The gap phenomenon in parabolic geometries

Boris Kruglikov; Dennis The

doi:10.1515/crelle-2014-0072

Article

The gap phenomenon in parabolic geometries

Boris Kruglikov and Dennis The

Published/Copyright: September 14, 2014

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2017 Issue 723

Abstract

The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a universal upper bound on the submaximal symmetry dimension. We use Kostant’s version of the Bott–Borel–Weil Theorem to show that this bound is in fact sharp in almost all complex and split-real cases by exhibiting (abstract) models. We explicitly compute all submaximal symmetry dimensions when G is any complex or split-real simple Lie group.

Funding source: Australian Research Council

Award Identifier / Grant number: DP110100416

Funding statement: Boris Kruglikov was supported by the University of Tromsø while visiting the Australian National University, where this work was initiated. The hospitality of ANU is gratefully acknowledged. Dennis The was supported under the Australian Research Council’s Discovery Projects funding scheme (project number DP110100416).

A Yamaguchi’s prolongation and rigidity theorems

Let 𝔤 be a complex simple Lie algebra and 𝔭 a parabolic subalgebra. In [50], Yamaguchi proved:^[19]

Theorem A.1

Theorem A.1 (Prolongation theorem)

We have g≅pr⁡(g-) except for:

1 -gradings: Aℓ/Pk, Bℓ/P1, Cℓ/Pℓ, Dℓ/P1, Dℓ/Pℓ, E6/P6, E7/P7.
contact gradings^[20]: Aℓ/P1,ℓ, Bℓ/P2, Cℓ/P1, Dℓ/P2, G2/P2, F4/P1, E6/P2, E7/P1, E8/P8.
(𝔤,𝔭)≅(Aℓ,P1,i) with ℓ≥3 and i≠1,ℓ, or (Cℓ,P1,ℓ) with ℓ≥2.

Moreover, we always have g≅pr⁡(g-,g0) except when (g,p)≅(Aℓ,P1) or (Cℓ,P1).

Theorem A.2

Theorem A.2 (Rigidity theorem)

We have H+2⁢(g-,g)≠0 if and only if (g,p) is:

1 -graded,
a contact gradation, or
listed in Table 8.

Table 8

Yamaguchi-nonrigid geometries, excluding 1-graded and parabolic contact geometries.

G	Range	2-graded	3-graded	4-graded	5-graded	6-graded
Aℓ	ℓ≥4	P1,s,P2,s,Ps,s+1	P1,2,s,P1,s,ℓ	P1,2,s,t	-	-
	ℓ=3	P1,2	P1,2,3	-	-	-
Bℓ	ℓ≥4	P3,Pℓ	P1,2	P2,3	-	-
	ℓ=3	P3	P1,2,P1,3	P2,3	P1,2,3	-
	ℓ=2	-	P1,2	-	-	-
Cℓ	ℓ≥4	P2,Pℓ-1	P1,ℓ,P2,ℓ,Pℓ-1,ℓ	P1,2	P1,2,ℓ	P1,2,s(s<ℓ)
	ℓ=3	P2	P1,3,P2,3	P1,2	P1,2,3	-
Dℓ	ℓ≥5	P3,P1,ℓ	P1,2	P2,3,P1,2,ℓ	-	-
	ℓ=4	P1,4	P1,2	P1,2,4	-	-
G2	-	-	P1	-	P1,2	-

B Dynkin diagram recipes

We use the notations of Section 3.2. The following Dynkin diagram recipes are well known [1, 11].

Recipe 1

Deleting the I𝔭 nodes from 𝒟⁢(𝔤,𝔭) yields 𝒟⁢(𝔤0ss), and dim⁡(𝔷⁢(𝔤0))=|I𝔭|. Also,

dim⁡(𝔤-)=12⁢(dim⁡(𝔤)-dim⁡(𝔤0)),dim⁡(𝔭)=12⁢(dim⁡(𝔤)+dim⁡(𝔤0)).

Recipe 2

Let λ∈𝔥* be the weight with coefficient ri=〈λ,αj∨〉 inscribed on the i-th node of 𝒟⁢(𝔤). To compute σj⁢(λ), add rj to adjacent coefficients, with multiplicity if there is a multiple edge directed towards the adjacent node; then replace rj by -rj.

Recipe 3

We have w=(j⁢k)∈W𝔭⁢(2) if and only if j∈I𝔭 and j≠k∈I𝔭∪𝒩⁢(j), where 𝒩⁢(j)={i∣ci⁢j≤-1}.

By the “minus lowest weight” convention, the 𝔤0-irrep 𝕍μ with lowest weight μ is denoted by the Dynkin diagram notation for -μ. Below is Kostant’s theorem^[21] for H2⁢(𝔤-,𝕌).

Recipe 4

Let 𝕌 be a 𝔤-irrep with minus lowest weight λ. Then

H2⁢(𝔤-,𝕌)≅⊕w∈W𝔭⁢(2)𝕍-w⋅λ

as 𝔤0-modules. Let w=(j⁢k)∈W𝔭⁢(2), so Φw={αj,σj⁢(αk)}. Via the isomorphism

H2⁢(𝔤-,𝕌)≅ker⁡(□)⊂⋀2𝔤-*⊗𝕌,

the 𝔤0-module 𝕍-w⋅λ has the unique (up to scale) lowest weight vector

(B.1)ϕ0:=eαj∧eσj⁢(αk)⊗v,

where eγ∈𝔤γ are root vectors, and v∈𝕌 is a weight vector with weight w⁢(-λ).

Let -μ be 𝔭-dominant, and define Jμ:={j∉I𝔭∣〈μ,αj∨〉≠0}.

Recipe 5

Let ϕ0∈𝕍μ be a lowest weight vector, 𝔞0=𝔞⁢𝔫⁢𝔫⁢(ϕ0), and 𝔭μop:=(𝔤0ss)≤0 be the (opposite) parabolic in 𝔤0ss in the ZJμ grading. Then

dim⁡(𝔞0)=max⁡{dim⁡(𝔞⁢𝔫⁢𝔫⁢(ϕ))∣0≠ϕ∈𝕍μ}

with

(B.2)𝔞0={H∈𝔥∣μ⁢(H)=0}⊕⊕γ∈Δ⁢(𝔤0,≤0)𝔤γ,

(B.3)dim⁡(𝔞0)=dim⁡(𝔭μop)+|I𝔭|-1.

The notations and recipes below are new (see Section 3). Define

Iμ:={j∈I𝔭∣〈μ,αj∨〉=0}.

Denote by 𝒟⁢(𝔤,𝔭,μ) the Dynkin diagram notation for -μ marked with an asterisk on uncrossed nodes with a nonzero coefficient, and a square on crossed nodes with a zero coefficient.
Let 𝔞⁢(μ):=pr𝔤⁡(𝔤-,𝔞⁢𝔫⁢𝔫⁢(ϕ0)), where ϕ0 is a lowest weight vector in 𝕍μ.
If 𝔤 is simple, w∈W𝔭⁢(2), and μ=-w⋅λ𝔤, write Jw:=Jμ, Iw:=Iμ, 𝔞⁢(w)=𝔞⁢(μ), etc.

Recipe 6

We have that (𝔤,𝔭,μ) is PR if and only if Iμ=∅ if and only if 𝒟⁢(𝔤,𝔭,μ) has no squares.

Recipe 7

If Iμ≠∅, then on 𝒟⁢(𝔤,𝔭,μ), remove all nodes corresponding to I𝔭\Iμ and Jμ, and any edges connected to these nodes. In the resulting diagram, remove any connected components which do not contain an Iμ node. This is 𝒟⁢(𝔤¯,𝔭¯) for the reduced geometry (𝔤¯,𝔭¯), where 𝔭¯ corresponds to crosses on the Iμ nodes, and

dim⁡(𝔞k⁢(μ))=dim⁡(𝔤¯k) for all k>0.

Combine with Recipes 1 and 5 to compute 𝔘μ=dim⁡(𝔞⁢(μ))=dim⁡(𝔤-)+dim⁡(𝔞0)+dim⁡(𝔞+).

Example B.1

Example B.1 (G2/P1)

The highest weight of 𝔤=Lie⁡(G2) is

For G2/P1, i.e.

the grading element is Z=Z1, 𝔤 is 3-graded, 𝔷⁢(𝔤0)=span⁢{Z1}, 𝔤0ss≅𝔰⁢𝔩2⁢(ℂ), and W𝔭⁢(2)={(12)}. Given w=(12), we compute w⋅λ𝔤=w⁢(λ𝔤+ρ)-ρ using Recipe 2:

so by the minus lowest weight convention,

has homogeneity +4, and W𝔭⁢(2)=W+𝔭⁢(2). Since the lowest root of 𝔤1 is α1=2⁢λ1-λ2, we have, as a 𝔤0≅𝔤⁢𝔩2⁢(ℂ) module,

This recovers Cartan’s result [12] stating that the fundamental (harmonic) curvature tensor for (2,3,5)-geometries is a binary quartic field defined on the 2-plane distribution. Furthermore,

Φw={α1,3⁢α1+α2},w⁢(-λ𝔤)=3⁢λ1-2⁢λ2=-α2.

Thus,

H2⁢(𝔤-,𝔤)≅ker⁡(□)⊂⋀2𝔤-*⊗𝔤

has lowest weight vector

ϕ0=eα1∧e3⁢α1+α2⊗e-α2.

For

Jw={2}, Iw=∅, i.e. no squares. Thus, G2/P1 is PR, so 𝔞⁢(w)=𝔤-⊕𝔞0. Hence, 𝔭wop≅𝔭1⊂A1, so

dim⁡(𝔞0)=dim⁡(𝔭wop)=2.

Thus,

𝔖=dim⁡(𝔞⁢(w))=dim⁡(𝔤-)+dim⁡(𝔞0)=5+2=7.

See the examples in Section 3.3 for applications of Recipe 7.

C Submaximal symmetry dimensions

The tables to follow summarize data associated to the gap problem for all regular, normal parabolic geometries of complex or split-real type (G,P) with Gsimple. Let 𝔖 be the submaximal symmetry dimension, and define 𝔖w similarly but require im⁢(κH)⊂𝕍-w⋅λ𝔤. Then

𝔖=maxw∈W+𝔭⁢(2)⁡𝔖w.

For each w∈W+𝔭⁢(2), compute 𝔘w=dim⁡(𝔞⁢(w)) by Recipe 7. Other than the exceptional A2/P and B2/P case (Table 9), we always have 𝔖w=𝔘w, cf. Theorem 4.1.6. Also, by Remark 4.1.10, we may simplify the calculation of 𝔖w by always passing to the minimal twistor space.

Note that 𝔞1=0 in Tables 9–11.

Table 9

Submaximal symmetry dimensions for geometries of type A2/P and B2/P.

G	P	dim⁡(G/P)	w	Jw	Twistor space type	𝔖w	𝔘w
A2	P1	2	(12)	{2}	-	3	4
	P1,2	3	(12)	∅	A2/P1	3	4
			(21)	∅	A2/P2	3	4
B2	P1	3	(12)	{2}	-	4	5
	P2	3	(21)	{1}	-	5	5
	P1,2	4	(12)	∅	B2/P1	4	5
			(21)	∅	B2/P2	5	5

Table 10

Submaximal symmetry dimensions for 1-graded geometries.

G	P	Range	dim⁡(G/P)	w	Jw	𝔖w=𝔘w
Aℓ	P1	ℓ≥3	ℓ	(12)	{2,3,ℓ}	(ℓ-1)2+4
	P2	ℓ≥3	2⁢(ℓ-1)	(21)	{3,ℓ}	(ℓ-1)2+5
				(23)	{{1,4,ℓ},ℓ≥4,{1},ℓ=3,	{ℓ2-3⁢ℓ+10,ℓ≥4,9,ℓ=3,
	Pk	3≤k≤⌈ℓ2⌉	k⁢(ℓ+1-k)	(k,k+1)	{1,k-1,k+2,ℓ}	ℓ⁢(ℓ-1)-k⁢(ℓ-k)+6
				(k,k-1)	{1,k-2,k+1,ℓ}	ℓ⁢(ℓ-k)+(k-1)2+6
Bℓ	P1	ℓ≥3	2⁢ℓ-1	(12)	{3}	2⁢ℓ2-5⁢ℓ+9
Cℓ	Pℓ	ℓ≥3	(ℓ+12)	(ℓ,ℓ-1)	{1,ℓ-2,ℓ-1}	ℓ⁢(3⁢ℓ-5)2+5
Dℓ	P1	ℓ≥4	2⁢ℓ-2	(12)	{{3},ℓ≥5,{3,4},ℓ=4,	2⁢ℓ2-7⁢ℓ+12
	Pℓ	ℓ≥5	(ℓ2)	(ℓ,ℓ-2)	{2,ℓ-3,ℓ-1}	ℓ⁢(3⁢ℓ-11)2+16
E6	P6	-	16	(65)	{2,4}	45
E7	P7	-	27	(76)	{1,5}	76

Table 11

Submaximal symmetry dimensions for parabolic contact geometries.

G	P	Range	dim⁡(G/P)	w	Jw	𝔖w=𝔘w
Aℓ	P1,ℓ	ℓ≥3	2⁢ℓ-1	(1,ℓ)	{2,ℓ-1}	(ℓ-1)2+4
				(12)	{{2,3},ℓ≥4,{2},ℓ=3,	(ℓ-1)2+4
				(ℓ,ℓ-1)	{{ℓ-2,ℓ-1},ℓ≥4,{2},ℓ=3,	(ℓ-1)2+4
Bℓ	P2	ℓ≥3	4⁢ℓ-5	(21)	{1,3}	2⁢ℓ2-5⁢ℓ+8
				(23)	{{1,3,4},ℓ≥4,{1,3},ℓ=3,	{2⁢ℓ2-7⁢ℓ+15,ℓ≥4,11,ℓ=3,
Cℓ	P1	ℓ≥2	2⁢ℓ-1	(12)	{{2,3},ℓ≥3,{2},ℓ=2,	{2⁢ℓ2-5⁢ℓ+8,ℓ≥3,5,ℓ=2,
Dℓ	P2	ℓ≥4	4⁢ℓ-7	(21)	{{1,3},ℓ≥5,{1,3,4},ℓ=4,	2⁢ℓ2-7⁢ℓ+11
				(23)	{{1,3,4},ℓ≠5,{1,3,4,5},ℓ=5,	2⁢ℓ2-9⁢ℓ+19
		ℓ=4		(24)	{1,3,4}	15
G2	P2	-	5	(21)	{1}	7
F4	P1	-	15	(12)	{2,3}	28
E6	P2	-	21	(24)	{3,4,5}	43
E7	P1	-	33	(13)	{3,4}	76
E8	P8	-	57	(87)	{6,7}	147

Table 12

Submaximal symmetry dimensions for Yamaguchi-nonrigid geometries in type A and B, excluding 1-graded and parabolic contact geometries.

G	P	Range	dim⁡(G/P)	w	Jw	Iw	(𝔤¯,𝔭¯)	𝔞1	𝔞2	Twistor space type	𝔖w=𝔘w
Aℓ	P1,2	ℓ≥3	2⁢ℓ-1	(21)	{3,ℓ}	{1}	A1/P1	✓	-	Aℓ/P2	(ℓ-1)2+5
				(12)	{3,ℓ}	∅	-	-	-	Aℓ/P1	(ℓ-1)2+4
		ℓ=3		(23)	∅	∅	-	-	-	A3/P2	9
	P1,3	ℓ≥4	3⁢ℓ-4	(31)	{2,4,ℓ}	∅	-	-	-	-	ℓ2-3⁢ℓ+9
				(12)	{2,ℓ}	∅	-	-	-	Aℓ/P1	(ℓ-1)2+4
		ℓ=4		(34)	{2}	∅	-	-	-	A4/P3	14
	P1,s	4≤s≤ℓ-1	ℓ⁢s-(s-1)2	(12)	{2,3,ℓ}	{s}	Aℓ-4/Ps-3	✓	-	Aℓ/P1	(ℓ-1)2+4
				(1,s)	{2,s-1,s+1,ℓ}	∅	-	-	-	-	(ℓ-2)⁢s+(ℓ-s)2+6
		4≤s=ℓ-1		(ℓ-1,ℓ)	{ℓ-2}	∅	-	-	-	Aℓ/Pℓ-1	(ℓ-1)2+5
	P2,s	4≤s≤ℓ-1	s⁢(ℓ+3-s)-4	(21)	{3,ℓ}	{s}	Aℓ-4/Ps-3	✓	-	Aℓ/P2	(ℓ-1)2+5
		4≤s=ℓ-1		(ℓ-1,ℓ)	{1,ℓ-2}	{2}	Aℓ-4/P1	✓	-	Aℓ/Pℓ-1	(ℓ-1)2+5
	Ps,s+1	2≤s≤ℓ-2	(ℓ-s)⁢(s+1)+s	(s,s+1)	{1,s-1,s+2,ℓ}	{s+1}	A1/P1	✓	-	Aℓ/Ps	ℓ⁢(ℓ-1)-s⁢(ℓ-s)+6
				(s+1,s)	{1,s-1,s+2,ℓ}	{s}	A1/P1	✓	-	Aℓ/Ps+1	ℓ⁢(ℓ-1)-s⁢(ℓ-s)+6
		s=2≤ℓ-2		(21)	{ℓ}	∅	-	-	-	Aℓ/P2	(ℓ-1)2+5
	P1,s,ℓ	3≤s≤ℓ-2	ℓ-1+(ℓ+1-s)⁢s	(1,ℓ)	{2,ℓ-1}	{s}	Aℓ-4/Ps-2	✓	-	Aℓ/P1,ℓ	(ℓ-1)2+4
	P1,2,3	ℓ≥3	3⁢(ℓ-1)	(21)	{ℓ}\{3}	{1}	A1/P1	✓	-	Aℓ/P2	(ℓ-1)2+5
				(12)	{ℓ}\{3}	∅	-	-	-	Aℓ/P1	(ℓ-1)2+4
		ℓ=3	6	(23)	∅	{3}	A1/P1	✓	-	A3/P2	9
				(32)	∅	∅	-	-	-	A3/P3	8
				(13)	∅	∅	-	-	-	A3/P1,3	8
	P1,2,s	4≤s<ℓ	s⁢(ℓ+3-s)-3	(21)	{3,ℓ}	{1,s}	A1/P1×Aℓ-4/Ps-3	✓	-	Aℓ/P2	(ℓ-1)2+5
				(12)	{3,ℓ}	{s}	Aℓ-4/Ps-3	✓	-	Aℓ/P1	(ℓ-1)2+4
	P1,2,ℓ	ℓ≥4	3⁢(ℓ-1)	(21)	{3}	{1}	A1/P1	✓	-	Aℓ/P2	(ℓ-1)2+5
				(12)	{3}	∅	-	-	-	Aℓ/P1	(ℓ-1)2+4
				(1,ℓ)	{ℓ-1}	∅	-	-	-	Aℓ/P1,ℓ	(ℓ-1)2+4
	P1,2,3,s	4≤s<ℓ	3⁢(ℓ-1)+(ℓ+1-s)⁢(s-3)	(21)	{ℓ}	{1,s}	A1/P1×Aℓ-4/Ps-3	✓	-	Aℓ/P2	(ℓ-1)2+5
	P1,2,3,ℓ	ℓ≥4	4⁢ℓ-6	(21)	∅	{1}	A1/P1	✓	-	Aℓ/P2	(ℓ-1)2+5
		ℓ=4		(34)	∅	{4}	A1/P1	✓	-	Aℓ/P3	14
	P1,2,s,t	4≤s<t<ℓ	2⁢ℓ-1+(ℓ+1-s)⁢(s-2)	(21)	{3,ℓ}	{1,s,t}	A1/P1×Aℓ-4/Ps-3,t-3	✓	✓	Aℓ/P2	(ℓ-1)2+5
			+(ℓ+1-t)⁢(t-s)
	P1,2,s,ℓ	4≤s<ℓ	3⁢(ℓ-1)+(ℓ-s)⁢(s-2)	(21)	{3}	{1,s}	A1/P1×Aℓ-4/Ps-3	✓	-	Aℓ/P2	(ℓ-1)2+5
		4≤s=ℓ-1		(ℓ-1,ℓ)	{ℓ-2}	{2,ℓ}	Aℓ-4/P1×A1/P1	✓	-	Aℓ/Pℓ-1	(ℓ-1)2+5
Bℓ	P3	ℓ≥4	6⁢ℓ-12	(32)	{1,4}	∅	-	-	-	-	2⁢ℓ2-7⁢ℓ+16
		ℓ=3		(32)	{1,2}	∅	-	-	-	-	11
	Pℓ	ℓ≥4	(ℓ+12)	(ℓ,ℓ-1)	{2,ℓ-2,ℓ-1}	∅	-	-	-	-	ℓ⁢(3⁢ℓ-7)2+10
	P1,2	ℓ≥3	4⁢ℓ-4	(12)	{3}	{2}	A1/P1	✓	-	Bℓ/P1	2⁢ℓ2-5⁢ℓ+9
				(21)	{3}	∅	-	-	-	Bℓ/P2	2⁢ℓ2-5⁢ℓ+8
	P2,3	ℓ≥4	6⁢ℓ-10	(32)	{1,4}	{2}	A1/P1	✓	-	Bℓ/P3	2⁢ℓ2-7⁢ℓ+16
		ℓ=3		(32)	{1}	∅	-	-	-	B3/P3	11
	P1,3	ℓ=3	8	(32)	{2}	∅	-	-	-	B3/P3	11
	P1,2,3	ℓ=3	9	(32)	∅	∅	-	-	-	B3/P3	11

Table 13

Submaximal symmetry dimensions for Yamaguchi-nonrigid geometries in type C, D, G, excluding 1-graded and parabolic contact geometries.

G	P	Range	dim⁡(G/P)	w	Jw	Iw	(𝔤¯,𝔭¯)	𝔞1	𝔞2	Twistor space type	𝔖w=𝔘w
Cℓ	P2	ℓ≥3	4⁢ℓ-5	(21)	{3}	∅	-	-	-	-	2⁢ℓ2-5⁢ℓ+9
		ℓ=3		(23)	{1,3}	∅	-	-	-	-	11
	Pℓ-1	ℓ≥4	(ℓ+4)⁢(ℓ-1)2	(ℓ-1,ℓ)	{1,ℓ-2,ℓ}	∅	-	-	-	-	ℓ⁢(3⁢ℓ-5)2+5
	P1,2	ℓ≥3	4⁢ℓ-4	(21)	{3}	{1}	A1/P1	✓	-	Cℓ/P2	2⁢ℓ2-5⁢ℓ+9
				(12)	{3}	∅	-	-	-	Cℓ/P1	2⁢ℓ2-5⁢ℓ+8
	P1,ℓ	ℓ≥3	ℓ2+3⁢ℓ-22	(1,ℓ)	{2,ℓ-1}	∅	-	-	-	-	ℓ⁢(3⁢ℓ-5)2+5
		ℓ≥4		(12)	{2,3}	{ℓ}	Cℓ-3/Pℓ-3	✓	-	Cℓ/P1	2⁢ℓ2-5⁢ℓ+8
		ℓ=3		(12)	{2}	∅	-	-	-	C3/P1	11
	P2,ℓ	ℓ≥4	ℓ2+5⁢ℓ-82	(21)	{3}	{ℓ}	Cℓ-3/Pℓ-3	✓	-	Cℓ/P2	2⁢ℓ2-5⁢ℓ+9
		ℓ=3		(21)	∅	∅	-	-	-	C3/P2	12
		ℓ=3		(23)	{1}	∅	-	-	-	C3/P2	11
	Pℓ-1,ℓ	ℓ≥4	ℓ2+3⁢ℓ-22	(ℓ-1,ℓ)	{1,ℓ-2}	∅	-	-	-	Cℓ/Pℓ-1	ℓ⁢(3⁢ℓ-5)2+5
	P1,2,3	ℓ≥3	6⁢ℓ-9	(21)	∅	{1}	A1/P1	✓	-	Cℓ/P2	2⁢ℓ2-5⁢ℓ+9
	P1,2,s	4≤s<ℓ	-6-3⁢s2+5⁢s+4⁢s⁢ℓ2	(21)	{3}	{1,s}	A1/P1×Cℓ-3/Ps-3	✓	✓	Cℓ/P2	2⁢ℓ2-5⁢ℓ+9
	P1,2,ℓ	ℓ≥4	ℓ2+5⁢ℓ-62	(21)	{3}	{1,ℓ}	A1/P1×Cℓ-3/Pℓ-3	✓	-	Cℓ/P2	2⁢ℓ2-5⁢ℓ+9
Dℓ	P3	ℓ≥5	6⁢ℓ-15	(32)	{{1,4},ℓ≥6,{1,4,5},ℓ=5,	∅	-	-	-	-	2⁢ℓ2-9⁢ℓ+20
	P1,2	ℓ≥4	4⁢ℓ-6	(12)	{{3},ℓ≥5,{3,4},ℓ=4,	{2}	A1/P1	✓	-	Dℓ/P1	2⁢ℓ2-7⁢ℓ+12
				(21)	{{3},ℓ≥5,{3,4},ℓ=4,	∅	-	-	-	Dℓ/P2	2⁢ℓ2-7⁢ℓ+11
	P1,ℓ	ℓ≥5	(ℓ+2)⁢(ℓ-1)2	(12)	{3}	{ℓ}	Dℓ-3/Pℓ-3	✓	-	Dℓ/P1	2⁢ℓ2-7⁢ℓ+12
		ℓ=4		(12)	{3}	∅	-	-	-	D4/P1	16
				(42)	{3}	∅	-	-	-	D4/P4	16
	P2,3	ℓ≥5	6⁢ℓ-13	(32)	{{1,4},ℓ≥6,{1,4,5},ℓ=5,	{2}	A1/P1	✓	-	Dℓ/P3	2⁢ℓ2-9⁢ℓ+20
	P1,2,ℓ	ℓ≥5	ℓ2+3⁢ℓ-62	(12)	{3}	{2,ℓ}	A1/P1×Dℓ-3/Pℓ-3	✓	-	Dℓ/P1	2⁢ℓ2-7⁢ℓ+12
		ℓ=4		(12)	{3}	{2}	A1/P1	✓	-	D4/P1	16
				(42)	{3}	{2}	A1/P1	✓	-	D4/P4	16
G2	P1	-	5	(12)	{2}	∅	-	-	-	-	7
	P1,2	-	6	(12)	∅	∅	-	-	-	G2/P1	7

D NPR geometries

The classification of NPR geometries in Table 4 follows immediately from the results in Appendix C, but as we do not show the extensive calculations there, we sketch here an independent proof.

By Proposition 3.4.7 and Corollary 3.4.8, if G is of exceptional type or |I𝔭|=1, then G/P is PR. Thus, we henceforth assume that G/P is a classical complex flag variety, with |I𝔭|≥2 (and G simple). Regularity Z⁢(-w⋅λ𝔤)≥1 is equivalent to (3.4) which simplifies in each case of (3.5) to

(D.1){(a)Z⁢(λ𝔤)≤rj-(rk+1)⁢ck⁢j,(b)Z⁢(λ𝔤)≤rj+(rk+1)⁢(1-ck⁢j),(c)Z⁢(λ𝔤)≤2⁢(ri+1).

The proof is simply an analysis of (3.5) and (D.1). We leave the Aℓ and Cℓ cases to the reader.

The 𝑩ℓ or 𝑫ℓ case

Here, ck⁢j∈{0,-1} if and only if 𝔤=Dℓ. Also, 0≤rj+rk≤1 (since j≠k) using

(D.2)λ𝔤=λ2={α1+2⁢α2+…+2⁢αℓ,𝔤=Bℓ(ℓ≥3),α1+2⁢α2+…+2⁢αℓ-2+αℓ-1+αℓ,𝔤=Dℓ(ℓ≥4).

(a) Assume ck⁢j=-2, so 𝔤=Bℓ and (k,j)=(ℓ-1,ℓ). From (3.5), ck⁢i=0, so ℓ≥4, rj=rk=0. From (D.1), |I𝔭|=Z⁢(λ𝔤)=2, which contradicts (D.2). Thus, ck⁢j∈{0,-1}. Then rj-(rk+1)⁢ck⁢j≤2, so from (D.1), |I𝔭|=Z⁢(λ𝔤)=2 and exactly one of j,k is 2. If 𝔤=Bℓ or j=2, then Z⁢(λ𝔤)≥3, so 𝔤=Dℓ and k=2∉I𝔭. Hence,

Dℓ/P1,ℓ,ℓ≥5,w=(12),i=ℓ.

(b) Assume 𝔤=Bℓ. From (D.1), |I𝔭|≥3, so Z⁢(λ𝔤)≥5. But since 0≤rj+rk≤1 and ck⁢j≤-1, we have rj+(rk+1)⁢(1-ck⁢j)≥5 only if rj=0, rk=1, ck⁢j=-2, and hence (k,j)=(ℓ-1,ℓ). From (3.5), ck⁢i=0, so ℓ≥4, rj=rk=0. From (D.1), Z⁢(λ𝔤)=3, a contradiction.

Thus, 𝔤=Dℓ. Assuming rj=1, then rk=0 and (D.1) implies |I𝔭|=Z⁢(λ𝔤)=3. But j=2∈I𝔭, so by (D.2), Z⁢(λ𝔤)≥4, a contradiction. Hence, rj=0, and (D.1) implies rk=1, ck⁢j=-1, and 3≤|I𝔭|≤Z⁢(λ𝔤)≤4. From (D.2), I𝔭={1,2,ℓ}, so

Dℓ/P1,2,ℓ,ℓ≥5,w=(12),i=ℓ.

(c) Assume ri=0, so from (D.1), |I𝔭|=Z⁢(λ𝔤)=2. Using (D.2), this is impossible given ci⁢j=-1 by (3.5). Thus, ri=1, and 2≤|I𝔭|≤Z⁢(λ𝔤)≤4. Keeping in mind ci⁢j=-1, we have:

Bℓ/P1,2, ℓ≥3, w=(12), i=2, and Dℓ/P1,2, ℓ≥4, w=(12), i=2.
Bℓ/P2,3, ℓ≥4, w=(32), i=2, and Dℓ/P2,3, ℓ≥4, w=(32), i=2.
Dℓ/P1,2,ℓ, ℓ≥5, w=(12), i=2. Also, D4/P1,2,4, w=(12) or (42), i=2.

Acknowledgements

We are grateful for many helpful discussions with Boris Doubrov, Mike Eastwood, Katharina Neusser, Katja Sagerschnig, and Travis Willse. Much progress on this paper was made during the conference “The Interaction of Geometry and Representation Theory: Exploring New Frontiers” devoted to Mike Eastwood’s 60th birthday, and held in Vienna in September 2012 at the Erwin Schrödinger Institute. Boris Doubrov gave some key insights during this conference which led to the proof of Proposition 3.1.1. The representation theory software LiE and Ian Anderson’s DifferentialGeometry package in Maple were invaluable tools for facilitating the analysis in this paper.

References

[1] Baston R. J. and Eastwood M. G., The Penrose transform: Its interaction with representation theory, Oxford Math. Monogr., Clarendon Press, Oxford 1989. Search in Google Scholar

[2] Bianchi L., Sugli spazi a tre dimensioni che ammettono un gruppo continuo di movimenti, Mem. Soc. Ital. Sci. (3) 11 (1898), 267–352. Search in Google Scholar

[3] Bourbaki N., Éléments de mathématique. Groupes et algèbres de Lie. Chapitres 4 à 6, Hermann, Paris 1968. Search in Google Scholar

[4] Bryant R. L., Some aspects of the local and global theory of Pfaffian systems, Ph.D. thesis, University of North Carolina at Chapel Hill, 1979. Search in Google Scholar

[5] Bryant R. L., Conformal geometry and 3-plane fields on 6-manifolds, Developments of Cartan geometry and related mathematical problems, RIMS Sympos. Proc. Kyoto Univ. 1502, Research Institute for Mathematical Sciences, Kyoto (2006), 1–15. Search in Google Scholar

[6] Bryant R. L., Eastwood M. G., Gover A. R. and Neusser K., Some differential complexes within and beyond parabolic geometry, preprint 2012, http://arxiv.org/abs/1112.2142. Search in Google Scholar

[7] Čap A., Automorphism groups of parabolic geometries, Rend. Circ. Mat. Palermo (2) Suppl. 75 (2005), 233–239. Search in Google Scholar

[8] Čap A., Correspondence spaces and twistor spaces for parabolic geometries, J. reine angew. Math 582 (2005), 143–172. 10.1515/crll.2005.2005.582.143Search in Google Scholar

[9] Čap A. and Neusser K., On automorphism groups of some types of generic distributions, Differential Geom. Appl. 27 (2009), no. 6, 769–779. 10.1016/j.difgeo.2009.05.003Search in Google Scholar

[10] Čap A. and Schichl H., Parabolic geometries and canonical Cartan connections, Hokkaido Math. J. 29 (2000), no. 3, 453–505. 10.14492/hokmj/1350912986Search in Google Scholar

[11] Čap A. and Slovák J., Parabolic geometries I: Background and general theory, Math. Surveys Monogr. 154, American Mathematical Society, Providence 2009. 10.1090/surv/154Search in Google Scholar

[12] Cartan E., Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Sci. Éc. Norm. Supér. (3) 27 (1910), 109–192. 10.24033/asens.618Search in Google Scholar

[13] Cartan E., Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes. I, Ann. Mat. Pura Appl. (4) 11 (1932), 17–90. 10.1007/BF02417822Search in Google Scholar

[14] Casey S., Dunajski M. and Tod P., Twistor geometry of a pair of second order ODEs, Comm. Math. Phys. 321 (2013), 681–701. 10.1007/s00220-013-1729-7Search in Google Scholar

[15] Chern S. S., The geometry of the differential equation y′′′=F⁢(x,y,y′,y′′), Sci. Rep. Nat. Tsing Hua Univ. (A) 4 (1940), 97–111. Search in Google Scholar

[16] Darboux G., Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. III partie et IV partie, Gauthier-Villar, Paris 1894/1896. Search in Google Scholar

[17] Doubrov B. and The D., Maximally degenerate Weyl tensors in Riemannian and Lorentzian signatures, Differential Geom. Appl. 34 (2014), 25–44. 10.1016/j.difgeo.2014.03.007Search in Google Scholar

[18] Egorov I. P., Collineations of projectively connected spaces (Russian), Dokl. Akad. Nauk SSSR 80 (1951), 709–712. Search in Google Scholar

[19] Egorov I. P., Maximally mobile Riemannian spaces V4 of nonconstant curvature (Russian), Dokl. Akad. Nauk SSSR 103 (1955), 9–12. Search in Google Scholar

[20] Egorov I. P., Riemannian spaces of second lacunarity (Russian), Dokl. Akad. Nauk SSSR 111 (1956), 276–279. Search in Google Scholar

[21] Egorov I. P., Maximally mobile Einstein spaces of nonconstant curvature (Russian), Dokl. Akad. Nauk SSSR 145 (1962), 975–978; translation in Dokl. Sov. Math. 3 (1962), 1124–1127. Search in Google Scholar

[22] Egorov I. P., Automorphisms in generalized spaces (Russian), Itogi Nauki Tekh. Probl. Geom. 10 (1978), 147–191; translation in J. Sov. Math. 14, no. 3 (1980), 1260–1287. 10.1007/BF01095473Search in Google Scholar

[23] Fels M. E., The equivalence problem for systems of second-order ordinary differential equations, Proc. Lond. Math. Soc. (3) 71 (1995), 221–240. 10.1112/plms/s3-71.1.221Search in Google Scholar

[24] Fubini G., Sugli spazi che ammettono un gruppo continuo di movimenti, Ann. Mat. Pure Appl. (3) 8 (1903), 39–81. 10.1007/BF02419264Search in Google Scholar

[25] Godlinski M. and Nurowski P., Geometry of third-order ODEs, preprint 2009, http://arxiv.org/abs/0902.4129. Search in Google Scholar

[26] Grossman D. A., Torsion-free path geometries and integrable second order ODE systems, Selecta Math. (N.S.) 6 (2000), no. 4, 399–342. 10.1007/PL00001394Search in Google Scholar

[27] Hammerl M., Homogeneous Cartan geometries, Arch. Math. (Brno) 43 (2007), no. 5, 431–442. Search in Google Scholar

[28] Kobayashi S., Transformation groups in differential geometry, Springer-Verlag, Berlin 1972. 10.1007/978-3-642-61981-6Search in Google Scholar

[29] Kostant B., Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. (2) 74 (1961), 329–387. 10.1007/b94535_13Search in Google Scholar

[30] Kruglikov B., Finite-dimensionality in Tanaka theory, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 1, 75–90. 10.1016/j.anihpc.2010.10.001Search in Google Scholar

[31] Kruglikov B., The gap phenomenon in the dimension study of finite type systems, Cent. Eur. J. Math. 10 (2012), no. 5, 1605–1618. 10.2478/s11533-012-0070-2Search in Google Scholar

[32] Lie S., Untersuchungen über geodätische Kurven, Math. Ann. 20 (1882), 357–454. 10.1007/BF01443601Search in Google Scholar

[33] LiE, available at http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/. Search in Google Scholar

[34] Morimoto T., On the intransitive Lie algebras whose transitive parts are infinite and primitive, J. Math. Soc. Japan 29 (1977), no. 1, 35–65. 10.2969/jmsj/02910035Search in Google Scholar

[35] Morimoto T., Geometric structures on filtered manifolds, Hokkaido Math. J. 22 (1993), 263–347. 10.14492/hokmj/1381413178Search in Google Scholar

[36] Nagano T., On conformal transformations of Riemannian spaces, J. Math. Soc. Japan 10 (1958), no. 1, 79–93. 10.2969/jmsj/01010079Search in Google Scholar

[37] Penrose R., A spinor approach to general relativity, Ann. Phys. 10 (1960), no. 2, 171–201. 10.1016/B978-0-08-017639-0.50017-0Search in Google Scholar

[38] Petrov A. Z., Classification of spaces defining gravitational fields (Russian), Kazan. Gos. Univ. Uč. Zap. 114 (1954), no. 8, 55–69; translation in Gen. Relativity Gravitation 32 (2000), no. 8, 1665–1685. 10.1023/A:1001910908054Search in Google Scholar

[39] Ricci G., Sur les groupes continus de mouvements d’une variété quelconque à trois dimensions, C. R. Acad. Sci. Paris 127 (1898), 344–346. Search in Google Scholar

[40] Sato H. and Yoshikawa A. Y., Third order ordinary differential equations and Legendre connections, J. Math. Soc. Japan 50 (1998), no. 4, 993–1013. 10.2969/jmsj/05040993Search in Google Scholar

[41] Šilhan J., Online LiE implementation of Kostant’s algorithm at http://bart.math.muni.cz/~silhan/lie/. Search in Google Scholar

[42] Stephani H., Kramer D., Maccallum M., Hoenselaers C. and Herlt E., Exact solutions of Einstein’s field equations, 2nd ed., Cambridge University Press, Cambridge 2003. 10.1017/CBO9780511535185Search in Google Scholar

[43] Strazzullo F., Symmetry analysis of general rank-3 Pfaffian systems in five variables, Ph.D. thesis, Utah State University, 2009. Search in Google Scholar

[44] Tanaka N., On differential systems, graded Lie algebras and pseudogroups, J. Math. Kyoto. Univ. 10 (1970), 1–82. 10.1215/kjm/1250523814Search in Google Scholar

[45] Tanaka N., On the equivalence problems associated with simple graded Lie algebras, Hokkaido Math. J. 8 (1979), 23–84. 10.14492/hokmj/1381758416Search in Google Scholar

[46] Tresse M. A., Déterminations des invariants ponctuels de l’équation différentielle ordinaire du second ordre y′′=ω⁢(x,y,y′), Hirzel, Leipzig 1896. Search in Google Scholar

[47] Wafo Soh C., Mahomed F. M. and Qu C., Contact symmetry algebras of scalar ordinary differential equations, Nonlinear Dynam. 28 (2002), 213–230. 10.1023/A:1015030931611Search in Google Scholar

[48] Wang H. C., On Finsler spaces with completely integrable equations of Killing, J. Lond. Math. Soc. 22 (1947), 5–9. 10.1112/jlms/s1-22.1.5Search in Google Scholar

[49] Wolf J. A., Spaces of constant curvature, 5th ed., Publish or Perish, Houston 1984. Search in Google Scholar

[50] Yamaguchi K., Differential systems associated with simple graded Lie algebras, Progress in differential geometry, Adv. Stud. Pure Math. 22, Kinokuniya Company, Tokyo (1993), 413–494. 10.2969/aspm/02210413Search in Google Scholar

[51] Yamaguchi K., G2-geometry of overdetermined systems of second order, Analysis and geometry in several complex variables (Katata 1997), Trends Math., Birkhäuser-Verlag, Boston (1999), 289–314. 10.1007/978-1-4612-2166-1_14Search in Google Scholar

[52] Zelenko I., On Tanaka’s prolongation procedure for filtered structures of constant type, SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 094. 10.3842/SIGMA.2009.094Search in Google Scholar

Received: 2013-5-21

Revised: 2014-5-21

Published Online: 2014-9-14

Published in Print: 2017-2-1

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