The compact picture of symmetry breaking operators for rank one - - PowerPoint PPT Presentation
The compact picture of symmetry breaking operators for rank one orthogonal and unitary groups Jan Mllers (FAU Erlangen-Nrnberg) joint w/ Bent rsted 50th Seminar Sophus Lie, Bedlewo September 26, 2016 Jan Mllers Symmetry breaking
Jan Möllers (FAU Erlangen-Nürnberg)
joint w/ Bent Ørsted 50th Seminar Sophus Lie, Bedlewo
September 26, 2016
Jan Möllers Symmetry breaking operators September 26, 2016 1 / 10
Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup.
Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10
Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup.
Definition (Symmetry breaking operator)
A symmetry breaking operator is a (continuous) intertwining operator from a representation π
HomG ′(π|G ′, τ).
Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10
Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup.
Definition (Symmetry breaking operator)
A symmetry breaking operator is a (continuous) intertwining operator from a representation π
HomG ′(π|G ′, τ). In the category of smooth admissible Fréchet representations of moderate growth:
Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10
Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup.
Definition (Symmetry breaking operator)
A symmetry breaking operator is a (continuous) intertwining operator from a representation π
HomG ′(π|G ′, τ). In the category of smooth admissible Fréchet representations of moderate growth:
1 (Kobayashi–Oshima 2013, Krötz–Schlichtkrull 2013) dim HomG ′(π|G ′, τ) < ∞ for all
irreducible π, τ if the homogeneous space (G × G ′)/diag(G ′) is real spherical.
Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10
Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup.
Definition (Symmetry breaking operator)
A symmetry breaking operator is a (continuous) intertwining operator from a representation π
HomG ′(π|G ′, τ). In the category of smooth admissible Fréchet representations of moderate growth:
1 (Kobayashi–Oshima 2013, Krötz–Schlichtkrull 2013) dim HomG ′(π|G ′, τ) < ∞ for all
irreducible π, τ if the homogeneous space (G × G ′)/diag(G ′) is real spherical.
2 (Sun–Zhu 2012) dim HomG ′(π|G ′, τ) ≤ 1 for all irreducible π, τ if (G, G ′) is one of the
multiplicity-one pairs (GL(n, R), GL(n − 1, R)), (O(p, q), O(p, q − 1)), (U(p, q), U(p, q − 1)), . . .
Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10
Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup.
Definition (Symmetry breaking operator)
A symmetry breaking operator is a (continuous) intertwining operator from a representation π
HomG ′(π|G ′, τ). In the category of smooth admissible Fréchet representations of moderate growth:
1 (Kobayashi–Oshima 2013, Krötz–Schlichtkrull 2013) dim HomG ′(π|G ′, τ) < ∞ for all
irreducible π, τ if the homogeneous space (G × G ′)/diag(G ′) is real spherical.
2 (Sun–Zhu 2012) dim HomG ′(π|G ′, τ) ≤ 1 for all irreducible π, τ if (G, G ′) is one of the
multiplicity-one pairs (GL(n, R), GL(n − 1, R)), (O(p, q), O(p, q − 1)), (U(p, q), U(p, q − 1)), . . .
Question
In case
2 , for which π, τ is the multiplicity = 1?
If the multiplicity is = 1, construct explicitly a symmetry breaking operator π|G ′ → τ.
Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10
Let G be a real reductive Lie group and G ′ ⊆ G a reductive subgroup.
Definition (Symmetry breaking operator)
A symmetry breaking operator is a (continuous) intertwining operator from a representation π
HomG ′(π|G ′, τ). In the category of smooth admissible Fréchet representations of moderate growth:
1 (Kobayashi–Oshima 2013, Krötz–Schlichtkrull 2013) dim HomG ′(π|G ′, τ) < ∞ for all
irreducible π, τ if the homogeneous space (G × G ′)/diag(G ′) is real spherical.
2 (Sun–Zhu 2012) dim HomG ′(π|G ′, τ) ≤ 1 for all irreducible π, τ if (G, G ′) is one of the
multiplicity-one pairs (GL(n, R), GL(n − 1, R)), (O(p, q), O(p, q − 1)), (U(p, q), U(p, q − 1)), . . .
Question
In case
2 , for which π, τ is the multiplicity = 1?
If the multiplicity is = 1, construct explicitly a symmetry breaking operator π|G ′ → τ. Idea: Study this question algebraically in a different category.
Jan Möllers Symmetry breaking operators September 26, 2016 2 / 10
Let G be a real reductive group, g its Lie algebra and K ⊆ G a maximal compact subgroup.
Jan Möllers Symmetry breaking operators September 26, 2016 3 / 10
Let G be a real reductive group, g its Lie algebra and K ⊆ G a maximal compact subgroup.
Construction (Harish-Chandra module)
To each smooth admissible representation (π, V ) of G one can associate the Harish-Chandra module (πHC, VHC) of (g, K) on VHC = {v ∈ V : dim span π(K)v < ∞} ≃
K
[π|K : σ]
· σ (K-finite vectors in V ).
Jan Möllers Symmetry breaking operators September 26, 2016 3 / 10
Let G be a real reductive group, g its Lie algebra and K ⊆ G a maximal compact subgroup.
Construction (Harish-Chandra module)
To each smooth admissible representation (π, V ) of G one can associate the Harish-Chandra module (πHC, VHC) of (g, K) on VHC = {v ∈ V : dim span π(K)v < ∞} ≃
K
[π|K : σ]
· σ (K-finite vectors in V ).
Theorem (Casselman–Wallach)
The functor π → πHC is an equivalence of categories:
Jan Möllers Symmetry breaking operators September 26, 2016 3 / 10
Let G be a real reductive group, g its Lie algebra and K ⊆ G a maximal compact subgroup.
Construction (Harish-Chandra module)
To each smooth admissible representation (π, V ) of G one can associate the Harish-Chandra module (πHC, VHC) of (g, K) on VHC = {v ∈ V : dim span π(K)v < ∞} ≃
K
[π|K : σ]
· σ (K-finite vectors in V ).
Theorem (Casselman–Wallach)
The functor π → πHC is an equivalence of categories:
Problem
For a smooth admissible representation π of G the restriction π|G ′ is in general not admissible. injective map: HomG ′(π|G ′, τ) ֒ → Hom(g′,K ′)(πHC|(g′,K ′), τHC).
Jan Möllers Symmetry breaking operators September 26, 2016 3 / 10
Let G be a real reductive group, g its Lie algebra and K ⊆ G a maximal compact subgroup.
Construction (Harish-Chandra module)
To each smooth admissible representation (π, V ) of G one can associate the Harish-Chandra module (πHC, VHC) of (g, K) on VHC = {v ∈ V : dim span π(K)v < ∞} ≃
K
[π|K : σ]
· σ (K-finite vectors in V ).
Theorem (Casselman–Wallach)
The functor π → πHC is an equivalence of categories:
Problem
For a smooth admissible representation π of G the restriction π|G ′ is in general not admissible. injective map: HomG ′(π|G ′, τ) ֒ → Hom(g′,K ′)(πHC|(g′,K ′), τHC). Remark: Surjectivity of this map is equivalent to the automatic continuity of invariant distribution vectors for the homogeneous space (G × G ′)/diag(G ′).
Jan Möllers Symmetry breaking operators September 26, 2016 3 / 10
Question
Construct and classify algebraic symmetry breaking operators in Hom(g′,K ′)(πHC|(g′,K ′), τHC) for irreducible Harish-Chandra modules π and τ of G and G ′ and multiplicity-one pairs (G, G ′).
Jan Möllers Symmetry breaking operators September 26, 2016 4 / 10
Question
Construct and classify algebraic symmetry breaking operators in Hom(g′,K ′)(πHC|(g′,K ′), τHC) for irreducible Harish-Chandra modules π and τ of G and G ′ and multiplicity-one pairs (G, G ′).
Casselman Embedding Theorem
Every irreducible Harish-Chandra module occurs as a subrepresentation inside a (degenerate) principal series representation.
Jan Möllers Symmetry breaking operators September 26, 2016 4 / 10
Question
Construct and classify algebraic symmetry breaking operators in Hom(g′,K ′)(πHC|(g′,K ′), τHC) for irreducible Harish-Chandra modules π and τ of G and G ′ and multiplicity-one pairs (G, G ′).
Casselman Embedding Theorem
Every irreducible Harish-Chandra module occurs as a subrepresentation inside a (degenerate) principal series representation. Study symmetry breaking operators between principal series representations.
Jan Möllers Symmetry breaking operators September 26, 2016 4 / 10
Question
Construct and classify algebraic symmetry breaking operators in Hom(g′,K ′)(πHC|(g′,K ′), τHC) for irreducible Harish-Chandra modules π and τ of G and G ′ and multiplicity-one pairs (G, G ′).
Casselman Embedding Theorem
Every irreducible Harish-Chandra module occurs as a subrepresentation inside a (degenerate) principal series representation. Study symmetry breaking operators between principal series representations. Let P = MAN ⊆ G and P′ = M′A′N′ ⊆ G ′ be maximal parabolic subgroups.
Jan Möllers Symmetry breaking operators September 26, 2016 4 / 10
Question
Construct and classify algebraic symmetry breaking operators in Hom(g′,K ′)(πHC|(g′,K ′), τHC) for irreducible Harish-Chandra modules π and τ of G and G ′ and multiplicity-one pairs (G, G ′).
Casselman Embedding Theorem
Every irreducible Harish-Chandra module occurs as a subrepresentation inside a (degenerate) principal series representation. Study symmetry breaking operators between principal series representations. Let P = MAN ⊆ G and P′ = M′A′N′ ⊆ G ′ be maximal parabolic subgroups. Fix ν ∈ a∗ such that Σ(n, a) = {ν, 2ν, . . . , qν}, and similarly ν′ ∈ (a′)∗.
Jan Möllers Symmetry breaking operators September 26, 2016 4 / 10
Question
Construct and classify algebraic symmetry breaking operators in Hom(g′,K ′)(πHC|(g′,K ′), τHC) for irreducible Harish-Chandra modules π and τ of G and G ′ and multiplicity-one pairs (G, G ′).
Casselman Embedding Theorem
Every irreducible Harish-Chandra module occurs as a subrepresentation inside a (degenerate) principal series representation. Study symmetry breaking operators between principal series representations. Let P = MAN ⊆ G and P′ = M′A′N′ ⊆ G ′ be maximal parabolic subgroups. Fix ν ∈ a∗ such that Σ(n, a) = {ν, 2ν, . . . , qν}, and similarly ν′ ∈ (a′)∗. For ξ ∈ M, ξ′ ∈ M′ and r, r′ ∈ C we consider the degenerate principal series representations πξ,r = IndG
P (ξ ⊗ erν ⊗ 1),
τξ′,r′ = IndG ′
P′(ξ′ ⊗ er′ν′ ⊗ 1).
Jan Möllers Symmetry breaking operators September 26, 2016 4 / 10
Question
Construct and classify algebraic symmetry breaking operators in Hom(g′,K ′)(πHC|(g′,K ′), τHC) for irreducible Harish-Chandra modules π and τ of G and G ′ and multiplicity-one pairs (G, G ′).
Casselman Embedding Theorem
Every irreducible Harish-Chandra module occurs as a subrepresentation inside a (degenerate) principal series representation. Study symmetry breaking operators between principal series representations. Let P = MAN ⊆ G and P′ = M′A′N′ ⊆ G ′ be maximal parabolic subgroups. Fix ν ∈ a∗ such that Σ(n, a) = {ν, 2ν, . . . , qν}, and similarly ν′ ∈ (a′)∗. For ξ ∈ M, ξ′ ∈ M′ and r, r′ ∈ C we consider the degenerate principal series representations πξ,r = IndG
P (ξ ⊗ erν ⊗ 1),
τξ′,r′ = IndG ′
P′(ξ′ ⊗ er′ν′ ⊗ 1).
Determine Hom(g′,K ′)((πξ,r)HC, (τξ′,r′)HC)
Jan Möllers Symmetry breaking operators September 26, 2016 4 / 10
Decompose πξ,r and τξ′,r′ into K-isotypic components: (πξ,r)HC =
K
E(α), (τξ′,r′)HC =
K ′
E′(α′). Note: The decomposition is independent of r, r′ ∈ C.
Jan Möllers Symmetry breaking operators September 26, 2016 5 / 10
Decompose πξ,r and τξ′,r′ into K-isotypic components: (πξ,r)HC =
K
E(α), (τξ′,r′)HC =
K ′
E′(α′). Note: The decomposition is independent of r, r′ ∈ C.
Multiplicity-freeness assumptions
1 dim HomK(α, E(α)) ≤ 1 for all α ∈
K (i.e. πξ,r is K-multiplicity-free),
Jan Möllers Symmetry breaking operators September 26, 2016 5 / 10
Decompose πξ,r and τξ′,r′ into K-isotypic components: (πξ,r)HC =
K
E(α), (τξ′,r′)HC =
K ′
E′(α′). Note: The decomposition is independent of r, r′ ∈ C.
Multiplicity-freeness assumptions
1 dim HomK(α, E(α)) ≤ 1 for all α ∈
K (i.e. πξ,r is K-multiplicity-free),
2 dim HomK ′(α′, E′(α′)) ≤ 1 for all α′ ∈
K ′, (i.e. τξ′,r′ is K ′-multiplicity-free),
Jan Möllers Symmetry breaking operators September 26, 2016 5 / 10
Decompose πξ,r and τξ′,r′ into K-isotypic components: (πξ,r)HC =
K
E(α), (τξ′,r′)HC =
K ′
E′(α′). Note: The decomposition is independent of r, r′ ∈ C.
Multiplicity-freeness assumptions
1 dim HomK(α, E(α)) ≤ 1 for all α ∈
K (i.e. πξ,r is K-multiplicity-free),
2 dim HomK ′(α′, E′(α′)) ≤ 1 for all α′ ∈
K ′, (i.e. τξ′,r′ is K ′-multiplicity-free),
3 dim HomK ′(E(α), E′(α′)) ≤ 1 for all α ∈
K and α′ ∈ K ′ (i.e. E(α) is K ′-multiplicity-free).
Jan Möllers Symmetry breaking operators September 26, 2016 5 / 10
Decompose πξ,r and τξ′,r′ into K-isotypic components: (πξ,r)HC =
K
E(α), (τξ′,r′)HC =
K ′
E′(α′). Note: The decomposition is independent of r, r′ ∈ C.
Multiplicity-freeness assumptions
1 dim HomK(α, E(α)) ≤ 1 for all α ∈
K (i.e. πξ,r is K-multiplicity-free),
2 dim HomK ′(α′, E′(α′)) ≤ 1 for all α′ ∈
K ′, (i.e. τξ′,r′ is K ′-multiplicity-free),
3 dim HomK ′(E(α), E′(α′)) ≤ 1 for all α ∈
K and α′ ∈ K ′ (i.e. E(α) is K ′-multiplicity-free). By
3 we can further decompose each K-type E(α) into K ′-representations:
E(α) =
K ′
E(α, α′) with E(α, α′) ≃
if α′ occurs in α, else.
Jan Möllers Symmetry breaking operators September 26, 2016 5 / 10
Decompose πξ,r and τξ′,r′ into K-isotypic components: (πξ,r)HC =
K
E(α), (τξ′,r′)HC =
K ′
E′(α′). Note: The decomposition is independent of r, r′ ∈ C.
Multiplicity-freeness assumptions
1 dim HomK(α, E(α)) ≤ 1 for all α ∈
K (i.e. πξ,r is K-multiplicity-free),
2 dim HomK ′(α′, E′(α′)) ≤ 1 for all α′ ∈
K ′, (i.e. τξ′,r′ is K ′-multiplicity-free),
3 dim HomK ′(E(α), E′(α′)) ≤ 1 for all α ∈
K and α′ ∈ K ′ (i.e. E(α) is K ′-multiplicity-free). By
3 we can further decompose each K-type E(α) into K ′-representations:
E(α) =
K ′
E(α, α′) with E(α, α′) ≃
if α′ occurs in α, else. For every pair (α, α′) with E(α, α′) = 0 and E′(α′) = 0 we fix a K ′-isomorphism Rα,α′ : E(α, α′) ∼ → E′(α′).
Jan Möllers Symmetry breaking operators September 26, 2016 5 / 10
By the multiplicity-free assumptions and Schur’s Lemma, every linear map T : (πξ,r)HC → (τξ′,r′)HC is K ′-intertwining if and only if there exist scalars tα,α′ ∈ C such that T|E(α,α′) =
for E′(α′) = 0, else.
Jan Möllers Symmetry breaking operators September 26, 2016 6 / 10
By the multiplicity-free assumptions and Schur’s Lemma, every linear map T : (πξ,r)HC → (τξ′,r′)HC is K ′-intertwining if and only if there exist scalars tα,α′ ∈ C such that T|E(α,α′) =
for E′(α′) = 0, else.
Theorem (M.–Ørsted 2014)
A linear map T : (πξ,r)HC → (τξ′,r′)HC given by scalars tα,α′ ∈ C is (g′, K ′)-intertwining if and
K and α′, β′ ∈ K ′ the following identity holds: (σ′
β′ − σ′ α′ + 2r′)tα,α′ =
K (α;α′)↔(β;β′)
λβ,β′
α,α′(σβ − σα + 2r)tβ,β′.
(⋆)
Jan Möllers Symmetry breaking operators September 26, 2016 6 / 10
By the multiplicity-free assumptions and Schur’s Lemma, every linear map T : (πξ,r)HC → (τξ′,r′)HC is K ′-intertwining if and only if there exist scalars tα,α′ ∈ C such that T|E(α,α′) =
for E′(α′) = 0, else.
Theorem (M.–Ørsted 2014)
A linear map T : (πξ,r)HC → (τξ′,r′)HC given by scalars tα,α′ ∈ C is (g′, K ′)-intertwining if and
K and α′, β′ ∈ K ′ the following identity holds: (σ′
β′ − σ′ α′ + 2r′)tα,α′ =
K (α;α′)↔(β;β′)
λβ,β′
α,α′(σβ − σα + 2r)tβ,β′.
(⋆) Proof: The LHS of (⋆) is obtained by first applying T to E(α, α′) and then acting on E′(α′) by g′, the RHS is obtained by first acting on E(α, α′) by g′ and then applying T. We write (α, α′) ↔ (β, β′) if E(β, β′) can be reached from E(α, α′) by the action of g′. The numbers σα, σ′
α′ ∈ R only depend on ξ, ξ′ and α, α′, not on r, r′ ∈ C, and can be
computed easily from the highest weights of α, α′. The constants λβ,β′
α,α′ depend only on the choice of the isomorphisms Rα,α′ and Rβ,β′.
Jan Möllers Symmetry breaking operators September 26, 2016 6 / 10
Let (G, G ′) = (O(n + 1, 1), O(n, 1)) and consider spherical principal series representations πr = π1,r and τr′ = τ 1,r′.
Jan Möllers Symmetry breaking operators September 26, 2016 7 / 10
Let (G, G ′) = (O(n + 1, 1), O(n, 1)) and consider spherical principal series representations πr = π1,r and τr′ = τ 1,r′. The maximal compact subgroups are K = O(n + 1) × O(1) and K ′ = O(n) × O(1), and (πr)HC|K ≃
∞
Hα(Rn+1) ⊗ sgnα
, (τr′)HC|K ′ ≃
∞
Hα′(Rn) ⊗ sgnα′
, where Hk(Rm) denotes the space of spherical harmonics on Rm of degree k.
Jan Möllers Symmetry breaking operators September 26, 2016 7 / 10
Let (G, G ′) = (O(n + 1, 1), O(n, 1)) and consider spherical principal series representations πr = π1,r and τr′ = τ 1,r′. The maximal compact subgroups are K = O(n + 1) × O(1) and K ′ = O(n) × O(1), and (πr)HC|K ≃
∞
Hα(Rn+1) ⊗ sgnα
, (τr′)HC|K ′ ≃
∞
Hα′(Rn) ⊗ sgnα′
, where Hk(Rm) denotes the space of spherical harmonics on Rm of degree k. Classical branching rules: Hα(Rn+1) ⊗ sgnα
α
Hα′(Rn) ⊗ sgnα
. ⇒ E(α, α′) ≃
for α − α′ even, for α − α′ odd.
Jan Möllers Symmetry breaking operators September 26, 2016 7 / 10
Let (G, G ′) = (O(n + 1, 1), O(n, 1)) and consider spherical principal series representations πr = π1,r and τr′ = τ 1,r′. The maximal compact subgroups are K = O(n + 1) × O(1) and K ′ = O(n) × O(1), and (πr)HC|K ≃
∞
Hα(Rn+1) ⊗ sgnα
, (τr′)HC|K ′ ≃
∞
Hα′(Rn) ⊗ sgnα′
, where Hk(Rm) denotes the space of spherical harmonics on Rm of degree k. Classical branching rules: Hα(Rn+1) ⊗ sgnα
α
Hα′(Rn) ⊗ sgnα
. ⇒ E(α, α′) ≃
for α − α′ even, for α − α′ odd. For α − α′ even we may choose Rα,α′ = restxn+1=0 : Hα(Rn+1) ∼ → Hα′(Rn).
Jan Möllers Symmetry breaking operators September 26, 2016 7 / 10
Scalar identities for tα,α′
(2α + n − 2)(2r′ + 2α′ + n − 2)tα,α′ = (α + α′ + n − 2)(2r + 2α + n − 1)tα+1,α′+1 + (α − α′)(2r − 2α − n + 3)tα−1,α′+1 (1) (2α + n − 2)(2r′ − 2α′ − n + 4)tα,α′ = (α − α′ + 1)(2r + 2α + n − 1)tα+1,α′−1 + (α + α′ + n − 3)(2r − 2α − n + 3)tα−1,α′−1. (2)
Jan Möllers Symmetry breaking operators September 26, 2016 8 / 10
Scalar identities for tα,α′
(2α + n − 2)(2r′ + 2α′ + n − 2)tα,α′ = (α + α′ + n − 2)(2r + 2α + n − 1)tα+1,α′+1 + (α − α′)(2r − 2α − n + 3)tα−1,α′+1 (1) (2α + n − 2)(2r′ − 2α′ − n + 4)tα,α′ = (α − α′ + 1)(2r + 2α + n − 1)tα+1,α′−1 + (α + α′ + n − 3)(2r − 2α − n + 3)tα−1,α′−1. (2) K-type picture: Scalar identities: × × × × × × α α′ (α, α′) (α − 1, α′ + 1) (α + 1, α′ + 1) (α, α′) (α − 1, α′ − 1) (α + 1, α′ − 1)
Jan Möllers Symmetry breaking operators September 26, 2016 8 / 10
Scalar identities for tα,α′
(2α + n − 2)(2r′ + 2α′ + n − 2)tα,α′ = (α + α′ + n − 2)(2r + 2α + n − 1)tα+1,α′+1 + (α − α′)(2r − 2α − n + 3)tα−1,α′+1 (1) (2α + n − 2)(2r′ − 2α′ − n + 4)tα,α′ = (α − α′ + 1)(2r + 2α + n − 1)tα+1,α′−1 + (α + α′ + n − 3)(2r − 2α − n + 3)tα−1,α′−1. (2) K-type picture: Scalar identities: × × × × × × α α′ (α, α′) (α − 1, α′ + 1) (α + 1, α′ + 1) (α, α′) (α − 1, α′ − 1) (α + 1, α′ − 1)
Theorem (M.–Ørsted 2014)
dim Hom(g′,K ′)((πr)HC, (τr′)HC) =
for (r, r′) = (− n
2 − i, − n−1 2
− j), i, j ∈ N, i − j ∈ 2N, 1 else.
Jan Möllers Symmetry breaking operators September 26, 2016 8 / 10
Final remarks
Multiplicity 2 in the previous theorem does not contradict the fact that (O(n + 1, 1), O(n, 1)) is a multiplicity-one pair: For those (r, r′) with multiplicity 2, the representations πr and τr′ are both reducible with two composition factors, and there exist independent intertwining operators between them.
Jan Möllers Symmetry breaking operators September 26, 2016 9 / 10
Final remarks
Multiplicity 2 in the previous theorem does not contradict the fact that (O(n + 1, 1), O(n, 1)) is a multiplicity-one pair: For those (r, r′) with multiplicity 2, the representations πr and τr′ are both reducible with two composition factors, and there exist independent intertwining operators between them. Comparing with the analogous results by Kobayashi–Speh in the smooth category we find that the map HomG ′(πr|G ′, τr′) ֒ → Hom(g′,K ′)((πr)HC, (τr′)HC) is a bijection.
Jan Möllers Symmetry breaking operators September 26, 2016 9 / 10
Final remarks
Multiplicity 2 in the previous theorem does not contradict the fact that (O(n + 1, 1), O(n, 1)) is a multiplicity-one pair: For those (r, r′) with multiplicity 2, the representations πr and τr′ are both reducible with two composition factors, and there exist independent intertwining operators between them. Comparing with the analogous results by Kobayashi–Speh in the smooth category we find that the map HomG ′(πr|G ′, τr′) ֒ → Hom(g′,K ′)((πr)HC, (τr′)HC) is a bijection. The method also works nicely for
(G, H) = (U(n + 1, 1), U(n, 1)) and spherical principal series, (G, H) = (Pin(n + 1, 1), Pin(n, 1)) and spinor-valued principal series, . . .
Jan Möllers Symmetry breaking operators September 26, 2016 9 / 10
Jan Möllers Symmetry breaking operators September 26, 2016 10 / 10