Branching rules for generalized Verma modules and their applications

Motivation 

Branching rules for (generalized) Verma modules 

The conformal case 

Construction of families 

The CR case 

Curved case 

Branching rules for generalized Verma modules 

and their applications 

Vladimír Souček 

Charles University, Praha 

Mathematical Institute 

April 2009, Göttingen, Conference in honor of Bent Ørsted 


Branching rules

Motivation 




The CR case 

Curved case 

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 



Motivation 




The CR case 

Curved case 

Contents 

1 Motivation 




5 The CR case 

6 Curved case 



Motivation 




The CR case 

Curved case 

Flat case, the classical correspondence 

interplay between differential geometry and representation 

theory 



Motivation 




The CR case 

Curved case 



theory 

1-1 map between the invariant differential operators and 

homomorphisms of the corresponding (generalized) Verma 

modules. 



Motivation 




The CR case 

Curved case 



theory 

1-1 map between the invariant differential operators and 

homomorphisms of the corresponding (generalized) Verma 

modules. 

G semisimple Lie group, P parabolic subgroup, X = G/P 

generalized flag manifold 

V, W two finite-dimensional irreducible P-modules 

there is a bijective correspondence between intertwining 

differential operators D : C ∞ (G, V) P ↦→ C ∞ (G, W) P and 

G-homomorphisms between (generalized) Verma modules 

D : M(W ∗ ) ↦→ M(V ∗ ). 



Motivation 




The CR case 

Curved case 

Invariant operators between different dimensions 

conformal geometry, (continuous) families D λ of invariant 

differential operators of a given order k between a manifold 

and its submanifold (of codimension one) 



Motivation 




The CR case 

Curved case 

Invariant operators between different dimensions 

conformal geometry, (continuous) families D λ of invariant 

differential operators of a given order k between a manifold 

and its submanifold (of codimension one) 

A. Juhl: Families of conformally covariant differential 

operators, Q-curvature and holography, Birkhäuser, Progress 

in Math., to appear in May 2009 

- motivation from Selberg zeta function and automorphic 

distributions 

- a curved version of those operators is related to the 

Q-curvature in conformal geometry 



Motivation 




The CR case 

Curved case 

T. Branson, R. Gover: Conformally invariant non-local 

operators, Pac. Jour. Math, 2001, 

- families D λ are used for a boundary value problem in a 

construction of conformally invariant (non-local) operators 



Motivation 




The CR case 

Curved case 

T. Branson, R. Gover: Conformally invariant non-local 

operators, Pac. Jour. Math, 2001, 

- families D λ are used for a boundary value problem in a 

construction of conformally invariant (non-local) operators 

R. Gover: Conformal Dirichlet-Neumann maps and 

Poincaré-Einstein manifolds, SIGMA, 3, 2007, 



Motivation 




The CR case 

Curved case 

the Juhl family - flat case 

Invariant differential operators between irreducible bundles but 

in different dimensions(!) 



Motivation 




The CR case 

Curved case 




G (n) = SO 0 (n, 1) ⊂ G (n+1) = SO 0 (n + 1, 1) 

X = S n−1 = G (n) /P (n) ⊂ M = S n = G (n+1) /P (n+1) 

let k be a fixed positive integer; 

let R w be a character for P (n+1) , resp. for P (n) , 

We want to construct a continuous family 

D w : C ∞ (G (n+1) , V −w+k ) P(n+1) ↦→ C ∞ (G (n) , V −w ) P(n) 

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



Motivation 




The CR case 

Curved case 




G (n) = SO 0 (n, 1) ⊂ G (n+1) = SO 0 (n + 1, 1) 

X = S n−1 = G (n) /P (n) ⊂ M = S n = G (n+1) /P (n+1) 

let k be a fixed positive integer; 

let R w be a character for P (n+1) , resp. for P (n) , 

We want to construct a continuous family 

D w : C ∞ (G (n+1) , V −w+k ) P(n+1) ↦→ C ∞ (G (n) , V −w ) P(n) 

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

Dual formulation: a family Φ w of g (n) -homomorphisms 

between (generalized) Verma modules 

Φ w : M (n) (V w ) ↦→ M (n+1) (V w−k ). 



Motivation 




The CR case 

Curved case 

Continuous families of homomorphisms between different 

dimensions 

Suppose that there are branching rules for a continuous family 

of (generalized) Verma modules depending on a parametr w : 

M (n+1) 

w (V µ ) ≃ ⊕ αw ∈AM (n) (V αw ) 

as G n -modules. 

Then in generic situation, all G n -homomorphisms of another 

(generalized) Verma module M (n) (W) to M w 

(n+1) (V µ ) are 

given by embeddings onto a component in the decomposition. 



Motivation 




The CR case 

Curved case 

Continuous families of homomorphisms between different 

dimensions 

Suppose that there are branching rules for a continuous family 

of (generalized) Verma modules depending on a parametr w : 

M (n+1) 

w (V µ ) ≃ ⊕ αw ∈AM (n) (V αw ) 

as G n -modules. 

Then in generic situation, all G n -homomorphisms of another 

(generalized) Verma module M (n) (W) to M w 

(n+1) (V µ ) are 

given by embeddings onto a component in the decomposition. 

Hence to understand constructions of continuous families, we 

have to understand branching rules for Verma modules. 



Motivation 




The CR case 

Curved case 

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 the 

branching 



Motivation 




The CR case 

Curved case 





branching 

An (explicit) description of corresponding invariant families of 

differential operators 



Motivation 




The CR case 

Curved case 





branching 



A study of curved analogues of these operators on curved 

versions of parabolic geometries 



Motivation 




The CR case 

Curved case 





branching 



A study of curved analogues of these operators on curved 

versions of parabolic geometries 

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

Somberg (Prague) 



Motivation 




The CR case 

Curved case 

Contents 

1 Motivation 




5 The CR case 

6 Curved case 



Motivation 




The CR case 

Curved case 

Generalized Verma modules 

Let g be a complex simple Lie algebra, p its parabolic subgroup, 

g = p ⊕ n − , let g 0 ⊂ p be the Levi factor of p and h the Cartan 

subalgebra. Consider Λ ∈ h ∗ dominant and integral for p and let 

V Λ be the corresponding finite dimensional irreducible 

representation of p with highest weight Λ. Then the (generalized) 

Verma module M Λ is defined as 

M Λ := U(g) ⊗ U(p) V Λ . 

It is a universal highest weight module with weight Λ. As a vector 

space, it is isomorphic to U(n − ). 

Multiplicity of a weight space U(n − ) λ is given by the Kostant 

partition function. 



Motivation 




The CR case 

Curved case 

the case SL 3 ↦→ SL 2 , Borel case 

g = sl 3 , b upper triangular (Borel) g = b ⊕ n − 

g ′ = sl 2 embedded to the left upper corner 

roots of sl 3 : ±α, ±β, ±γ; γ = α + β, 

roots in n − : −α, −β, −γ, 

root elements Y α , Y β , Y γ for negative roots 

roots of sl 2 : are ±α 

a basis in the Verma module with a highest weight vector v : 

{Y j αY k γ Y l β | ≡ αj β k γ l , v ∈ M λ ; j, k, l ∈ N 0 } 

M 0 := . . . α 3 v α 2 v αv v 

M 1 := . . . α3 βv α 2 βv αβv βv 

. . . α 2 γv α γ v γv 



Motivation 




The CR case 

Curved case 

1 

1 

2 1 

2 1 

3 2 1 

3 2 1 

4 3 2 1 

4 3 2 

Y γ 

ւ 

Y β 

↓ 

5 4 3 1 Yα 

տ 

5 4 2 

5 3 1 

4 2 

5 3 1 

Vladimír Souček 4 Branching 2 rules

Motivation 




The CR case 

Curved case 

1 

2 1 

2 1 

3 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 



Motivation 




The CR case 

Curved case 

1 

2 1 

2 1 

3 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 



Motivation 




The CR case 

Curved case 

Theorem 

Let λ = (λ 1 , λ 2 , λ 3 ) be a highest weight of the Verma module M (3) 

λ 

for l 3 . Then it decomposes (as l 2 -module) for a generic weight as 

M (3) 

λ 

= ∑ 

M (2) 

(λ 1 −m,λ 2 −n) 

m,n∈N 0 

Indeed, α = (1, −1, 0), β = (0, 1, −1), γ = (1, 0, −1), hence 

λ − nβ − mγ = (λ 1 − m, λ 2 − n, ∗). 



Motivation 




The CR case 

Curved case 

A simple computation implies that there is a singular vector in 

each weight in boxes. 

Exceptional cases - individual Verma modules are included one in 

another 

When M λ is not irreducible for l 3 , we can make a quotient with the 

maximal podmodule - it is possible to deduce the usual branching 

rules for finite dimensional modules 



Motivation 




The CR case 

Curved case 

the case SL 3 ↦→ SL 2 , parabolic case 

1 

2 1 

2 1 

2 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 

Vladimír Souček Branching rules

Motivation 




The CR case 

Curved case 

Theorem 

Let λ = (λ 1 |λ 2 , λ 3 ) be a p-dominant weight ane let the generalized 

(3) 

Verma module M p λ 

for l 3 . Then it decomposes (as l 2 -module) for 

a generic weight as 

(3) 

M p λ = ∑ 

(2) 

M p (λ 1 −m,λ 2 −n) 

m,n∈N 0 ,λ 2 −n≥λ 3 



Motivation 




The CR case 

Curved case 

General result 

A careful study of multiplicity-one theorems and branching 

rules in general situation (in particular for unitary highest 

weight modules) can be found in 

T. Kobayashi: Multiplicity-free theorems of the restrictions of 

unitary highest weight modules with respect to reductive 

symmetric pairs, Progress in Math., Birkhäuser, 2006 

T. Kobayashi: Discrete decomposability of the restriction of 

A q (λ) with respect to reductive subgroup, Invent. Math. III, 

131, 1997, 229-256 



Motivation 




The CR case 

Curved case 

Contents 

1 Motivation 




5 The CR case 

6 Curved case 



Motivation 




The CR case 

Curved case 


The sphere S n is a homogeneous space G/P, with 

G = G (n) = SO(n + 1, 1) and it contains the sphere S n−1 given as 

a homogeneous space G/P with G = G (n−1) . 

Let g (n) = so(n + 1, 1), g (n−1) = so(n, 1) with the standard 

embedding g (n−1) ⊂ g (n) 

Let g (n) = gp (n) ⊕ n (n) 

− 

and the same in one dimension below. 

Let {Y 1 , . . .,Y n } a basis of n (n) 

− such that {Y 1, . . .,Y n−1 } is a 

basis of n (n−1) 

− . 

Irreducible p (n) -modules are labeled by weights Λ = (λ 1 |λ 2 , . . .,λ k ) 

dominant integral for p. 

Let us split Λ as λ = (w, λ) with w ∈ R being the conformal 

weight and λ being a dominant integral weight for so(n). 

Similarly for Λ ′ = (w ′ , λ ′ ) for dimension n − 1 instead of n. 



Motivation 




The CR case 

Curved case 

Branching rules in conformal case 

Theorem 

(T. Kobayashi) Let V w,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 V λ. 

In particular 

M (n) 

w,0 ≃ ⊕ k∈Z ≥0 

M (n−1) 

w−k,0 . 



Motivation 




The CR case 

Curved case 

Generic case 

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

not singular vectors for j > 1 

Denote ∆ = ∑ n−1 

1 

(Y j ) 2 

. . . ∆ 2 v . . . ∆v . . . v 

. . . ∆ 2 vY n v . . . ∆vY n v . . . Y n v 

. . . ∆ 2 v(Y n ) 2 v . . . d ∆ v(Y n ) 2 v . . . (Y n ) 2 v 

. . . ∆ 2 v(Y n ) 3 v . . . d∆v(Y n ) 3 v . . . (Y n ) 3 v 

. . . ∆ 2 v(Y n ) 4 v . . . d∆v(Y n ) 4 v . . . (Y n ) 4 v 



Motivation 




The CR case 

Curved case 

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 2 n v +∆ 2 v, 

where ∆ = ∑ n−1 

1 

(Y j ) 2 and v is the highest weight vector in 

the ’big’ Verma module. 



Motivation 




The CR case 

Curved case 

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 2 n v +∆ 2 v, 

where ∆ = ∑ n−1 

1 

(Y j ) 2 and v is the highest weight vector in 

the ’big’ Verma module. 

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

happens 



Motivation 




The CR case 

Curved case 

Contents 

1 Motivation 




5 The CR case 

6 Curved case 



Motivation 




The CR case 

Curved case 

Splitting operators 

Splitting operators are particular types of invariant differential 

operators between irreducible and non-decomposable P-modules. 

Let V = V λ be an irreducible G-module with the highest weight 

λ = (λ 1 , λ 2 , . . .,λ k ). It is a non-decomposable P-module and it 

has a filtration by P-modules. The smallest element in the 

filtration W is an irreducible P-module. The same is true for the 

factor module W ′ of V by the biggest nontrivial P-submodule. 

So we have an invariant projection π of V onto W and an invariant 

embedding ι of W ′ into V. Let V, W and W ′ be the associated 

homogeneous bundles on G/P. 

Typical splitting operators: 

D λ : Γ(W) → Γ(V λ ), π ◦ D λ = α id, α ≠ 0, 

E λ : Γ(V λ ) → Γ(W ′ ), E λ ◦ ι = α id, α ≠ 0. 

We can twist V by a one dimensional module R[w]. 



Motivation 




The CR case 

Curved case 

Example - standard tractor bundle. 

Consider defining g-module with highest weight Λ 0 = (1, 0, . . .,0). 

It splits under reduction to g 0 into three components with highest 

weights (1|0, . . .,0) ⊕ (0|1, 0, . . .,0) ⊕ (−1|0, . . .,0). 

Denote 

E A .. sections of (1, 0, . . .,0) 

E[1] .. sections of (1|0, . . .,0) 

E a [1] .. sections of (0|1, 0, . . .,0) 

E[−1] .. sections of (−1|0, . . .,0) 

Then E A [w] ≃ E[w + 1] ⊕ E a [w + 1] ⊕ E[w − 1]. 

E[w + 1] is the irreducible quotion, E[−1] is irreducible subbundle. 



Motivation 




The CR case 

Curved case 

Standard splitting operator 

D A : E[w + 1] ↦→ E A [w] 

⎛ 

⎞ 

w(n + 2w − 2)σ 

Dσ = ⎝(n + 2w − 2)∇ a σ⎠ 

−∆σ 

E A : E A [w] ↦→ E[w − 1] 

⎛ ⎞ 

σ 

E ⎝µ a 

⎠ = (w + 2n − 2)(w + n)ρ + (w + 2n − 2)∇ a µ a − ∆σ 

ρ 



Motivation 




The CR case 

Curved case 

The second order family 

See Let Π A A 

denote the projection (induced by finite-dimensional 

′ 

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

Let E A′ is the splitting operator for g (n) . Then the composition 

D 2 (w) := E A′ ◦ Π A A 

◦ D ′ A maps E[1] to cE ′ [−1] and is conformally 

invariant. 

Explicitely 

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



Motivation 




The CR case 

Curved case 

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 



Motivation 




The CR case 

Curved case 




w−2k,λ 

embedded 

′ 

into M (n) 

w,λ 

Find a splitting operator mapping sections E w,λ of bundle 

associated V w,λ to a tractor bundle twisted by a density 



Motivation 




The CR case 

Curved case 




w−2k,λ 

embedded 

′ 

into M (n) 

w,λ 



Apply a suitably chosen projection coming from branching 

rules for the pair (g (n+1) , g (n) ) (twisted by a density 

representation). 



Motivation 




The CR case 

Curved case 




w−2k,λ 

embedded 

′ 

into M (n) 

w,λ 



Apply a suitably chosen projection coming from branching 

rules for the pair (g (n+1) , g (n) ) (twisted by a density 

representation). 

All should be arranged in such a way that there is a 

differential splitting back to E w−2k,λ ′ 



Motivation 




The CR case 

Curved case 

Contents 

1 Motivation 




5 The CR case 

6 Curved case 



Motivation 




The CR case 

Curved case 

The CR case 

The sphere S 2n+1 ⊂ C n 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 embedding 

g (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 {X 1 , . . .,X n , Y 1 , . . .,Y n , X }, resp. 

{X 1 , . . .,X n−1 , Y 1 , . . .,Y n−1 , X }, 



Motivation 




The CR case 

Curved case 

Theorem 

(B.Ørsted, P. Somberg, VS) Let M (n+1) (λ, λ ′ ) be the generalized 

Verma module induced by a character of g (n+1) , similarly for 

M (n) (λ, λ ′ ). 

Then for every N ∈ N there is a family of g (n) -homomorphisms 

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



Motivation 




The CR case 

Curved case 

The singular vector w for D 4 has a form 

(λ+n−1)(λ+n−2)(X 2 n +Y 2 n ) 2 −2(λ+n−2)(X 2 n +Y 2 n )(∆ CR −2λ ′ X) 

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

where ∆ = ∑ n−1 

1 

(Y j ) 2 . 



Motivation 




The CR case 

Curved case 



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

where ∆ = ∑ n−1 

1 

(Y j ) 2 . 

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



Motivation 




The CR case 

Curved case 



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

where ∆ = ∑ n−1 

1 

(Y j ) 2 . 

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

Out of these special cases, the corresponding singular vectors 

generate the Verma modules, their sum is direct. 



Motivation 




The CR case 

Curved case 

Contents 

1 Motivation 




5 The CR case 

6 Curved case 



Motivation 




The CR case 

Curved case 

Curved case 

Manifolds with a given parabolic structure 

(G, ω), where G is P-principal bundle over M and ω is a 

Cartan connection on G 

invariant differential operators on homogeneous models have, 

as a rule, curved analogues 



Motivation 




The CR case 

Curved case 

Curved case 






Example Conformal case. 

manifold (M, [g]) with conformal structure ≃ 

fiber bundle G 0 → M with the structure group G 0 = CO(n) 

(resp. G 0 = CSpin(n)) prolongation to (G, ω) 

construction of (families of) natural differential operators 

acting between bundles associated to G 0 -nodules, resp. 

P-modules 



Motivation 




The CR case 

Curved case 

Curved case 






Example Conformal case. 

manifold (M, [g]) with conformal structure ≃ 

fiber bundle G 0 → M with the structure group G 0 = CO(n) 

(resp. G 0 = CSpin(n)) prolongation to (G, ω) 

construction of (families of) natural differential operators 

acting between bundles associated to G 0 -nodules, resp. 

P-modules 

Problem: To construct curved analogues of the flat families of 

invariant differential operators discussed above.

Branching rules for generalized Verma modules and their applications

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