Conformal BGG Sequences via Poincaré Metrics
Robin Graham
New Directions in Exterior Differential Systems
in Honor of Robert Bryant’s 60th Birthday
Estes Park
July 19, 2013
Hyperbolic Space and the Sphere
Hyperbolic space:
Hn+1 = {|x|2 < 1} ⊂ Rn+1 , with metric
gH =
Sphere:
4|dx|2
(1 − |x|2 )2
S n = {|x| = 1}, with conformal class
[h],
h = |dx|2 |TS n
Isometry group (Hn+1 ) = conformal group (S n )= G = O(n + 1, 1)
Hn+1 = G /K ,
K = O(n + 1),
S n = G /P,
where
P = isotropy group of a null line
Correspondence is useful in understanding conformal geometry
Curved Poincaré Metrics
Curved version of the correspondence:
Let [h] be a conformal class of metrics on M n , n ≥ 3.
A Poincaré metric for (M, [h]) is a metric g on M × (0, ǫ) such that
1. g has (M, [h]) as conformal infinity:
if r denotes the variable in (0, ǫ), then
r 2 g ∈ C (M × [0, ǫ)) and (r 2 g )|T (M×{0}) ∈ [h].
2. Ric(g ) = −ng asymptotically as r → 0
n odd: Infinite order solutions exist with r 2 g ∈ C ∞ (M × [0, ǫ))
n even: Formal obstruction at order n to smooth solutions
Solution is unique to appropriate order up to a diffeomorphism
equal to identity on M (impose evenness condition if n odd)
Curved Poincaré Metrics
Construct a Poincaré metric in the form
g=
dr 2 + hr
r2
h0 = h,
where h is an arbitrarily chosen metric in the conformal class on
M. Upper-half-space realization for Hn+1 : hr = |dx|2 on Rn .
The equation Ric(g ) = −ng can be written in terms of hr and
analyzed in formal power series. One finds:
hr = h − Ph r 2 + . . .
where
Ph =
1 Rh
h
Rich −
n−2
2(n − 1)
(Schouten tensor)
The obstruction at order n in even dimensions is a conformally
invariant trace-free, symmetric natural 2-tensor.
Bach tensor when n = 4
Conformally invariant powers of the Laplacian
Example of the use of Poincaré metrics in conformal geometry:
Derivation of the conformally invariant powers of the Laplacian.
Given f ∈ C ∞ (M), does there exist u ∈ C ∞ (M × [0, ǫ)) with
u|M = f
and
∆g u = 0 to infinite order?
Answer: Yes if n is odd, no in general if n is even.
If n is even, there is an obstruction at order n:
there exists a unique u mod O(r n ) for which ∆g u = O(r n ).
Moreover r −n ∆g u |r =0 is independent of the choice of such u.
f → r −n ∆g u |r =0 is a conformally invariant differential operator
n/2
on M of order n, with principal part ∆h .
Conformally invariant powers of the Laplacian
Generalization to other powers of the Laplacian
Modify extension problem. Choose s ∈ R. Require asymptotic
behavior:
r −s u ∈ C ∞ (M × [0, ǫ)),
r −s u |M = f
Replace equation ∆g u = O(r ∞ ) by eigenequation:
∆g − s(n − s) u = O(r ∞ )
For s ∈
/ n/2 − N, formal power series solutions exist.
If s = n/2 − k, k ∈ N, there is an obstruction at order 2k
Defines a conformally invariant operator Pk with principal part ∆kh .
Must impose k ≤ n/2 for n even because of the obstruction in the
Poincaré metric
Condition u ∼ r s f has the interpretation that f is a conformal
density on M of weight −s.
Analog for BGG Sequences
Today: Describe analogous derivation of BGG sequences for
conformal geometry
Joint work in progress with Olivier Biquard
Begin by recalling conformal BGG sequences
(M n , [h]) conformal manifold
The form of the de Rham sequence depends on parity of n:
n odd:
Λ0 - Λ1 - · · · - Λn−1 - Λn
n even:
n
n
3
Λ0 - Λ1 - · · · -Λ 2 −1 Q
Λ+2 Q
s
Q Λ n2 +1 - · · · - Λn
n
3
s
Q Λ2 −
BGG Sequences
The conformal BGG sequences are a family of sequences of
conformally invariant differential operators having the same
pattern.
For each n ≥ 3, the set of sequences is parametrized by the
finite-dimensional irreducible representations Uλ of
g = so(n + 2, C). Each Uλ determines a sequence of weighted
irreducible bundles together with conformally invariant differential
operators between them:
n odd:
W 0 - W 1 - · · · - W n−1 - W n
n even:
W 0 - W 1 - · · · -W
n
W+2 Q
3
s
Q W n2 +1 - · · · - W n
n
Q
3
s
QW 2 n
−1
2
−
Each W i is determined by Uλ and the arrows are differential
operators of various orders.
BGG Sequences
The deRham sequence corresponds to Uλ = trivial representation.
If (M, [h]) is locally conformally flat, then
1. The sequences are complexes.
2. The local kernel of the first operator W 0 → W 1 is isomorphic
to the finite-dimensional vector space Uλ .
Recall: The finite-dimensional irreducible representations of g are
parametrized by dominant integral weights λ ∈ h∗ , where h ⊂ g is
a Cartan subalgebra.
The bundles W i are determined by the affine action of the Weyl
group on λ.
Middle operator for n odd is W
n−1
2
→W
Middle operators for n even are
n+1
2
n
n
W+2 Q
3
s
Q W n2 +1
n
3
s
QW 2 W 2 −1
Q
Plan of Construction
Given λ, will define a sequence of first-order differential operators
D on (X , g ), where X = M × (0, ǫ). BGG operators on M will
arise as obstructions at infinity to solving Du = 0 on X .
Motivating example is the deRham sequence.
The exterior derivative on Λk T ∗ M obstructs the solvability of
du = 0 on Λk T ∗ X :
If f section of Λk T ∗ M is given and we ask to find u section of
Λk T ∗ X such that i ∗ u = f and du = 0, a necessary condition is
df = 0.
In the general construction, the interior first order operators which
play the role of d on X are known as generalized gradients.
The obstructions on the boundary are the BGG operators.
Generalized Gradients
Let Vα , Vβ be finite-dimensional irreducible representations of
Spin(n + 1). Let T denote the standard representation of
SO(n + 1). Can decompose T ⊗ Vα into irreducibles.
Fact: All summands have multiplicity 1.
Suppose that Vβ occurs in the decomposition. There is a
well-defined projection πα,β : T ⊗ Vα → Vβ .
Let (X n+1 , g ) be a (spin) Riemannian manifold.
Representations Vα , Vβ induce associated bundles on (X , g ).
Use the same notation Vα , Vβ for the associated bundles and their
spaces of sections.
The induced first-order differential operator
∇
πα,β
Dα,β : Vα → T ∗ M ⊗ Vα → Vβ
is called a generalized gradient.
Twisted Projected deRham Sequences
Recall BGG sequences are parametrized by dominant integral
weights λ for g = so(n + 2, C). Choose such a λ.
Have k = so(n + 1, C) ⊂ so(n + 2, C) = g.
Cartan subalgebras hk ⊂ hg . Now λ ∈ h∗g .
Set λ′ = λ|hk ∈ h∗k . Then λ′ is a dominant integral weight for k ↔
irreducible representation Vλ′ for Spin(n + 1).
Twist the deRham bundles on (X , g ): consider Λk ⊗ Vλ′
n+1
(replace by Λ±2 ⊗ Vλ′ if n + 1 even and k =
n+1
2 )
Decompose Λk ⊗ Vλ′ into irreducibles. Denote by Λk ⊛ Vλ′ the
highest weight irreducible summand (Cartan product).
For each k there is a generalized gradient
Dk : Λk−1 ⊛ Vλ′ → Λk ⊛ Vλ′ .
Denote νk = highest weight for Λk .
Then Dk = Dνk−1 +λ′ ,νk +λ′ in previous notation.
Twisted Projected deRham Sequences
Obtain interior sequences of generalized gradients:
n + 1 odd:
Vλ′ - Λ1 ⊛ Vλ′ - · · · - Λn ⊛ Vλ′ - Λn+1 ⊛ Vλ′
n + 1 even:
Vλ′ - · · · - Λ
n+1
n−1
2
Λ 2 ⊛ Vλ′ Q n+3
3
+
s
QΛ 2 ⊛ V ′ ⊛ Vλ′ Q n+1
λ
3
s
QΛ 2 ⊛V ′ λ
−
· · · - Λn+1 ⊛ Vλ′
BGG sequence with parameter λ will be constructed from this
interior sequence for λ′ .
Away from the middle, each BGG operator is an obstruction for
the corresponding interior operator.
For simplicity, consider first half of BGG sequence before the
middle. Second half after the middle is similar.
Branching
Interested in BGG operator Pk : W k−1 → W k with 1 ≤ k ≤ [ n−1
2 ]
Ignoring the conformal weight, W k is an irreducible bundle on M
associated to an irreducible representation of Spin(n). Write
W k = Wρk , where ρk is the highest weight for Spin(n).
Corresponding interior operator is the generalized gradient
Dk : Λk−1 ⊛ Vλ′ → Λk ⊛ Vλ′
Can decompose Λk ⊛ Vλ′ under restriction to Spin(n). Branching
rules determine the decomposition into irreducibles under Spin(n):
M
Wρ
Λk ⊛ V λ ′ =
All multiplicities are 1.
Main observation: Wρk−1 occurs in the decomposition for
Λk−1 ⊛ Vλ′ , but Wρk does not.
So in solving Dk u = 0, cannot correct a component in Wρk . Gives
rise to an obstruction in Wρk = target of BGG operator Pk
BGG Operators as Obstructions
Set
sk−1 = k − 1 + (νk−1 − νk , λ′ )
dk = sk − sk−1
where (·, ·) is the Killing form.
sk−1 measures the conformal weight of W k−1 , and sk that of W k .
dk is the order of the BGG operator Pk .
Theorem. Given f section of Wρk−1 on M. There exists a section
u of Λk−1 ⊛ Vλ′ on X = M × (0, ǫ) such that
u = r sk−1 f + O(r sk−1 +1 )
and
Dk u = O(r sk−1 +dk ).
The Wρk -component of r −(sk−1 +dk ) Dk u |M is independent of the
choice of u. The map
f → r −(sk−1 +dk ) Dk u |M
defines a conformally invariant operator Pk : Wρk−1 → Wρk .
Disclosure Statement
Our lawyers advise us that for our protection we must inform you
of certain considerations.
1. Have suppressed conformal weights in the Wρk
2. There are implicit identifications arising from Λk ⊛ Vλ′ =
Among other things, this affects what O(r s ) means
L
3. Statement is slight oversimplification: different components
vanish to different orders.
Wρ .
Second Order Example
Next simplest BGG sequence after deRham is parametrized by the
standard representation Cn+2 of so(n + 2, C).
First operator is P1 : C ∞ → S02 given by P1 f = tf(∇2 f + Pf ).
Now Vλ′ = Cn+1 = standard representation of SO(n + 1).
Have Λ0 ⊛ Vλ′ = Cn+1 and Λ1 ⊛ Vλ′ = S02 Cn+1 , so corresponding
gradient is D : T ∗ X → S02 T ∗ X . This is the conformal Killing
operator D = tf Sym ∇.
Prescription instructs to decompose under SO(n):
Cn+1 = C ⊕ Cn
S02 Cn+1 = C ⊕ Cn ⊕ S02 Cn
So embed C ∞ (M) → Λ1 X |∂X ; obstruction lies in S02 T ∗ M.
Write g = r −2 (dr 2 + hr ). Take u = r −2 f (x)dr + O(r −1 ).
Try to solve Du = 0. Highest order is automatic, next order
determines O(r −1 ) term in u, next order after that gives
obstruction lying in S02 T ∗ M. (O(r s ) notation differs from before)
Remarks
1. Theorem above concerns first half of BGG sequence before
middle. Analogous construction works for second half after the
middle. One difference: for n even, there is a technical issue for
first operator after the middle. Proof not complete in this case.
2. Construction breaks down for middle operators. Extension to
middle operators is under investigation.
3. Have not yet shown that our Pk are always nonzero. In flat
case, Pk is a composition of generalized gradients on M. Must
show the composition is nonzero. Reduces to particular
representation-theoretic question. Trying to prove it.
4. Have not shown that our Pk form a complex in flat case. Have
an outline of an argument; working on details.
5. Construction only uses Poincaré metric expansion up to second
order: hr = h − Ph r 2 + . . .. In particular, works for arbitrarily high
order operators for n even, unlike construction of conformal powers
of Laplacian.
Remarks
6. Further comparison of BGG operator construction versus powers
of Laplacian construction
7. Comparison to Čap-Slovak-Soucek construction of BGG
operators
Application: Discrete Series for SO(n + 1, 1)
G a Lie group. An irreducible unitary representation of G is
discrete if it is unitarily equivalent to a subrepresentation of L2 (G ).
If discrete representations exist, they are arguably the most
fundamental irreducible unitary representations.
If G is compact, the Peter-Weyl Theorem says all are discrete.
Many noncompact groups have no discrete representations.
SO(n + 1, 1) has discrete representations if and only if n is odd.
Example: Realize Hn+1 = G /K . Then
n+1
ω ∈ L2 (Hn+1 , Λ+2 ) : dω = 0
is a discrete representation.
For G semisimple, the set of discrete representations of G is called
its discrete series.
Discrete Series for SO(n + 1, 1), n odd
There are various constructions of the discrete series of semisimple
G . Interested here in that of Atiyah-Schmid (1977).
Atiyah-Schmid realize the discrete series as spaces of L2 harmonic
spinors for twisted Dirac operators on G /K .
Let S± be the half-spin representations of K = Spin(n + 1). Let Vµ
be a finite-dimensional irreducible representation of Spin(n + 1).
There is a twisted Dirac operator ð+ : S+ ⊗ Vµ → S− ⊗ Vµ .
Atiyah-Schmid show that for appropriate µ,
Vµ := u ∈ L2 (Hn+1 , S+ ⊗ Vµ ) : ð+ u = 0
is a discrete representation, and the full discrete series is obtained
upon varying µ. They also show that any such u is actually a
section of S+ ⊛ Vµ . It follows that any such u is also in the kernel
of the gradient D : S+ ⊛ Vµ → S− ⊛ Vµ . Now this D is one of the
gradients in our construction of BGG operators. It is the first
operator after the middle in its sequence. By our construction, its
boundary value bu must be in the kernel of this BGG operator.
Conclusion
This is the main observation in the proof of the following:
Theorem. Let Vµ be the discrete series representation with
parameter µ. There is a SO(n + 1, 1)-equivariant boundary value
map b : Vµ → Γ(S n , Wρ ), where Wρ is the domain bundle of the
first BGG operator P after the middle in the BGG sequence with
parameter λ = µ − σ+ . Moreover, the range of b is contained in
the kernel of P.
Here σ+ denotes the highest weight of S+ .
Remark. The condition that µ generates a discrete series
representation is equivalent to the condition that λ = µ − σ+ is a
dominant integral weight of so(n + 2, C). This induces a bijection
between discrete series representations and BGG sequences.
We have formulated conditions which we anticipate characterize
the range of b. This would give an explicit embedding of the
discrete series in the principal series for SO(n + 1, 1), n odd.