Fact 2: For most irreducible representation Π the restriction of Π to a non compact subgroup H is not a direct sum. (Kobayashi)
6
Fact 2: For most irreducible representation Π the restriction of Π to a non compact subgroup H is not a direct sum. (Kobayashi) For representations Π a representation of G π a representation of H we refer to an operator in HomH (Π|H , π) as symmetry breaking operator
6
Fact 2: For most irreducible representation Π the restriction of Π to a non compact subgroup H is not a direct sum. (Kobayashi) For representations Π a representation of G π a representation of H we refer to an operator in HomH (Π|H , π) as symmetry breaking operator Example: For a subgroup H ⊂ G we have H/P ∩ H ⊂ G/P . The restriction f → f|H/P ∩H defines a symmetry breaking operator (Π, S(G/P )) → (π, S(G/P )) 6
We define the multiplicity of π in the restriction of Π to H m(Π, π) = dim HomH (Π|H , π) This coincides with the definition of multiplicity if Π|H is a direct sum of irreducible representations.
7
We define the multiplicity of π in the restriction of Π to H m(Π, π) = dim HomH (Π|H , π) This coincides with the definition of multiplicity if Π|H is a direct sum of irreducible representations.
Note that m(Π, π) = dim HomH (Π|H ⊗ π ∨, C) i.e. it counts the continuous H-equivariant linear forms on Π|H ⊗ π ∨. 7
A long but interesting example, joint work with T.Kobayashi Symmetry breaking for representations of O(n+1,1) and O(n,1)
8
A long but interesting example, joint work with T.Kobayashi Symmetry breaking for representations of O(n+1,1) and O(n,1) G= O(n+1,1) P = M AN parabolic subgroup with M = O(n) × Z2 , A = R∗, N = Rn H =O(n,1) = Stab(en) and P 0 = M 0A0N 0 = H ∩ P We consider principal series representations I(V, ν) of G and Jδ (W, ν) of H induced from a maximal parabolic subgroup P = M AN . Here V × , W × δ are representation of M and M 0 respectively, and λ, ν are characters of A and A0, We use non normalized induction. 8
Consider the symmetry breaking operators HomO(n,1)(I(V, λ), Jδ (W, ν))
9
Consider the symmetry breaking operators HomO(n,1)(I(V, λ), Jδ (W, ν))
Theorem 1. Suppose [V : W ] 6= 0. Then there exists a family of symmetry breaking operators V,W
for δε = +1
V,W
for δε = −1,
Aλ,ν,+ : I(V, λ) → Jδ (W, ν) and Aλν− : I(V, λ) → Jδ (W, ν)
that depend holomorphically for entire (λ, ν) ∈ C2
9
In particular we can conclude
Consequence: Suppose [V : W ] 6= 0 and that I(V, λ) and Jδ (W, ν) are tempered. Then there exists a family of symmetry breaking operators I(V, λ) → Jδ (W, λ)
10
We can do better if Vi =
i ^
(Rn)
and Wj =
j ^
(Rn−1)
11
We can do better if Vi =
i ^
(Rn)
and Wj =
j ^
(Rn−1)
Theorem 2 (multiplicities for principal series representations). We have m(I(Vi, λ), Jδ (Wj , ν)) ≤ 2.
11
Assume now that both Π and π are irreducible representations
12
Assume now that both Π and π are irreducible representations Theorem by Sun and Zhu: m(Π, π) ≤ 1
12
Assume now that both Π and π are irreducible representations Theorem by Sun and Zhu: m(Π, π) ≤ 1
Open problem: For which pairs (Π, π) of irreducible representations do we have multiplicity 1.
12
Assume now that both Π and π are irreducible representations Theorem by Sun and Zhu: m(Π, π) ≤ 1
Open problem: For which pairs (Π, π) of irreducible representations do we have multiplicity 1.
I want discuss an interesting case: 12
Let Πi be the irreducible subrepresentation of I(V−i, i).
13
Let Πi be the irreducible subrepresentation of I(V−i, i). The representations Πi are irreducible unitary representations with nontrivial (g, K)-cohomology and every irreducible unitary representations with nontrivial (g, K)-cohomology is equivalent to one of the representations Πi.
13
Let Πi be the irreducible subrepresentation of I(V−i, i). The representations Πi are irreducible unitary representations with nontrivial (g, K)-cohomology and every irreducible unitary representations with nontrivial (g, K)-cohomology is equivalent to one of the representations Πi. Example: Π0 trivial representation. If n=2m or n=2m+1 then Πm is tempered.
13
Let Πi be the irreducible subrepresentation of I(V−i, i). The representations Πi are irreducible unitary representations with nontrivial (g, K)-cohomology and every irreducible unitary representations with nontrivial (g, K)-cohomology is equivalent to one of the representations Πi. Example: Π0 trivial representation. If n=2m or n=2m+1 then Πm is tempered. We denote the corresponding representations of H by πj . 13
Theorem 3. The following tables summarize our results about symmetry breaking Πi → πj The first row are representations of G, the second row are representations of H. Symmetry breaking operators are represented by black arrows .
14
Symmetry breaking for O(2m+1,1), O(2m.1) Π0 ↓ π0
.
Π1 ↓ π1
.
... ... ...
...
...
...
...
Πm−1 ↓ πm−1
.
Πm ↓ πm
Symmetry breaking for O(2m+2,1), O(2m+1.1) Π0 ↓ π0
.
Π1 ↓ π1
.
... ... ...
... ...
... ...
Πm−1 ↓ πm−1
.
Πm ↓ πm
Πm+1 .
15
Remark 1: The red downward pointing arrows define continuous homomorphisms between unitary representations. They were first obtained in joint work with T.N. Venkataramana. Symmetry breaking for O(2m+1,1), O(2m.1) Π0 ↓ π0
.
Π1 ↓ π1
.
... ... ...
...
...
...
...
Πm−1 ↓ πm−1
.
Πm ↓ πm
Symmetry breaking for O(2m+2,1), O(2m+1.1) Π0 ↓ π0
.
Π1 ↓ π1
.
... ... ...
... ...
... ...
Πm−1 ↓ πm−1
.
Πm ↓ πm
Πm+1 . 17
Remark 2: The symmetry breaking operator Ai : Πi → πi−1 is obtained using the meromorphic continuation of an integral operator V ,W
i i−1 : I(Vi, λ) → J−(Wi−1, ν) Aλ,ν,−,
18
Remark 2: The symmetry breaking operator Ai : Πi → πi−1 is obtained using the meromorphic continuation of an integral operator V ,W
i i−1 : I(Vi, λ) → J−(Wi−1, ν) Aλ,ν,−,
Ai is not trivial on the lowest K-type. As corollary we obtain the following results on invariant forms H-invariant forms 18
Corollary 4 (periods). Another way to phrase our results for G=O(2m+1,1) Corollary 5 (periods). HomO(2m,1)(Πi ⊗ πi, C) = C and HomO(2m,1)(Πi ⊗ πi−1, C) = C Furthermore HomO(2m−i,1)(Πi, C) = C [periods] Remark: Use this to obtain a different proof of early results of Millson-Kudla on cycles on hyperbolic space. 19
Main ingredients in proof of the theorem :
20
Main ingredients in proof of the theorem : V,i ,Wj • Functional equations for the integral operator:s Aλ,ν,+ and V,i ,Wj Aλ,ν,− obtained by composing them with Knapp Stein inter-
twining operators.
20
Main ingredients in proof of the theorem : V,i ,Wj • Functional equations for the integral operator:s Aλ,ν,+ and V,i ,Wj Aλ,ν,− obtained by composing them with Knapp Stein inter-
twining operators.
• Vanishing of the integral operator: V ,W
i i =0 A−i,−i,+
20
We expect that a similar statement is true in general
If S0 is a finite dimensional representation of O(2m+1,1) and T0 a finite dimensional representation of O(2m, 1) with HomO(2m,1)(S0, T0) 6= 0 There is a natural way to order the irreducible representations Si and Ti with the same infinitesimal character as U0 respectively W0 as in the theorem using nontrivial Ext
21
Conjecture HomO(n,1)(Si ⊗ Tj , C) = C iff j= i, i-1 A similar statement should be true for G=O(2m,1) and H=O(2n-1,1)
22
Conjecture HomO(n,1)(Si ⊗ Tj , C) = C iff j= i, i-1 A similar statement should be true for G=O(2m,1) and H=O(2n-1,1) Remark: This conjecture would also allow to obtain periods for unitary representations with H ∗(g, K, Π ⊗ F ) 6= 0 for a finite dimensional representation F .
22
Invariant trilinear forms on tensor products of spherical principal representations of PGL(2, R) joint work with Raul Gomez G = P GL(2, F) ⊗ P GL(2, F) ⊗ P GL(2, F) H = P GL(2, R) diagonally embedded into G
Π = I(ν1) ⊗ I(ν2) ⊗ I(ν3) is triple tensor product of spherical principal series representations I(νi) which are induced from a character νi a Borel subgroup B using normalized induction
We consider P GL(2, R) invariant forms on Π = I(ν1) ⊗ I(ν2) ⊗ I(ν3) 23
History of the problem; In his thesis Dipendra Prasad determined the invariant trilinear forms for a triple product of irreducible unitary representations of GL(2,Q) .
24
History of the problem; In his thesis Dipendra Prasad determined the invariant trilinear forms for a triple product of irreducible unitary representations of GL(2,Q) . Loke showed show that under some very general assumptions there is exactly one invariant trilinear form on the (g, K) modules of the principal series representations of Gl(2,R) and Gl(2,C)
24
History of the problem; In his thesis Dipendra Prasad determined the invariant trilinear forms for a triple product of irreducible unitary representations of GL(2,Q) . Loke showed show that under some very general assumptions there is exactly one invariant trilinear form on the (g, K) modules of the principal series representations of Gl(2,R) and Gl(2,C) Bernstein and Reznikov an integral representations of invariant forms on principal series representations.
24
History of the problem; In his thesis Dipendra Prasad determined the invariant trilinear forms for a triple product of irreducible unitary representations of GL(2,Q) . Loke showed show that under some very general assumptions there is exactly one invariant trilinear form on the (g, K) modules of the principal series representations of Gl(2,R) and Gl(2,C) Bernstein and Reznikov an integral representations of invariant forms on principal series representations. Rankin Cohen brackets for define invariant trilinear forms for holomorphic discrete series 24
Our problem: Instead of of obtaining formulas for the invariant forms geometric ideas to
use
determine the dimension of the space of invariant forms,
25
Our problem: Instead of of obtaining formulas for the invariant forms geometric ideas to
use
determine the dimension of the space of invariant forms,
The geometry naturally connected to this problem are the orbits of H on P GL(2, F)/B ⊗ P GL(2, F)/B ⊗ P GL(2, F)/B
25
The representation Π = I(ν1) ⊗ I(ν2) ⊗ I(ν3) defines an action G on line bundles on P GL(2, F)/B ⊗ P GL(2, F)/B ⊗ P GL(2, F)/B and on the Schwartz space of sections and on the dual space of distributions.
26
The representation Π = I(ν1) ⊗ I(ν2) ⊗ I(ν3) defines an action G on line bundles on P GL(2, F)/B ⊗ P GL(2, F)/B ⊗ P GL(2, F)/B and on the Schwartz space of sections and on the dual space of distributions. We want to determine the dimension of the space H invariant distributions and their support.
26
The representation Π = I(ν1) ⊗ I(ν2) ⊗ I(ν3) defines an action G on line bundles on P GL(2, F)/B ⊗ P GL(2, F)/B ⊗ P GL(2, F)/B and on the Schwartz space of sections and on the dual space of distributions. We want to determine the dimension of the space H invariant distributions and their support. This can be reduced to an equivalent problem for action of the Borel subgroup B ⊂ H on a Schwarz space on P GL(2, F)/B ⊗ P GL(2, F)/B. 26
Summary of the results:
27
Summary of the results:
The dimension of the space of invariant forms is either 1, 2, 3.
The following graphic shows the multiplicities under the assumption that ν1 = ν2 and that the ν3 defines a trivial character of B/B 0. 27
1
1 3
1 3
2
3
2
1
3
2
2
Remark: We have not yet completed the case P GL(2, C). So far it looks to be quite different from PGL(2,R) and very similar to the case of invariant trilinear forms on principal series representations of O(n,1), n larger than 2.
28
Remark: We have not yet completed the case P GL(2, C). So far it looks to be quite different from PGL(2,R) and very similar to the case of invariant trilinear forms on principal series representations of O(n,1), n larger than 2.
For P GL(2, Qp) we just discovered a problem in our argument and in the moment we conjecture that the dimension of the space of invariant forms is always 1.
28
Thank you
29
Simons Symposium April. 2016
1
Outline of the talk
2
Outline of the talk
• Review of the restriction problem of infinite dimensional representations.
2
Outline of the talk
• Review of the restriction problem of infinite dimensional representations.
• Example 1. Representations of rank one orthogonal groups. (joint work with T. Kobayashi)
2
Outline of the talk
• Review of the restriction problem of infinite dimensional representations.
• Example 1. Representations of rank one orthogonal groups. (joint work with T. Kobayashi)
• Example 2: Invariant trilinear forms on principal series representations of PGL(2, R) (joint work R. Gomez)
2
Outline of the talk
• Review of the restriction problem of infinite dimensional representations.
• Example 1. Representations of rank one orthogonal groups. (joint work with T. Kobayashi)
• Example 2: Invariant trilinear forms on principal series representations of PGL(2, R) (joint work R. Gomez) • Open problems: 2
Short review of restrictions of representations Π to a subgroup H ⊂ G: What do mean by multiplicity m(Π, π)?
3
Short review of restrictions of representations Π to a subgroup H ⊂ G: What do mean by multiplicity m(Π, π)?
Assume that V is a topological vector space, G a reductive noncompact Lie group and H a noncompact subgroup . A representation Π of G is a continuous homomorphism of G into End(V ). I do not assume that the representation is unitary but I assume that it is admissible.
3
Fact 1 There are many different realization of the same irreducible representation (i.e the same representation appears in different disguises) since Two representations are equivalent if their underlying Harish Chandra modules are equivalent
4
Fact 1 There are many different realization of the same irreducible representation (i.e the same representation appears in different disguises) since Two representations are equivalent if their underlying Harish Chandra modules are equivalent Example: P a parabolic subgroup of G, Then G/P is compact. G acts on G/P and hence on functions by (Π(g0)F )(gP ) = F (go−1gP ) so we have equivalent representations on VS = S(G/P ) Schwartz space V ∞ = C ∞(G/P ) Vc = C(G/P ) . 4
Which realization of representations are we considering?
5
Which realization of representations are we considering? Consider Π : G → End(HΠ) continuous irreducible representation on a Banach space HΠ
5
Which realization of representations are we considering? Consider Π : G → End(HΠ) continuous irreducible representation on a Banach space HΠ Extend the action of G to the action of the convolution algebra of S(G) Schwartz functions on G.
5
Which realization of representations are we considering? Consider Π : G → End(HΠ) continuous irreducible representation on a Banach space HΠ Extend the action of G to the action of the convolution algebra of S(G) Schwartz functions on G. Endow ∞ := Π(S(G))H HΠ Π
with a Fr´ echet topology. By a result of Casselman Wallach, this gives rise to a continuous representation Π∞ of G on a nuclear Fr´ echet space, which also naturally a module for the Lie algebra. 5
Fact 2: For most irreducible representation Π the restriction of Π to a non compact subgroup H is not a direct sum. (Kobayashi)
6
Fact 2: For most irreducible representation Π the restriction of Π to a non compact subgroup H is not a direct sum. (Kobayashi) For representations Π a representation of G π a representation of H we refer to an operator in HomH (Π|H , π) as symmetry breaking operator
6
Fact 2: For most irreducible representation Π the restriction of Π to a non compact subgroup H is not a direct sum. (Kobayashi) For representations Π a representation of G π a representation of H we refer to an operator in HomH (Π|H , π) as symmetry breaking operator Example: For a subgroup H ⊂ G we have H/P ∩ H ⊂ G/P . The restriction f → f|H/P ∩H defines a symmetry breaking operator (Π, S(G/P )) → (π, S(G/P )) 6
We define the multiplicity of π in the restriction of Π to H m(Π, π) = dim HomH (Π|H , π) This coincides with the definition of multiplicity if Π|H is a direct sum of irreducible representations.
7
We define the multiplicity of π in the restriction of Π to H m(Π, π) = dim HomH (Π|H , π) This coincides with the definition of multiplicity if Π|H is a direct sum of irreducible representations.
Note that m(Π, π) = dim HomH (Π|H ⊗ π ∨, C) i.e. it counts the continuous H-equivariant linear forms on Π|H ⊗ π ∨. 7
A long but interesting example, joint work with T.Kobayashi Symmetry breaking for representations of O(n+1,1) and O(n,1)
8
A long but interesting example, joint work with T.Kobayashi Symmetry breaking for representations of O(n+1,1) and O(n,1) G= O(n+1,1) P = M AN parabolic subgroup with M = O(n) × Z2 , A = R∗, N = Rn H =O(n,1) = Stab(en) and P 0 = M 0A0N 0 = H ∩ P We consider principal series representations I(V, ν) of G and Jδ (W, ν) of H induced from a maximal parabolic subgroup P = M AN . Here V × , W × δ are representation of M and M 0 respectively, and λ, ν are characters of A and A0, We use non normalized induction. 8
Consider the symmetry breaking operators HomO(n,1)(I(V, λ), Jδ (W, ν))
9
Consider the symmetry breaking operators HomO(n,1)(I(V, λ), Jδ (W, ν))
Theorem 1. Suppose [V : W ] 6= 0. Then there exists a family of symmetry breaking operators V,W
for δε = +1
V,W
for δε = −1,
Aλ,ν,+ : I(V, λ) → Jδ (W, ν) and Aλν− : I(V, λ) → Jδ (W, ν)
that depend holomorphically for entire (λ, ν) ∈ C2
9
In particular we can conclude
Consequence: Suppose [V : W ] 6= 0 and that I(V, λ) and Jδ (W, ν) are tempered. Then there exists a family of symmetry breaking operators I(V, λ) → Jδ (W, λ)
10
We can do better if Vi =
i ^
(Rn)
and Wj =
j ^
(Rn−1)
11
We can do better if Vi =
i ^
(Rn)
and Wj =
j ^
(Rn−1)
Theorem 2 (multiplicities for principal series representations). We have m(I(Vi, λ), Jδ (Wj , ν)) ≤ 2.
11
Assume now that both Π and π are irreducible representations
12
Assume now that both Π and π are irreducible representations Theorem by Sun and Zhu: m(Π, π) ≤ 1
12
Assume now that both Π and π are irreducible representations Theorem by Sun and Zhu: m(Π, π) ≤ 1
Open problem: For which pairs (Π, π) of irreducible representations do we have multiplicity 1.
12
Assume now that both Π and π are irreducible representations Theorem by Sun and Zhu: m(Π, π) ≤ 1
Open problem: For which pairs (Π, π) of irreducible representations do we have multiplicity 1.
I want discuss an interesting case: 12
Let Πi be the irreducible subrepresentation of I(V−i, i).
13
Let Πi be the irreducible subrepresentation of I(V−i, i). The representations Πi are irreducible unitary representations with nontrivial (g, K)-cohomology and every irreducible unitary representations with nontrivial (g, K)-cohomology is equivalent to one of the representations Πi.
13
Let Πi be the irreducible subrepresentation of I(V−i, i). The representations Πi are irreducible unitary representations with nontrivial (g, K)-cohomology and every irreducible unitary representations with nontrivial (g, K)-cohomology is equivalent to one of the representations Πi. Example: Π0 trivial representation. If n=2m or n=2m+1 then Πm is tempered.
13
Let Πi be the irreducible subrepresentation of I(V−i, i). The representations Πi are irreducible unitary representations with nontrivial (g, K)-cohomology and every irreducible unitary representations with nontrivial (g, K)-cohomology is equivalent to one of the representations Πi. Example: Π0 trivial representation. If n=2m or n=2m+1 then Πm is tempered. We denote the corresponding representations of H by πj . 13
Theorem 3. The following tables summarize our results about symmetry breaking Πi → πj The first row are representations of G, the second row are representations of H. Symmetry breaking operators are represented by black arrows .
14
Symmetry breaking for O(2m+1,1), O(2m.1) Π0 ↓ π0
.
Π1 ↓ π1
.
... ... ...
...
...
...
...
Πm−1 ↓ πm−1
.
Πm ↓ πm
Symmetry breaking for O(2m+2,1), O(2m+1.1) Π0 ↓ π0
.
Π1 ↓ π1
.
... ... ...
... ...
... ...
Πm−1 ↓ πm−1
.
Πm ↓ πm
Πm+1 .
15
Remark 1: The red downward pointing arrows define continuous homomorphisms between unitary representations. They were first obtained in joint work with T.N. Venkataramana. Symmetry breaking for O(2m+1,1), O(2m.1) Π0 ↓ π0
.
Π1 ↓ π1
.
... ... ...
...
...
...
...
Πm−1 ↓ πm−1
.
Πm ↓ πm
Symmetry breaking for O(2m+2,1), O(2m+1.1) Π0 ↓ π0
.
Π1 ↓ π1
.
... ... ...
... ...
... ...
Πm−1 ↓ πm−1
.
Πm ↓ πm
Πm+1 . 17
Remark 2: The symmetry breaking operator Ai : Πi → πi−1 is obtained using the meromorphic continuation of an integral operator V ,W
i i−1 : I(Vi, λ) → J−(Wi−1, ν) Aλ,ν,−,
18
Remark 2: The symmetry breaking operator Ai : Πi → πi−1 is obtained using the meromorphic continuation of an integral operator V ,W
i i−1 : I(Vi, λ) → J−(Wi−1, ν) Aλ,ν,−,
Ai is not trivial on the lowest K-type. As corollary we obtain the following results on invariant forms H-invariant forms 18
Corollary 4 (periods). Another way to phrase our results for G=O(2m+1,1) Corollary 5 (periods). HomO(2m,1)(Πi ⊗ πi, C) = C and HomO(2m,1)(Πi ⊗ πi−1, C) = C Furthermore HomO(2m−i,1)(Πi, C) = C [periods] Remark: Use this to obtain a different proof of early results of Millson-Kudla on cycles on hyperbolic space. 19
Main ingredients in proof of the theorem :
20
Main ingredients in proof of the theorem : V,i ,Wj • Functional equations for the integral operator:s Aλ,ν,+ and V,i ,Wj Aλ,ν,− obtained by composing them with Knapp Stein inter-
twining operators.
20
Main ingredients in proof of the theorem : V,i ,Wj • Functional equations for the integral operator:s Aλ,ν,+ and V,i ,Wj Aλ,ν,− obtained by composing them with Knapp Stein inter-
twining operators.
• Vanishing of the integral operator: V ,W
i i =0 A−i,−i,+
20
We expect that a similar statement is true in general
If S0 is a finite dimensional representation of O(2m+1,1) and T0 a finite dimensional representation of O(2m, 1) with HomO(2m,1)(S0, T0) 6= 0 There is a natural way to order the irreducible representations Si and Ti with the same infinitesimal character as U0 respectively W0 as in the theorem using nontrivial Ext
21
Conjecture HomO(n,1)(Si ⊗ Tj , C) = C iff j= i, i-1 A similar statement should be true for G=O(2m,1) and H=O(2n-1,1)
22
Conjecture HomO(n,1)(Si ⊗ Tj , C) = C iff j= i, i-1 A similar statement should be true for G=O(2m,1) and H=O(2n-1,1) Remark: This conjecture would also allow to obtain periods for unitary representations with H ∗(g, K, Π ⊗ F ) 6= 0 for a finite dimensional representation F .
22
Invariant trilinear forms on tensor products of spherical principal representations of PGL(2, R) joint work with Raul Gomez G = P GL(2, F) ⊗ P GL(2, F) ⊗ P GL(2, F) H = P GL(2, R) diagonally embedded into G
Π = I(ν1) ⊗ I(ν2) ⊗ I(ν3) is triple tensor product of spherical principal series representations I(νi) which are induced from a character νi a Borel subgroup B using normalized induction
We consider P GL(2, R) invariant forms on Π = I(ν1) ⊗ I(ν2) ⊗ I(ν3) 23
History of the problem; In his thesis Dipendra Prasad determined the invariant trilinear forms for a triple product of irreducible unitary representations of GL(2,Q) .
24
History of the problem; In his thesis Dipendra Prasad determined the invariant trilinear forms for a triple product of irreducible unitary representations of GL(2,Q) . Loke showed show that under some very general assumptions there is exactly one invariant trilinear form on the (g, K) modules of the principal series representations of Gl(2,R) and Gl(2,C)
24
History of the problem; In his thesis Dipendra Prasad determined the invariant trilinear forms for a triple product of irreducible unitary representations of GL(2,Q) . Loke showed show that under some very general assumptions there is exactly one invariant trilinear form on the (g, K) modules of the principal series representations of Gl(2,R) and Gl(2,C) Bernstein and Reznikov an integral representations of invariant forms on principal series representations.
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History of the problem; In his thesis Dipendra Prasad determined the invariant trilinear forms for a triple product of irreducible unitary representations of GL(2,Q) . Loke showed show that under some very general assumptions there is exactly one invariant trilinear form on the (g, K) modules of the principal series representations of Gl(2,R) and Gl(2,C) Bernstein and Reznikov an integral representations of invariant forms on principal series representations. Rankin Cohen brackets for define invariant trilinear forms for holomorphic discrete series 24
Our problem: Instead of of obtaining formulas for the invariant forms geometric ideas to
use
determine the dimension of the space of invariant forms,
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Our problem: Instead of of obtaining formulas for the invariant forms geometric ideas to
use
determine the dimension of the space of invariant forms,
The geometry naturally connected to this problem are the orbits of H on P GL(2, F)/B ⊗ P GL(2, F)/B ⊗ P GL(2, F)/B
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The representation Π = I(ν1) ⊗ I(ν2) ⊗ I(ν3) defines an action G on line bundles on P GL(2, F)/B ⊗ P GL(2, F)/B ⊗ P GL(2, F)/B and on the Schwartz space of sections and on the dual space of distributions.
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The representation Π = I(ν1) ⊗ I(ν2) ⊗ I(ν3) defines an action G on line bundles on P GL(2, F)/B ⊗ P GL(2, F)/B ⊗ P GL(2, F)/B and on the Schwartz space of sections and on the dual space of distributions. We want to determine the dimension of the space H invariant distributions and their support.
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The representation Π = I(ν1) ⊗ I(ν2) ⊗ I(ν3) defines an action G on line bundles on P GL(2, F)/B ⊗ P GL(2, F)/B ⊗ P GL(2, F)/B and on the Schwartz space of sections and on the dual space of distributions. We want to determine the dimension of the space H invariant distributions and their support. This can be reduced to an equivalent problem for action of the Borel subgroup B ⊂ H on a Schwarz space on P GL(2, F)/B ⊗ P GL(2, F)/B. 26
Summary of the results:
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Summary of the results:
The dimension of the space of invariant forms is either 1, 2, 3.
The following graphic shows the multiplicities under the assumption that ν1 = ν2 and that the ν3 defines a trivial character of B/B 0. 27
1
1 3
1 3
2
3
2
1
3
2
2
Remark: We have not yet completed the case P GL(2, C). So far it looks to be quite different from PGL(2,R) and very similar to the case of invariant trilinear forms on principal series representations of O(n,1), n larger than 2.
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Remark: We have not yet completed the case P GL(2, C). So far it looks to be quite different from PGL(2,R) and very similar to the case of invariant trilinear forms on principal series representations of O(n,1), n larger than 2.
For P GL(2, Qp) we just discovered a problem in our argument and in the moment we conjecture that the dimension of the space of invariant forms is always 1.
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Thank you
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