Branching rules for generalized Verma modules and their ... · PDF fileBranching rules for generalized Verma modules ... SIGMA, 3, 2007, Vladim´ır ... formulation and understanding

MotivationBranching rules for (generalized) Verma modules

The conformal caseConstruction of families

The CR caseCurved case

Branching rules for generalized Verma modules

and their applications

Vladimır Soucek

Charles University, PrahaMathematical Institute

April 2009, Gottingen, Conference in honor of Bent Ørsted

Vladimır Soucek Branching rules




Contents

1 Motivation

2 Branching rules for (generalized) Verma modules

3 The conformal case

4 Construction of families

5 The CR case

6 Curved case





Contents

1 Motivation




5 The CR case

6 Curved case





Flat case, the classical correspondence

interplay between differential geometry and representationtheory






interplay between differential geometry and representationtheory1-1 map between the invariant differential operators andhomomorphisms of the corresponding (generalized) Vermamodules.






interplay between differential geometry and representationtheory1-1 map between the invariant differential operators andhomomorphisms of the corresponding (generalized) Vermamodules.G semisimple Lie group, P parabolic subgroup, X = G/Pgeneralized flag manifoldV, W two finite-dimensional irreducible P-modulesthere is a bijective correspondence between intertwiningdifferential operators D : C∞(G , V)P 7→ C∞(G , W)P andG -homomorphisms between (generalized) Verma modules

D : M(W∗) 7→ M(V∗).





Invariant operators between different dimensions

conformal geometry, (continuous) families Dλ of invariantdifferential operators of a given order k between a manifoldand its submanifold (of codimension one)





Invariant operators between different dimensions

conformal geometry, (continuous) families Dλ of invariantdifferential operators of a given order k between a manifoldand its submanifold (of codimension one)

A. Juhl: Families of conformally covariant differentialoperators, Q-curvature and holography, Birkhauser, Progressin Math., to appear in May 2009- motivation from Selberg zeta function and automorphicdistributions- a curved version of those operators is related to theQ-curvature in conformal geometry





T. Branson, R. Gover: Conformally invariant non-localoperators, Pac. Jour. Math, 2001,- families Dλ are used for a boundary value problem in aconstruction of conformally invariant (non-local) operators





T. Branson, R. Gover: Conformally invariant non-localoperators, Pac. Jour. Math, 2001,- families Dλ are used for a boundary value problem in aconstruction of conformally invariant (non-local) operators

R. Gover: Conformal Dirichlet-Neumann maps andPoincare-Einstein manifolds, SIGMA, 3, 2007,





the Juhl family - flat case

Invariant differential operators between irreducible bundles butin different dimensions(!)







G (n) = SO0(n, 1) ⊂ G (n+1) = SO0(n + 1, 1)X = Sn−1 = G (n)/P(n) ⊂ M = Sn = G (n+1)/P(n+1)

let k be a fixed positive integer;let Rw be a character for P(n+1), resp. for P(n),We want to construct a continuous family

Dw : C∞(G (n+1), V−w+k)P(n+1)

7→ C∞(G (n), V−w )P(n)

of differential operators commuting with the action of G (n).







G (n) = SO0(n, 1) ⊂ G (n+1) = SO0(n + 1, 1)X = Sn−1 = G (n)/P(n) ⊂ M = Sn = G (n+1)/P(n+1)

let k be a fixed positive integer;let Rw be a character for P(n+1), resp. for P(n),We want to construct a continuous family

Dw : C∞(G (n+1), V−w+k)P(n+1)

7→ C∞(G (n), V−w )P(n)

of differential operators commuting with the action of G (n).

Dual formulation: a family Φw of g(n)-homomorphismsbetween (generalized) Verma modules

Φw : M(n)(Vw ) 7→ M(n+1)(Vw−k).





Continuous families of homomorphisms between different

dimensions

Suppose that there are branching rules for a continuous familyof (generalized) Verma modules depending on a parametr w :

M(n+1)w (Vµ) ≃ ⊕αw∈AM(n)(Vαw )

as Gn-modules.Then in generic situation, all Gn-homomorphisms of another

(generalized) Verma module M(n)(W) to M(n+1)w (Vµ) are

given by embeddings onto a component in the decomposition.





Continuous families of homomorphisms between different

dimensions

Suppose that there are branching rules for a continuous familyof (generalized) Verma modules depending on a parametr w :

M(n+1)w (Vµ) ≃ ⊕αw∈AM(n)(Vαw )

as Gn-modules.Then in generic situation, all Gn-homomorphisms of another

(generalized) Verma module M(n)(W) to M(n+1)w (Vµ) are

given by embeddings onto a component in the decomposition.

Hence to understand constructions of continuous families, wehave to understand branching rules for Verma modules.





Formulation of the problem

Goal - formulation and understanding of branching rules for(generalized) Verma modules and, in particular, a description(explicit, if possible) of homomorphisms realizing thebranching







An (explicit) description of corresponding invariant families ofdifferential operators








A study of curved analogues of these operators on curvedversions of parabolic geometries








A study of curved analogues of these operators on curvedversions of parabolic geometries

On going research project with B. Ørsted (Aarhus) and P.Somberg (Prague)





Contents

1 Motivation




5 The CR case

6 Curved case





Generalized Verma modules

Let g be a complex simple Lie algebra, p its parabolic subgroup,g = p ⊕ n−, let g0 ⊂ p be the Levi factor of p and h the Cartansubalgebra. Consider Λ ∈ h∗ dominant and integral for p and letVΛ be the corresponding finite dimensional irreduciblerepresentation of p with highest weight Λ. Then the (generalized)Verma module MΛ is defined as

MΛ := U(g) ⊗U(p) VΛ.

It is a universal highest weight module with weight Λ. As a vectorspace, it is isomorphic to U(n−).Multiplicity of a weight space U(n−)λ is given by the Kostantpartition function.





the case SL3 7→ SL2, Borel case

g = sl3, b upper triangular (Borel) g = b ⊕ n−g′ = sl2 embedded to the left upper cornerroots of sl3 : ±α,±β,±γ; γ = α + β,roots in n− : −α,−β,−γ,root elements Yα, Yβ , Yγ for negative rootsroots of sl2 : are ±αa basis in the Verma module with a highest weight vector v :

{Y jαY k

γ Y ℓβ | ≡ αjβkγℓ, v ∈ Mλ; j , k , l ∈ N0}

M0 := . . . α3v α2v αv v

M1 :=. . . α3βv α2βv αβv βv

. . . α2γv αγv γvVladimır Soucek Branching rules




11

2 12 1

3 2 1Yγ

ւ3 2 1

4 3 2 1Yβ

↓4 3 2

5 4 3 1Yα

տ5 4 2

5 3 14 2

5 3 14 2Vladimır Soucek Branching rules




12 1

2 13 2 1

3 2 1

4 3 2 14 3 2

5 4 3 15 4 2

5 3 14 2

5 3 14 2





12 1

2 13 2 1

3 2 1

4 3 2 1

4 3 2

5 4 3 1

5 4 2

5 3 1

4 2

5 3 1

4 2





Theorem

Let λ = (λ1, λ2, λ3) be a highest weight of the Verma module M(3)λ

for l3. Then it decomposes (as l2-module) for a generic weight as

M(3)λ =

∑

m,n∈N0

M(2)(λ1−m,λ2−n)

Indeed, α = (1,−1, 0), β = (0, 1,−1), γ = (1, 0,−1), henceλ − nβ − mγ = (λ1 − m, λ2 − n, ∗).





A simple computation implies that there is a singular vector ineach weight in boxes.Exceptional cases - individual Verma modules are included one inanotherWhen Mλ is not irreducible for l3, we can make a quotient with themaximal podmodule - it is possible to deduce the usual branchingrules for finite dimensional modules





the case SL3 7→ SL2, parabolic case

12 1

2 12 2 1

2 2 1

2 2 2 1

2 2 2

2 2 2 1

2 2 1

2 1 0

1 0

1 0 0

1 0 0Vladimır Soucek Branching rules




Theorem

Let λ = (λ1|λ2, λ3) be a p-dominant weight ane let the generalized

Verma module Mp(3)λ for l3. Then it decomposes (as l2-module) for

a generic weight as

Mp(3)λ =

∑

m,n∈N0,λ2−n≥λ3

Mp(2)(λ1−m,λ2−n)





General result

A careful study of multiplicity-one theorems and branchingrules in general situation (in particular for unitary highestweight modules) can be found in

T. Kobayashi: Multiplicity-free theorems of the restrictions ofunitary highest weight modules with respect to reductivesymmetric pairs, Progress in Math., Birkhauser, 2006T. Kobayashi: Discrete decomposability of the restriction ofAq(λ) with respect to reductive subgroup, Invent. Math. III,131, 1997, 229-256





Contents

1 Motivation




5 The CR case

6 Curved case





The conformal case

The sphere Sn is a homogeneous space G/P, withG = G (n) = SO(n + 1, 1) and it contains the sphere Sn−1 given asa homogeneous space G/P with G = G (n−1).Let g(n) = so(n + 1, 1), g(n−1) = so(n, 1) with the standardembedding g(n−1) ⊂ g(n)

Let g(n) = gp(n) ⊕ n(n)− and the same in one dimension below.

Let {Y1, . . . ,Yn} a basis of n(n)− such that {Y1, . . . ,Yn−1} is a

basis of n(n−1)− .

Irreducible p(n)-modules are labeled by weights Λ = (λ1|λ2, . . . , λk)dominant integral for p.Let us split Λ as λ = (w , λ) with w ∈ R being the conformalweight and λ being a dominant integral weight for so(n).Similarly for Λ′ = (w ′, λ′) for dimension n − 1 instead of n.





Branching rules in conformal case

Theorem

(T. Kobayashi) Let Vw ,λ be an irreducible p(n)-module and M(n)w ,λ

the corresponding generalized Verma module.Then we have in generic situation (i.e. up to a discrete set of w ′s)the branching rules

M(n)w ,λ ≃ ⊕k∈Z≥0

⊕λ′րλ M(n−1)w−k,λ′ ,

where λ′ ր λ means that V ′λ appears in the branching rules for Vλ.

In particular

M(n)w ,0 ≃ ⊕k∈Z≥0

M(n−1)w−k,0.





Generic case

Let v ∈ M(n+1)(λ) be the highest weight vector Then (Yn)jv are

not singular vectors for j > 1Denote ∆ =

∑n−11 (Yj)

2

. . . ∆2v . . . ∆v . . . v

. . . ∆2vYnv . . . ∆vYnv . . . Ynv

. . . ∆2v(Yn)2v . . . d ∆ v(Yn)

2v . . . (Yn)2v

. . . ∆2v(Yn)3v . . . d∆v(Yn)

3v . . . (Yn)3v

. . . ∆2v(Yn)4v . . . d∆v(Yn)

4v . . . (Yn)4v





The order 4

the singular vector u for the embedding in degree 4 has a form

u =1

3(2λ+n−7)(2λ+n−5)Y 4

n v−2(2λ+n−5)∆Y 2n v +∆2v ,

where ∆ =∑n−1

1 (Yj)2 and v is the highest weight vector in

the ’big’ Verma module.





The order 4

the singular vector u for the embedding in degree 4 has a form

u =1

3(2λ+n−7)(2λ+n−5)Y 4

n v−2(2λ+n−5)∆Y 2n v +∆2v ,

where ∆ =∑n−1

1 (Yj)2 and v is the highest weight vector in

the ’big’ Verma module.

for 2λ + n − 7 = 0 or 2λ + n − 5 = 0, something specialhappens





Contents

1 Motivation




5 The CR case

6 Curved case





Splitting operators

Splitting operators are particular types of invariant differentialoperators between irreducible and non-decomposable P-modules.Let V = Vλ be an irreducible G -module with the highest weightλ = (λ1, λ2, . . . , λk). It is a non-decomposable P-module and ithas a filtration by P-modules. The smallest element in thefiltration W is an irreducible P-module. The same is true for thefactor module W

′ of V by the biggest nontrivial P-submodule.So we have an invariant projection π of V onto W and an invariantembedding ι of W

′ into V. Let V , W and W ′ be the associatedhomogeneous bundles on G/P.Typical splitting operators:Dλ : Γ(W ) → Γ(Vλ), π ◦ Dλ = α id, α 6= 0,Eλ : Γ(Vλ) → Γ(W ′), Eλ ◦ ι = α id, α 6= 0.We can twist V by a one dimensional module R[w ].





Example - standard tractor bundle.

Consider defining g-module with highest weight Λ0 = (1, 0, . . . , 0).It splits under reduction to g0 into three components with highestweights (1|0, . . . , 0) ⊕ (0|1, 0, . . . , 0) ⊕ (−1|0, . . . , 0).DenoteEA .. sections of (1, 0, . . . , 0)E [1] .. sections of (1|0, . . . , 0)Ea[1] .. sections of (0|1, 0, . . . , 0)E [−1] .. sections of (−1|0, . . . , 0)Then EA[w ] ≃ E [w + 1] ⊕ Ea[w + 1] ⊕ E [w − 1].E [w + 1] is the irreducible quotion, E [−1] is irreducible subbundle.





Standard splitting operator

DA : E [w + 1] 7→ EA[w ]

Dσ =

w(n + 2w − 2)σ(n + 2w − 2)∇aσ

−∆σ

EA : EA[w ] 7→ E [w − 1]

E

σµa

ρ

= (w + 2n − 2)(w + n)ρ + (w + 2n − 2)∇aµa − ∆σ





The second order family

See Let ΠAA′ denote the projection (induced by finite-dimensional

branching rules from g(n+1) to g(n)) twisted by identity on R[w ].Let EA′

is the splitting operator for g(n). Then the compositionD2(w) := EA′

◦ ΠAA′ ◦ DA maps E [1] to cE ′[−1] and is conformally

invariant.Explicitely

D2(w)(σ) = (n + w − 3)[(n + 2w − 2)∆′σ + (n + 2w − 3)∆σ]





General strategy

We want to construct an operator corresponding to a

particular piece of the branching rules: M(n−1)w−2k,λ′ embedded

into M(n)w ,λ





General strategy



into M(n)w ,λ

Find a splitting operator mapping sections Ew ,λ of bundleassociated Vw ,λ to a tractor bundle twisted by a density





General strategy



into M(n)w ,λ


Apply a suitably chosen projection coming from branchingrules for the pair (g(n+1), g(n)) (twisted by a densityrepresentation).





General strategy



into M(n)w ,λ


Apply a suitably chosen projection coming from branchingrules for the pair (g(n+1), g(n)) (twisted by a densityrepresentation).

All should be arranged in such a way that there is adifferential splitting back to Ew−2k,λ′





Contents

1 Motivation




5 The CR case

6 Curved case





The CR case

The sphere S2n+1 ⊂ Cn is a homogeneous space G/P, with

G = SU(n + 1, 1).g(n+1) = su(n + 1, 1), g(n) = su(n, 1) with the standard embeddingg(n) ⊂ g(n+1)

consider the corresponding parabolic subalgebras gp(n+1) and gp(n)

nilpotent parts n(n+1)− and n

(n)− have dimensions 2n + 1, resp.

2n − 1 with a suitable bases {X1, . . . ,Xn, Y1, . . . ,Yn, X}, resp.{X1, . . . ,Xn−1, Y1, . . . ,Yn−1, X},





Theorem

(B.Ørsted, P. Somberg, VS) Let M(n+1)(λ, λ′) be the generalizedVerma module induced by a character of g(n+1), similarly forM(n)(λ, λ′).Then for every N ∈ N there is a family of g(n)-homomorphisms

D : M(n)(λ − N, λ′) 7→ M(n+1)(λ, λ′), λ, λ′ ∈ C.





The singular vector w for D4 has a form

(λ+n−1)(λ+n−2)(X 2n +Y 2

n )2−2(λ+n−2)(X 2n +Y 2

n )(∆CR−2λ′X )

+∆2CR + 4λ′∆CRX + 4(λ′2 + λ + n − 1)X 2,

where ∆ =∑n−1

1 (Yj)2.






(λ+n−1)(λ+n−2)(X 2n +Y 2

n )2−2(λ+n−2)(X 2n +Y 2

n )(∆CR−2λ′X )

+∆2CR + 4λ′∆CRX + 4(λ′2 + λ + n − 1)X 2,

where ∆ =∑n−1

1 (Yj)2.

for λ + n− 1 = 0 or λ + n− 2 = 0, something special happens






(λ+n−1)(λ+n−2)(X 2n +Y 2

n )2−2(λ+n−2)(X 2n +Y 2

n )(∆CR−2λ′X )

+∆2CR + 4λ′∆CRX + 4(λ′2 + λ + n − 1)X 2,

where ∆ =∑n−1

1 (Yj)2.

for λ + n− 1 = 0 or λ + n− 2 = 0, something special happens

Out of these special cases, the corresponding singular vectorsgenerate the Verma modules, their sum is direct.





Contents

1 Motivation




5 The CR case

6 Curved case





Curved case

Manifolds with a given parabolic structure(G, ω), where G is P-principal bundle over M and ω is aCartan connection on Ginvariant differential operators on homogeneous models have,as a rule, curved analogues





Curved case

Manifolds with a given parabolic structure(G, ω), where G is P-principal bundle over M and ω is aCartan connection on Ginvariant differential operators on homogeneous models have,as a rule, curved analoguesExample Conformal case.manifold (M, [g ]) with conformal structure ≃fiber bundle G0 → M with the structure group G0 = CO(n)(resp. G0 = CSpin(n)) prolongation to (G, ω)construction of (families of) natural differential operatorsacting between bundles associated to G0-nodules, resp.P-modules





Curved case

Manifolds with a given parabolic structure(G, ω), where G is P-principal bundle over M and ω is aCartan connection on Ginvariant differential operators on homogeneous models have,as a rule, curved analoguesExample Conformal case.manifold (M, [g ]) with conformal structure ≃fiber bundle G0 → M with the structure group G0 = CO(n)(resp. G0 = CSpin(n)) prolongation to (G, ω)construction of (families of) natural differential operatorsacting between bundles associated to G0-nodules, resp.P-modulesProblem: To construct curved analogues of the flat families ofinvariant differential operators discussed above.


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Branching rules for generalized Verma modules and their ... · PDF fileBranching rules for generalized Verma modules ... SIGMA, 3, 2007, Vladim´ır ... formulation and understanding