Invariant calculus for parabolic geometries

Invariant calculus for parabolic geometries Jan Slovák Masaryk University, Brno, Czech Republic joint work with Andreas Čap and others over years V. Souček, M.G. Eastwood, A.R. Gover April 8, 2009 Bent Ørsted s conference Göttingen

1 Important sources 2 Parabolic Geometries Cartan connections as analogies to affine geometry on manifolds Underlying structures 3 The BGG calculus 4 Weyl connections and Rho tensors 5 Nearly invariant calculus Invariant operators Invariant and fundamental derivatives

old masters Cartan, Thomas, Veblen, Schouten,... (conformal and projective geometries, and more) Eastwood s group Baston, Gover, Graham... (invariant operators) Russian school (induced modules etc.) Tanaka s school Morimoto, Yamaguchi,... (Cartan s methods for filtered manifolds) T. Branson, Second order conformal covariants, Proc. Amer. Math Soc. 126 (1998), 1031 1042. P. Gauduchon, Structures de Weyl-Einstein, espaces de twisteurs et variétés de type S 1 S 3, J. reine angew. Math. 469 (1995), 1 50. V. Wünsch, On conformally invariant differential operators, Math. Nachr. 129 (1986), 269 281. and many more...

our own... A. Čap, J. Slovák, V. Souček, Bernstein-Gelfand-Gelfand Sequences, Annals of Mathematics, 154 (2001), 97-113. M.G. Eastwood, J. Slovák, Semi-holonomic Verma modules, J. of Algebra 197 (1997), 424 448. A. Čap, J. Slovák, V. Souček, Invariant operators on manifolds with almost Hermitian symmetric structures, I. Invariant differentiation, II. Normal Cartan connections, Acta Math. Univ. Commenianae, 66 (1997), 33 69; 203 220; III. Standard Operators, Diff. Geom. Appl. 12 (2000), 203-220. A. Čap, J. Slovák, Parabolic Geometries I: Background and General Theory, Mathematical Surveys and Monographs, AMS Publishing House, cca 639pp, to appear, Summer 2009.

Cartan connections Definition Cartan geometries of type G/P are curved deformations of the homogeneous space G G/P with the Maurer Cartan form ω Ω 1 (G; g).

Cartan connections Definition Cartan geometries of type G/P are curved deformations of the homogeneous space G G/P with the Maurer Cartan form ω Ω 1 (G; g). Thus, a Cartan connection is an absolute parallelism ω Ω 1 (G, g) on a principal fiber bundle G M with structure group P, enjoying suitable invariance properties with respect to the principal action of P: ω(ζ X )(u) = X for all X p, u G (the connection reproduces the fundamental vertical fields) (r b ) ω = Ad(g 1 ) ω (the connection form is equivariant with respect to the principal action) ω TuG : T u G g is a linear isomorphism for all u G (the absolute parallelism condition).

Parabolic geometries Definition Parabolic geometries are Cartan connections with the choice of parabolic P in semisimple real G. Morphisms of the parabolic geometries are principal fiber bundle morphisms ϕ : G G with ϕ (ω ) = ω. Important facts: Fixed grading g = g k g 1 g 0 g 1 g k on Lie algebra g, p = g 0 g 1 g k, g 0 is the reductive part in p, also write P + = exp p + and G = exp g for the corresponding nilpotent subgroups.

Curvature Definition The structure equations K = dω + 1 2 [ω, ω] define the curvature K of the Cartan connection ω.

Curvature Definition The structure equations K = dω + 1 2 [ω, ω] define the curvature K of the Cartan connection ω. ω defines constant vector fields ω 1 (X ), X g. Tr g ω 1 (X )(u) = ω 1 (Ad g 1 X )(u g). Curvature function κ(u)(x, Y ) = K(ω 1 (X )(u), ω 1 (Y )(u)) Curvature is a horizontal 2 form. = [X, Y ] ω(u)([ω 1 (X ), ω 1 (Y )]).

Underlying structures Definition Weyl structure on a parabolic geometry (G, ω) is a G 0 equivariant section σ : G 0 = G/P + G.

Underlying structures Definition Weyl structure on a parabolic geometry (G, ω) is a G 0 equivariant section σ : G 0 = G/P + G. θ = σ ω = θ k + + θ 1 : T G 0 g k g 1 γ = σ ω 0 : T G 0 g 0 P = σ ω + = P 1 + + P k : T G 0 p + = g 1 g k. On the underlying manifold M we obtain: θ is a soldering form for M γ is a linear connection form on M, θ + γ is the affine connection, called Weyl connection P is a one form on M valued in T M

Natural bundles and tractor bundles Natural bundles P representation λ on a vector space V provides the homogeneous vector bundle G P V and, more generally, the associated vector bundles V = G P V with standard fiber V over all manifolds with a parabolic geometry of the type G/P. These are the natural bundles V. Tractor bundles The natural bundles with G modules V are called the tractor bundles.

.. traction comes after tension and also credit to Tracy Thomas! Notice: The tractor bundles come equipped with the natural connection obtained from the extension of ω to G = G P G. The adjoint representation G provides the adjoint tractor bundles A, the standard representation of a matrix group on R n provides the standard tractors T. Natural bundles and tractor bundles Natural bundles P representation λ on a vector space V provides the homogeneous vector bundle G P V and, more generally, the associated vector bundles V = G P V with standard fiber V over all manifolds with a parabolic geometry of the type G/P. These are the natural bundles V. Tractor bundles The natural bundles with G modules V are called the tractor bundles.

The invariant operators between natural bundles may exist only if they exist at the homogeneous spaces. For homogeneous bundles, the invariant operators correspond to singular vectors in induced modules. Thus, the Harish Chandra s restriction on the singular characters of the modules applies. Consequently, the complete structure of the complexes of bundles possibly linked by operators are described by cohomological arguments. The Jantzen Zuckermann principle allows to translate the operators between the patterns (in the flat Klein s world). There is a curved version: whatever may be traslated in the homogeneous model translates in the curved world as well (Semi holonomic Verma modules)

The BGG sequenses idea is very simple starting from the twisted De Rham complex by a G module V we seek for suitable natural operators L: Ω 0 (M; VM) V L Γ(HV 0M) d V Ω 1 (M; VM) L D V Γ(H 1 V M) d V... D V... such that d V L has values in ker.

Theorem Let (G, ω) be a real parabolic geometry of the type G/P on a manifold M, V be a G module. If the twisted de Rham sequence 0 Ω 0 (M; VM) d V Ω 1 (M; VM) d V... d V Ω dim(g/p) (M; VM) 0. is a complex, then also the Bernstein Gelfand Gelfand sequence 0 Γ(H 0 V M) DV Γ(H 1 V M) DV... D V Γ(H dim(g/p) V M) 0 is a complex, and they both compute the same cohomology.

Example: a Lychagin Rumin complex For all (integrable) 5 dimensional CR manifolds, there is the resolution of the sheaf of constant complex functions C E[0, 0] T 1,0 T 0,1 Λ 2 T1,0 Q Λ 2 T1,0 2 Q T1,0 (ker L) Q (ker L) 3 Q 2 Q T0,1 Λ 2 T0,1 Q Λ 2 T0,1 which computes the de Rham cohomology with complex coefficients. The orders of the operators in the column in the middle of the diagram are two, while all the other ones are of first order.

Weyl connections G Weyl connections π exact Weyl connections L G 0 reductions to G 0 M G 0 / exp te θ = σ ω = θ k + + θ 1 : T G 0 g k g 1 γ = σ ω 0 : T G 0 g 0 P = σ ω + = P 1 + + P k : T G 0 p + = g 1 g k. Exact Weyl connections reductions to G 0, normal Weyl connections exponential coordinates.

Transformation rules Change of Weyl connections If ˆσ = σ exp Υ, and q = q 1 + + q l is a section of a natural bundle, then: ˆq l = i +j=l 1 i! ( 1)i λ(υ 1 ) i 1... λ(υ k ) i k q j ˆ ξ s = ξ s + i +j=0 1 i! ( 1)i( ad(υ 1 ) i 1... ad(υ k ) i k (ξ j ) ) s Here i is a multiindex (i 1,..., i k ) with i j 0. We put i! = i 1! i k! and i = i 1 + 2i 2 + + ki k, while ( 1) i = ( 1) i 1+ +i k For 1 graded algebras and irreducible bundles ˆq = q, ˆ ξ s = ξ s + [Υ, ξ] s

Tranformations continued Change of Rho tensors If ˆσ = σ exp Υ, then: ˆP i (ξ) = ( 1) j j +l=i j! ad(υ k ) j k ad(υ 1 ) j 1 (ξ l ) + k ( 1) j m=1 j +m=i j!(j ad(υ m+1) k) j k ad(υ m ) jm ( ξ Υ m ) + j +l=i j 1 = =j m 1 =0 ( 1) j j! ad(υ k ) j k ad(υ 1 ) j 1 (P l (ξ)). For 1 graded algebras ˆP(ξ) = 1 2 [Υ, [Υ, ξ]] + ξυ + P(ξ)

Invariant operators Beside the obvious categorical definition of natural operations, there is also the following concept: Definition Invariant operators for a parabolic geometry of the type G G/P are the affine invariants of the Weyl connections which are independent of the choice of a particular Weyl structure σ. The transformation formula for the Rho tensor always starts with the derivative of Υ and so this seems to be the perfect candidate to enter correction terms to the covariant derivatives so that the transformation rule remains algebraic in Υ s.

The derivatives Invariant derivative In analogy to the covariant derivatives we define ω : C (G, V) C (G, g V), s(u)(x ) = ω 1 (X )(u) s V. This operation does not transform sections into sections, in general! Fundamental derivative The extension of the invariant derivative to argument X g D ω : C (G, V) C (G, g V), s(u)(x ) = ω 1 (X )(u) s V is and invariant differential operator A V V called the fundamental derivative.

Invariant jet operator D ω ξ+ζ s = ξs + P(ξ) s ζ s, ξ a vector in TM, ζ a vertical vector on G 0. on all tractor bundles, the fundamental derivative is related to the invariant linear connection V ξ s = ξs + P(ξ) s + ξ s. 1st order invariant jets C (G, V) s (s, ω s) C (G, V (g V)) defines a natural operator (action given on J 1 V). higher orders C (G, V) P s j k ωs = (s, ω s,..., ( ω ) k s) C (G, V ( k g V)) P is an invariant operator valued in semi holonomic jets.

Bianchi and Ricci identities Bianchi In terms of the invariant differential: 0 = ) ([κ(x, Y ), Z]+κ([X, Y ], Z) κ(κ(x, Y ), Z) ωz κ(x, Y ) cycl

Bianchi and Ricci identities Bianchi In terms of the invariant differential: 0 = ) ([κ(x, Y ), Z]+κ([X, Y ], Z) κ(κ(x, Y ), Z) ωz κ(x, Y ) cycl Ricci ω X ω Y s ω Y ω X s = ω [X,Y ] ω κ (X,Y ) s + κ 0(X, Y ) s

The expansion Proposition Every G 0 invariant expression built of the invariant semiholonomic jets of sections and the Cartan curvatures in terms of the invariant derivatives (or the fundamental derivatives) expands into universal expressions built of the iterated covariant derivatives with respect to the Weyl connections, the iterated covariant derivatives of the curvature tensor of the Cartan connection and of the iterated covariant derivatives of the P tensor.

Proposition The transformation properties of any expansion of a formula in invariant derivatives and Cartan curvatures are algebraic in Υ s.

Proposition The transformation properties of any expansion of a formula in invariant derivatives and Cartan curvatures are algebraic in Υ s. Technically, we consider two P modules W and V and we claim that the differential operators defined for all s C (G, V) P and G 0 equivariant maps F on the images of the semi holonomic jet operators, D(s) = F ( j k ω(s σ), j k ωκ) σ = F ( j k γ (s σ), j k ωρ) : G 0 W, transform algebraically in terms of Υ under the change of the Weyl structures ˆσ = σ r exp Υ.

Theorem Every operator D expressed in affine invariants of the Weyl connections with algebraic transformation rules in Υ s is an expansion of an expression in the invariant derivatives and the Cartan curvature. In other words: if we get D(s) = F (j k γ s, j k γ ρ) with algebraic transformation rules, then D(s) = F (j k ω s, j k ωκ) with universal F. Note: We should not aim at getting F back from F by expansion there are many formal expressions for the same operator!