Branching rules for generalized Verma modules and their applications
Branching rules for generalized Verma modules and their applications
Branching rules for generalized Verma modules and their applications
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Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> <strong>generalized</strong> <strong>Verma</strong> <strong>modules</strong><br />
<strong>and</strong> <strong>their</strong> <strong>applications</strong><br />
Vladimír Souček<br />
Charles University, Praha<br />
Mathematical Institute<br />
April 2009, Göttingen, Conference in honor of Bent Ørsted<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Contents<br />
1 Motivation<br />
2 <strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
3 The con<strong>for</strong>mal case<br />
4 Construction of families<br />
5 The CR case<br />
6 Curved case<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Contents<br />
1 Motivation<br />
2 <strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
3 The con<strong>for</strong>mal case<br />
4 Construction of families<br />
5 The CR case<br />
6 Curved case<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Flat case, the classical correspondence<br />
interplay between differential geometry <strong>and</strong> representation<br />
theory<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Flat case, the classical correspondence<br />
interplay between differential geometry <strong>and</strong> representation<br />
theory<br />
1-1 map between the invariant differential operators <strong>and</strong><br />
homomorphisms of the corresponding (<strong>generalized</strong>) <strong>Verma</strong><br />
<strong>modules</strong>.<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Flat case, the classical correspondence<br />
interplay between differential geometry <strong>and</strong> representation<br />
theory<br />
1-1 map between the invariant differential operators <strong>and</strong><br />
homomorphisms of the corresponding (<strong>generalized</strong>) <strong>Verma</strong><br />
<strong>modules</strong>.<br />
G semisimple Lie group, P parabolic subgroup, X = G/P<br />
<strong>generalized</strong> flag manifold<br />
V, W two finite-dimensional irreducible P-<strong>modules</strong><br />
there is a bijective correspondence between intertwining<br />
differential operators D : C ∞ (G, V) P ↦→ C ∞ (G, W) P <strong>and</strong><br />
G-homomorphisms between (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
D : M(W ∗ ) ↦→ M(V ∗ ).<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Invariant operators between different dimensions<br />
con<strong>for</strong>mal geometry, (continuous) families D λ of invariant<br />
differential operators of a given order k between a manifold<br />
<strong>and</strong> its submanifold (of codimension one)<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Invariant operators between different dimensions<br />
con<strong>for</strong>mal geometry, (continuous) families D λ of invariant<br />
differential operators of a given order k between a manifold<br />
<strong>and</strong> its submanifold (of codimension one)<br />
A. Juhl: Families of con<strong>for</strong>mally covariant differential<br />
operators, Q-curvature <strong>and</strong> holography, Birkhäuser, Progress<br />
in Math., to appear in May 2009<br />
- motivation from Selberg zeta function <strong>and</strong> automorphic<br />
distributions<br />
- a curved version of those operators is related to the<br />
Q-curvature in con<strong>for</strong>mal geometry<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
T. Branson, R. Gover: Con<strong>for</strong>mally invariant non-local<br />
operators, Pac. Jour. Math, 2001,<br />
- families D λ are used <strong>for</strong> a boundary value problem in a<br />
construction of con<strong>for</strong>mally invariant (non-local) operators<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
T. Branson, R. Gover: Con<strong>for</strong>mally invariant non-local<br />
operators, Pac. Jour. Math, 2001,<br />
- families D λ are used <strong>for</strong> a boundary value problem in a<br />
construction of con<strong>for</strong>mally invariant (non-local) operators<br />
R. Gover: Con<strong>for</strong>mal Dirichlet-Neumann maps <strong>and</strong><br />
Poincaré-Einstein manifolds, SIGMA, 3, 2007,<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
the Juhl family - flat case<br />
Invariant differential operators between irreducible bundles but<br />
in different dimensions(!)<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
the Juhl family - flat case<br />
Invariant differential operators between irreducible bundles but<br />
in different dimensions(!)<br />
G (n) = SO 0 (n, 1) ⊂ G (n+1) = SO 0 (n + 1, 1)<br />
X = S n−1 = G (n) /P (n) ⊂ M = S n = G (n+1) /P (n+1)<br />
let k be a fixed positive integer;<br />
let R w be a character <strong>for</strong> P (n+1) , resp. <strong>for</strong> P (n) ,<br />
We want to construct a continuous family<br />
D w : C ∞ (G (n+1) , V −w+k ) P(n+1) ↦→ C ∞ (G (n) , V −w ) P(n)<br />
of differential operators commuting with the action of G (n) .<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
the Juhl family - flat case<br />
Invariant differential operators between irreducible bundles but<br />
in different dimensions(!)<br />
G (n) = SO 0 (n, 1) ⊂ G (n+1) = SO 0 (n + 1, 1)<br />
X = S n−1 = G (n) /P (n) ⊂ M = S n = G (n+1) /P (n+1)<br />
let k be a fixed positive integer;<br />
let R w be a character <strong>for</strong> P (n+1) , resp. <strong>for</strong> P (n) ,<br />
We want to construct a continuous family<br />
D w : C ∞ (G (n+1) , V −w+k ) P(n+1) ↦→ C ∞ (G (n) , V −w ) P(n)<br />
of differential operators commuting with the action of G (n) .<br />
Dual <strong>for</strong>mulation: a family Φ w of g (n) -homomorphisms<br />
between (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
Φ w : M (n) (V w ) ↦→ M (n+1) (V w−k ).<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Continuous families of homomorphisms between different<br />
dimensions<br />
Suppose that there are branching <strong>rules</strong> <strong>for</strong> a continuous family<br />
of (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong> depending on a parametr w :<br />
M (n+1)<br />
w (V µ ) ≃ ⊕ αw ∈AM (n) (V αw )<br />
as G n -<strong>modules</strong>.<br />
Then in generic situation, all G n -homomorphisms of another<br />
(<strong>generalized</strong>) <strong>Verma</strong> module M (n) (W) to M w<br />
(n+1) (V µ ) are<br />
given by embeddings onto a component in the decomposition.<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Continuous families of homomorphisms between different<br />
dimensions<br />
Suppose that there are branching <strong>rules</strong> <strong>for</strong> a continuous family<br />
of (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong> depending on a parametr w :<br />
M (n+1)<br />
w (V µ ) ≃ ⊕ αw ∈AM (n) (V αw )<br />
as G n -<strong>modules</strong>.<br />
Then in generic situation, all G n -homomorphisms of another<br />
(<strong>generalized</strong>) <strong>Verma</strong> module M (n) (W) to M w<br />
(n+1) (V µ ) are<br />
given by embeddings onto a component in the decomposition.<br />
Hence to underst<strong>and</strong> constructions of continuous families, we<br />
have to underst<strong>and</strong> branching <strong>rules</strong> <strong>for</strong> <strong>Verma</strong> <strong>modules</strong>.<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Formulation of the problem<br />
Goal - <strong>for</strong>mulation <strong>and</strong> underst<strong>and</strong>ing of branching <strong>rules</strong> <strong>for</strong><br />
(<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong> <strong>and</strong>, in particular, a description<br />
(explicit, if possible) of homomorphisms realizing the<br />
branching<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Formulation of the problem<br />
Goal - <strong>for</strong>mulation <strong>and</strong> underst<strong>and</strong>ing of branching <strong>rules</strong> <strong>for</strong><br />
(<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong> <strong>and</strong>, in particular, a description<br />
(explicit, if possible) of homomorphisms realizing the<br />
branching<br />
An (explicit) description of corresponding invariant families of<br />
differential operators<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Formulation of the problem<br />
Goal - <strong>for</strong>mulation <strong>and</strong> underst<strong>and</strong>ing of branching <strong>rules</strong> <strong>for</strong><br />
(<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong> <strong>and</strong>, in particular, a description<br />
(explicit, if possible) of homomorphisms realizing the<br />
branching<br />
An (explicit) description of corresponding invariant families of<br />
differential operators<br />
A study of curved analogues of these operators on curved<br />
versions of parabolic geometries<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Formulation of the problem<br />
Goal - <strong>for</strong>mulation <strong>and</strong> underst<strong>and</strong>ing of branching <strong>rules</strong> <strong>for</strong><br />
(<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong> <strong>and</strong>, in particular, a description<br />
(explicit, if possible) of homomorphisms realizing the<br />
branching<br />
An (explicit) description of corresponding invariant families of<br />
differential operators<br />
A study of curved analogues of these operators on curved<br />
versions of parabolic geometries<br />
On going research project with B. Ørsted (Aarhus) <strong>and</strong> P.<br />
Somberg (Prague)<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Contents<br />
1 Motivation<br />
2 <strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
3 The con<strong>for</strong>mal case<br />
4 Construction of families<br />
5 The CR case<br />
6 Curved case<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Generalized <strong>Verma</strong> <strong>modules</strong><br />
Let g be a complex simple Lie algebra, p its parabolic subgroup,<br />
g = p ⊕ n − , let g 0 ⊂ p be the Levi factor of p <strong>and</strong> h the Cartan<br />
subalgebra. Consider Λ ∈ h ∗ dominant <strong>and</strong> integral <strong>for</strong> p <strong>and</strong> let<br />
V Λ be the corresponding finite dimensional irreducible<br />
representation of p with highest weight Λ. Then the (<strong>generalized</strong>)<br />
<strong>Verma</strong> module M Λ is defined as<br />
M Λ := U(g) ⊗ U(p) V Λ .<br />
It is a universal highest weight module with weight Λ. As a vector<br />
space, it is isomorphic to U(n − ).<br />
Multiplicity of a weight space U(n − ) λ is given by the Kostant<br />
partition function.<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
the case SL 3 ↦→ SL 2 , Borel case<br />
g = sl 3 , b upper triangular (Borel) g = b ⊕ n −<br />
g ′ = sl 2 embedded to the left upper corner<br />
roots of sl 3 : ±α, ±β, ±γ; γ = α + β,<br />
roots in n − : −α, −β, −γ,<br />
root elements Y α , Y β , Y γ <strong>for</strong> negative roots<br />
roots of sl 2 : are ±α<br />
a basis in the <strong>Verma</strong> module with a highest weight vector v :<br />
{Y j αY k γ Y l β | ≡ αj β k γ l , v ∈ M λ ; j, k, l ∈ N 0 }<br />
M 0 := . . . α 3 v α 2 v αv v<br />
M 1 := . . . α3 βv α 2 βv αβv βv<br />
. . . α 2 γv α γ v γv<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
1<br />
1<br />
2 1<br />
2 1<br />
3 2 1<br />
3 2 1<br />
4 3 2 1<br />
4 3 2<br />
Y γ<br />
ւ<br />
Y β<br />
↓<br />
5 4 3 1 Yα<br />
տ<br />
5 4 2<br />
5 3 1<br />
4 2<br />
5 3 1<br />
Vladimír Souček 4 <strong>Branching</strong> 2 <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
1<br />
2 1<br />
2 1<br />
3 2 1<br />
3 2 1<br />
4 3 2 1<br />
4 3 2<br />
5 4 3 1<br />
5 4 2<br />
5 3 1<br />
4 2<br />
5 3 1<br />
4 2<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
1<br />
2 1<br />
2 1<br />
3 2 1<br />
3 2 1<br />
4 3 2 1<br />
4 3 2<br />
5 4 3 1<br />
5 4 2<br />
5 3 1<br />
4 2<br />
5 3 1<br />
4 2<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Theorem<br />
Let λ = (λ 1 , λ 2 , λ 3 ) be a highest weight of the <strong>Verma</strong> module M (3)<br />
λ<br />
<strong>for</strong> l 3 . Then it decomposes (as l 2 -module) <strong>for</strong> a generic weight as<br />
M (3)<br />
λ<br />
= ∑<br />
M (2)<br />
(λ 1 −m,λ 2 −n)<br />
m,n∈N 0<br />
Indeed, α = (1, −1, 0), β = (0, 1, −1), γ = (1, 0, −1), hence<br />
λ − nβ − mγ = (λ 1 − m, λ 2 − n, ∗).<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
A simple computation implies that there is a singular vector in<br />
each weight in boxes.<br />
Exceptional cases - individual <strong>Verma</strong> <strong>modules</strong> are included one in<br />
another<br />
When M λ is not irreducible <strong>for</strong> l 3 , we can make a quotient with the<br />
maximal podmodule - it is possible to deduce the usual branching<br />
<strong>rules</strong> <strong>for</strong> finite dimensional <strong>modules</strong><br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
the case SL 3 ↦→ SL 2 , parabolic case<br />
1<br />
2 1<br />
2 1<br />
2 2 1<br />
2 2 1<br />
2 2 2 1<br />
2 2 2<br />
2 2 2 1<br />
2 2 1<br />
2 1 0<br />
1 0<br />
1 0 0<br />
Vladimír Souček <strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Theorem<br />
Let λ = (λ 1 |λ 2 , λ 3 ) be a p-dominant weight ane let the <strong>generalized</strong><br />
(3)<br />
<strong>Verma</strong> module M p λ<br />
<strong>for</strong> l 3 . Then it decomposes (as l 2 -module) <strong>for</strong><br />
a generic weight as<br />
(3)<br />
M p λ = ∑<br />
(2)<br />
M p (λ 1 −m,λ 2 −n)<br />
m,n∈N 0 ,λ 2 −n≥λ 3<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
General result<br />
A careful study of multiplicity-one theorems <strong>and</strong> branching<br />
<strong>rules</strong> in general situation (in particular <strong>for</strong> unitary highest<br />
weight <strong>modules</strong>) can be found in<br />
T. Kobayashi: Multiplicity-free theorems of the restrictions of<br />
unitary highest weight <strong>modules</strong> with respect to reductive<br />
symmetric pairs, Progress in Math., Birkhäuser, 2006<br />
T. Kobayashi: Discrete decomposability of the restriction of<br />
A q (λ) with respect to reductive subgroup, Invent. Math. III,<br />
131, 1997, 229-256<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Contents<br />
1 Motivation<br />
2 <strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
3 The con<strong>for</strong>mal case<br />
4 Construction of families<br />
5 The CR case<br />
6 Curved case<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
The con<strong>for</strong>mal case<br />
The sphere S n is a homogeneous space G/P, with<br />
G = G (n) = SO(n + 1, 1) <strong>and</strong> it contains the sphere S n−1 given as<br />
a homogeneous space G/P with G = G (n−1) .<br />
Let g (n) = so(n + 1, 1), g (n−1) = so(n, 1) with the st<strong>and</strong>ard<br />
embedding g (n−1) ⊂ g (n)<br />
Let g (n) = gp (n) ⊕ n (n)<br />
−<br />
<strong>and</strong> the same in one dimension below.<br />
Let {Y 1 , . . .,Y n } a basis of n (n)<br />
− such that {Y 1, . . .,Y n−1 } is a<br />
basis of n (n−1)<br />
− .<br />
Irreducible p (n) -<strong>modules</strong> are labeled by weights Λ = (λ 1 |λ 2 , . . .,λ k )<br />
dominant integral <strong>for</strong> p.<br />
Let us split Λ as λ = (w, λ) with w ∈ R being the con<strong>for</strong>mal<br />
weight <strong>and</strong> λ being a dominant integral weight <strong>for</strong> so(n).<br />
Similarly <strong>for</strong> Λ ′ = (w ′ , λ ′ ) <strong>for</strong> dimension n − 1 instead of n.<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
<strong>Branching</strong> <strong>rules</strong> in con<strong>for</strong>mal case<br />
Theorem<br />
(T. Kobayashi) Let V w,λ be an irreducible p (n) -module <strong>and</strong> M (n)<br />
w,λ<br />
the corresponding <strong>generalized</strong> <strong>Verma</strong> module.<br />
Then we have in generic situation (i.e. up to a discrete set of w ′ s)<br />
the branching <strong>rules</strong><br />
M (n)<br />
w,λ ≃ ⊕ k∈Z ≥0<br />
⊕ λ ′ րλ M (n−1)<br />
w−k,λ ′ ,<br />
where λ ′ ր λ means that V<br />
λ ′ appears in the branching <strong>rules</strong> <strong>for</strong> V λ.<br />
In particular<br />
M (n)<br />
w,0 ≃ ⊕ k∈Z ≥0<br />
M (n−1)<br />
w−k,0 .<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Generic case<br />
Let v ∈ M (n+1) (λ) be the highest weight vector Then (Y n ) j v are<br />
not singular vectors <strong>for</strong> j > 1<br />
Denote ∆ = ∑ n−1<br />
1<br />
(Y j ) 2<br />
. . . ∆ 2 v . . . ∆v . . . v<br />
. . . ∆ 2 vY n v . . . ∆vY n v . . . Y n v<br />
. . . ∆ 2 v(Y n ) 2 v . . . d ∆ v(Y n ) 2 v . . . (Y n ) 2 v<br />
. . . ∆ 2 v(Y n ) 3 v . . . d∆v(Y n ) 3 v . . . (Y n ) 3 v<br />
. . . ∆ 2 v(Y n ) 4 v . . . d∆v(Y n ) 4 v . . . (Y n ) 4 v<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
The order 4<br />
the singular vector u <strong>for</strong> the embedding in degree 4 has a <strong>for</strong>m<br />
u = 1 3 (2λ+n−7)(2λ+n−5)Y 4 n v −2(2λ+n−5)∆Y 2 n v +∆ 2 v,<br />
where ∆ = ∑ n−1<br />
1<br />
(Y j ) 2 <strong>and</strong> v is the highest weight vector in<br />
the ’big’ <strong>Verma</strong> module.<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
The order 4<br />
the singular vector u <strong>for</strong> the embedding in degree 4 has a <strong>for</strong>m<br />
u = 1 3 (2λ+n−7)(2λ+n−5)Y 4 n v −2(2λ+n−5)∆Y 2 n v +∆ 2 v,<br />
where ∆ = ∑ n−1<br />
1<br />
(Y j ) 2 <strong>and</strong> v is the highest weight vector in<br />
the ’big’ <strong>Verma</strong> module.<br />
<strong>for</strong> 2λ + n − 7 = 0 or 2λ + n − 5 = 0, something special<br />
happens<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Contents<br />
1 Motivation<br />
2 <strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
3 The con<strong>for</strong>mal case<br />
4 Construction of families<br />
5 The CR case<br />
6 Curved case<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Splitting operators<br />
Splitting operators are particular types of invariant differential<br />
operators between irreducible <strong>and</strong> non-decomposable P-<strong>modules</strong>.<br />
Let V = V λ be an irreducible G-module with the highest weight<br />
λ = (λ 1 , λ 2 , . . .,λ k ). It is a non-decomposable P-module <strong>and</strong> it<br />
has a filtration by P-<strong>modules</strong>. The smallest element in the<br />
filtration W is an irreducible P-module. The same is true <strong>for</strong> the<br />
factor module W ′ of V by the biggest nontrivial P-submodule.<br />
So we have an invariant projection π of V onto W <strong>and</strong> an invariant<br />
embedding ι of W ′ into V. Let V, W <strong>and</strong> W ′ be the associated<br />
homogeneous bundles on G/P.<br />
Typical splitting operators:<br />
D λ : Γ(W) → Γ(V λ ), π ◦ D λ = α id, α ≠ 0,<br />
E λ : Γ(V λ ) → Γ(W ′ ), E λ ◦ ι = α id, α ≠ 0.<br />
We can twist V by a one dimensional module R[w].<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Example - st<strong>and</strong>ard tractor bundle.<br />
Consider defining g-module with highest weight Λ 0 = (1, 0, . . .,0).<br />
It splits under reduction to g 0 into three components with highest<br />
weights (1|0, . . .,0) ⊕ (0|1, 0, . . .,0) ⊕ (−1|0, . . .,0).<br />
Denote<br />
E A .. sections of (1, 0, . . .,0)<br />
E[1] .. sections of (1|0, . . .,0)<br />
E a [1] .. sections of (0|1, 0, . . .,0)<br />
E[−1] .. sections of (−1|0, . . .,0)<br />
Then E A [w] ≃ E[w + 1] ⊕ E a [w + 1] ⊕ E[w − 1].<br />
E[w + 1] is the irreducible quotion, E[−1] is irreducible subbundle.<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
St<strong>and</strong>ard splitting operator<br />
D A : E[w + 1] ↦→ E A [w]<br />
⎛<br />
⎞<br />
w(n + 2w − 2)σ<br />
Dσ = ⎝(n + 2w − 2)∇ a σ⎠<br />
−∆σ<br />
E A : E A [w] ↦→ E[w − 1]<br />
⎛ ⎞<br />
σ<br />
E ⎝µ a<br />
⎠ = (w + 2n − 2)(w + n)ρ + (w + 2n − 2)∇ a µ a − ∆σ<br />
ρ<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
The second order family<br />
See Let Π A A<br />
denote the projection (induced by finite-dimensional<br />
′<br />
branching <strong>rules</strong> from g (n+1) to g (n) ) twisted by identity on R[w].<br />
Let E A′ is the splitting operator <strong>for</strong> g (n) . Then the composition<br />
D 2 (w) := E A′ ◦ Π A A<br />
◦ D ′ A maps E[1] to cE ′ [−1] <strong>and</strong> is con<strong>for</strong>mally<br />
invariant.<br />
Explicitely<br />
D 2 (w)(σ) = (n + w − 3)[(n + 2w − 2)∆ ′ σ + (n + 2w − 3)∆σ]<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
General strategy<br />
We want to construct an operator corresponding to a<br />
particular piece of the branching <strong>rules</strong>: M (n−1)<br />
w−2k,λ<br />
embedded<br />
′<br />
into M (n)<br />
w,λ<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
General strategy<br />
We want to construct an operator corresponding to a<br />
particular piece of the branching <strong>rules</strong>: M (n−1)<br />
w−2k,λ<br />
embedded<br />
′<br />
into M (n)<br />
w,λ<br />
Find a splitting operator mapping sections E w,λ of bundle<br />
associated V w,λ to a tractor bundle twisted by a density<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
General strategy<br />
We want to construct an operator corresponding to a<br />
particular piece of the branching <strong>rules</strong>: M (n−1)<br />
w−2k,λ<br />
embedded<br />
′<br />
into M (n)<br />
w,λ<br />
Find a splitting operator mapping sections E w,λ of bundle<br />
associated V w,λ to a tractor bundle twisted by a density<br />
Apply a suitably chosen projection coming from branching<br />
<strong>rules</strong> <strong>for</strong> the pair (g (n+1) , g (n) ) (twisted by a density<br />
representation).<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
General strategy<br />
We want to construct an operator corresponding to a<br />
particular piece of the branching <strong>rules</strong>: M (n−1)<br />
w−2k,λ<br />
embedded<br />
′<br />
into M (n)<br />
w,λ<br />
Find a splitting operator mapping sections E w,λ of bundle<br />
associated V w,λ to a tractor bundle twisted by a density<br />
Apply a suitably chosen projection coming from branching<br />
<strong>rules</strong> <strong>for</strong> the pair (g (n+1) , g (n) ) (twisted by a density<br />
representation).<br />
All should be arranged in such a way that there is a<br />
differential splitting back to E w−2k,λ ′<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Contents<br />
1 Motivation<br />
2 <strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
3 The con<strong>for</strong>mal case<br />
4 Construction of families<br />
5 The CR case<br />
6 Curved case<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
The CR case<br />
The sphere S 2n+1 ⊂ C n is a homogeneous space G/P, with<br />
G = SU(n + 1, 1).<br />
g (n+1) = su(n + 1, 1), g (n) = su(n, 1) with the st<strong>and</strong>ard embedding<br />
g (n) ⊂ g (n+1)<br />
consider the corresponding parabolic subalgebras gp (n+1) <strong>and</strong> gp (n)<br />
nilpotent parts n (n+1)<br />
−<br />
<strong>and</strong> n (n)<br />
−<br />
have dimensions 2n + 1, resp.<br />
2n − 1 with a suitable bases {X 1 , . . .,X n , Y 1 , . . .,Y n , X }, resp.<br />
{X 1 , . . .,X n−1 , Y 1 , . . .,Y n−1 , X },<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Theorem<br />
(B.Ørsted, P. Somberg, VS) Let M (n+1) (λ, λ ′ ) be the <strong>generalized</strong><br />
<strong>Verma</strong> module induced by a character of g (n+1) , similarly <strong>for</strong><br />
M (n) (λ, λ ′ ).<br />
Then <strong>for</strong> every N ∈ N there is a family of g (n) -homomorphisms<br />
D : M (n) (λ − N, λ ′ ) ↦→ M (n+1) (λ, λ ′ ), λ, λ ′ ∈ C.<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
The singular vector w <strong>for</strong> D 4 has a <strong>for</strong>m<br />
(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)<br />
+∆ 2 CR + 4λ′ ∆ CR X + 4(λ ′2 + λ + n − 1)X 2 ,<br />
where ∆ = ∑ n−1<br />
1<br />
(Y j ) 2 .<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
The singular vector w <strong>for</strong> D 4 has a <strong>for</strong>m<br />
(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)<br />
+∆ 2 CR + 4λ′ ∆ CR X + 4(λ ′2 + λ + n − 1)X 2 ,<br />
where ∆ = ∑ n−1<br />
1<br />
(Y j ) 2 .<br />
<strong>for</strong> λ + n − 1 = 0 or λ + n − 2 = 0, something special happens<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
The singular vector w <strong>for</strong> D 4 has a <strong>for</strong>m<br />
(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)<br />
+∆ 2 CR + 4λ′ ∆ CR X + 4(λ ′2 + λ + n − 1)X 2 ,<br />
where ∆ = ∑ n−1<br />
1<br />
(Y j ) 2 .<br />
<strong>for</strong> λ + n − 1 = 0 or λ + n − 2 = 0, something special happens<br />
Out of these special cases, the corresponding singular vectors<br />
generate the <strong>Verma</strong> <strong>modules</strong>, <strong>their</strong> sum is direct.<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Contents<br />
1 Motivation<br />
2 <strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
3 The con<strong>for</strong>mal case<br />
4 Construction of families<br />
5 The CR case<br />
6 Curved case<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Curved case<br />
Manifolds with a given parabolic structure<br />
(G, ω), where G is P-principal bundle over M <strong>and</strong> ω is a<br />
Cartan connection on G<br />
invariant differential operators on homogeneous models have,<br />
as a rule, curved analogues<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Curved case<br />
Manifolds with a given parabolic structure<br />
(G, ω), where G is P-principal bundle over M <strong>and</strong> ω is a<br />
Cartan connection on G<br />
invariant differential operators on homogeneous models have,<br />
as a rule, curved analogues<br />
Example Con<strong>for</strong>mal case.<br />
manifold (M, [g]) with con<strong>for</strong>mal structure ≃<br />
fiber bundle G 0 → M with the structure group G 0 = CO(n)<br />
(resp. G 0 = CSpin(n)) prolongation to (G, ω)<br />
construction of (families of) natural differential operators<br />
acting between bundles associated to G 0 -nodules, resp.<br />
P-<strong>modules</strong><br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>
Motivation<br />
<strong>Branching</strong> <strong>rules</strong> <strong>for</strong> (<strong>generalized</strong>) <strong>Verma</strong> <strong>modules</strong><br />
The con<strong>for</strong>mal case<br />
Construction of families<br />
The CR case<br />
Curved case<br />
Curved case<br />
Manifolds with a given parabolic structure<br />
(G, ω), where G is P-principal bundle over M <strong>and</strong> ω is a<br />
Cartan connection on G<br />
invariant differential operators on homogeneous models have,<br />
as a rule, curved analogues<br />
Example Con<strong>for</strong>mal case.<br />
manifold (M, [g]) with con<strong>for</strong>mal structure ≃<br />
fiber bundle G 0 → M with the structure group G 0 = CO(n)<br />
(resp. G 0 = CSpin(n)) prolongation to (G, ω)<br />
construction of (families of) natural differential operators<br />
acting between bundles associated to G 0 -nodules, resp.<br />
P-<strong>modules</strong><br />
Problem: To construct curved analogues of the flat families of<br />
invariant differential operators discussed above.<br />
Vladimír Souček<br />
<strong>Branching</strong> <strong>rules</strong>