Non-classical flag domains and Spencer resolutions

Transcription

1 1

2 Non-classical flag domains and Spencer resolutions Phillip Griffiths 1 Talk based on joint work with Mark Green

3 Outline I. Introduction II. Notations and terminology III. Equivalent forms of non-classical IV. Realization of V µ as solutions to a PDE V. Spencer sequences VI. The Spencer sequence of V µ VII. Examples

4 I. Introduction This talk will be about properties of non-classical flag domains. Following the introduction of notations and terminology it will have two parts. A list of some of the special features present in the non-classical case;

5 One of these is the identification of very regular Harish-Chandra modules, realized as the coherent cohomology of line bundles over non-classical flag domains, as the solutions to a canonical Spencer sequence from the theory of overdetermined linear partial differential equations. Properties of the Spencer sequence then translate into information about the Harish-Chandra module; e.g., the localization of its characteristic sheaf gives the K-type. Conversely, the dictionary for special Harish-Chandra modules, such as those arising from degenerate limits of discrete series, raises new and interesting questions in linear PDE theory.

6 II. Notations and terminology G C will be a semi-simple complex Lie group; G R G C will be a connected real form containing a compact maximal torus T ; K R G R is the unique maximal compact subgroup containing T ; K C and T C are the complexifications; ( ) g C = h g α is the root space decomposition of g C α Φ relative to the Cartan sub-algebra h = t C ;

7 g R = k R p R is the Cartan-decomposition; Φ = Φ c Φ nc are the compact and non-compact roots; for a system of Φ + of positive roots, or equivalently a choice of Weyl chamber, and setting p = p C { ρ = 1 2 α Φ α + p = p + p, p = p + ; For a choice of Borel subgroup B G C with T C B Ď = G C /B is a flag variety;

8 A flag domain is given equivalently by an open GR -orbit D in Ď, a choice of positive roots giving an integrable almost complex structure on G R /T = D with T 1,0 x 0 D = α Φ + g α ; Z = K R /T is a maximal compact subvariety of D and U = {gz : g G C, gz D} is the cycle space; note that Z = K C /B K where B K = B K C ;

9 N Z/D Z is the normal bundle of Z D; a weight µ it gives rise to a holomorphic character of B and then to the holomorphic line bundle L µ =: G C B C µ ; V µ = H q (D, L µ ); we will usually take q = d where d = dim Z; W G C /T C will be the correspondence space, defined below, and the basic diagrams are D W U W D D where D, D are open G R -orbits in Ď.

10 Definition The flag domain is classical if it fibres holomorphically or anti-holomorphically over an Hermitian symmetric domain. Otherwise it is non-classical. then classical is equivalent to [k C, p + ] p +, and using the Cartan-Killing form non-classical is equivalent to [p +, p + ] 0; for Γ G R a co-compact, neat subgroup X = Γ\D and the H q (X, L µ ) are automorphic cohomology groups;

11 Hirzebruch s proportionality principle gives χ(x, L µ ) = ± vol X χ(ď, L µ), where χ(ď, L µ) is known by the Borel-Weil-Bott theorem; of particular interest are the automorphic cohomology groups when µ + ρ is singular ( χ(x, L µ ) = 0), and of very particular interest are the H q (X, L ρ ) corresponding to totally degenerate limits of discrete series (TDLD s) (Henri Carayol s talk).

12 III. Equivalent forms of non-classical (assume g C is simple) Geometric p H 0 (Z, N Z/D ), U G C /K C D is Z-connected.

13 Notes g C are holomorphic vector fields on Ď and the first property is that p injects into (fibre-generating) holomorphic normal vector fields. This gives a map e : Z Gr L (c, p), dim p = 2c whose geometry measures the non-classicalness of D. 2 2 For example, e is an immersion TDLDS.

14 Z-connectivity means that any two points x, x D may be joined by a chain of Z u s, u U x x (cf. Colleen Robles talk).

15 complex geometric L D has special properties d-pseudo-concavity

16 Notes The intrinsic Levi form L D will be discussed in Mark Green s talk. Recall that pseduo-convexity means H = complex-analytic hypersurface D D

17 d-pseudo-concavity means Z dim Z = d D D In fact, it seems likely that we will have Z Z D = spherical shell in C d? = dim H q (D, L) < for 0 q < d.

18 Hodge theoretic for any realization of D as a Mumford-Tate domain the IPR 0 for the period map at infinity Φ : B(N) D, the image of Φ Gr(LMHS) s

19 Note B(N) = {limiting mixed Hodge structures (W (N), F )}, and in Mark Green s and Matt Kerr s talks they will define Φ (W (N), F ) = and explain the above. lim exp(zn) F G R-orbit in D Im z

20 H 0 (X, L µ ) = 0 for µ regular (maybe just µ 0) Cohomological lots of H q (X, L k µ ) for µ regular, q 0 for a Harish-Chandra module V the (E 1, d 1 ) term of the Serre-Hochschild spectral sequence is not a bi-complex

21 Note n = g α and n c = g α. Then the HSSS is α Φ + α Φ + c E p,q 1 = H q (n c, p p + V ) H p+q (n, V ). The complex (E 1, d 1 ) is constructed from q n c p p + (p, q).

22 A priori the coboundary is δ : (p, q) (p 1, q + 2) + (p, q + 1) + (p + 1, q) + (p + 2, q 1) out because n c is a sub-algebra out only in the classical case

23 It is the far right term that gives E p,q 1 d 2 E p+2,q 1 1. In a few examples, for V a TDLDS its vanishing picks out V from among the Harish-Chandra modules with the same K-type.

24 Representation theoretic arithmetic automorphic representations whose infinite component is a TDLDS corresponding to H d (X, L ρ ) {connections with PDE} Note The first will be discussed in Henri Carayol s and Wushi Goldring s talks. The second is the next topic.

25 IV. Realization of V µ as solutions to a linear PDE We begin with a result about U and the definition of W. Recall that D is assumed to be non-classical, so that U G C /K C. Let A R p R be a maximal abelian subalgebra with Σ A R the restricted roots of ad A R acting on g R. Set ω 0 = {Y p R : λ, Y < π/2 for λ Σ}.

26 Then the theorem of Akheiser-Gindikin is U = G R exp(iω 0 ) u 0, u 0 = ek C G C /K C. This leads to the G R -orbit structure of U, which will be described in Mark s talk. Also it gives Universality: U is independent of the non-classical D Ď; U is Stein; in fact, Γ\U is Stein (Burns-Halverscheid-Hind).

27 The correspondence space W is defined by W G C /T C = enhanced flag variety U G C /K C

28 Then we get the earlier diagrams π D D G C /B W G C /T C π u U W D D (due to universality) The fibres of π D are = B/T C, and are contractable.

29 Example: SU(2, 1) Ď = flags in P 2 P 2 P 2 l p

30 D B D = non-classical D B

31 Z (P,L) = {(p, l)} = P 1 P p l U = B B L

32 W = p p p p p p P = p p L = pp

33 Definitions V µ = H q (D, L µ ); F p,q µ U has fibres H q (Z u, p N Zu/D L µ ). Theorem There exists a spectral sequence {Er p,q, d r } with E p,q 1 = H 0 (U, F p,q µ ); V µ = ker d 1 ker 2 on E 0,q 1.

34 Idea H q (D, L µ ) = H q DR( Γ(W, Ω πd π DL µ ; d πd ) ) (EGW theorem) = H q (n, OG W ) µ where OG W = Γ(W, O W ). Now use the HSSS and result that U is Stein.

35 Notes The theorem and proof work when we quotient by Γ. The d r are linear differential operators of order r. If µ + ρ is anti-dominant and µ 0 (very regular case), then the F p,q µ = 0 for q < d and the spectral sequence looks like

36 The fibres at u 0 and with Z = Z u0 are H d (Z, L µ ), H d (Z, N L/D L µ ),..., H d (Z, c N Z/D L µ ). Identifying p = T u0 U and p = p via the Cartan-Killing form the symbol map in the first spot is H d (Z, L µ ) p H d (Z, N Z/D L µ ) H d (Z, L µ ) H 0 (Z, N Z/D ) cup-product

37 For L µ Z and p H 0 (Z, N Z/D ) a fibre generating subspace, there is a highly developed theory, due to Mark Green and others, involving the sheaves p (m) p N Z/D L µ and their cohomology groups. When L µ Z is very negative, vanishing theorems kick in and the theory simplifies. This is what we shall turn to next. Perhaps the most interesting case is when µ + ρ is close to or on a wall, including µ + ρ = 0 (TDLDS). This situation has yet to be understood; there are some suggestive examples.

38 V. Spencer sequences In the late 1950 s and 1960 s, following his work with Kodaira which laid the foundations for modern deformation theory, Don Spencer became interested in the general theory of deformation of manifolds M having the structure defined by a transitive, continuous pseudogroup. In the complex analytic case the pseudogroup is the local biholomorphic transformations in C n.

39 The Lie algebra of a manifold with this structure is the sheaf Θ M of holomorphic vector fields. The subsequent Kodaira-Spencer-Kuranishi theory uses the representation of the (Zariski) tangent space H 1 (M, Θ M ) to the deformations of M by Dolbeault cohomology H 0,1 (M, TM). In the general case this raises the question of finding a Dolbeault-like resolution for the corresponding Lie algebra of vector fields. Typically Spencer posed the following even more general question:

40 Given a manifold M, vector bundles E M and F M and an arbitrary linear 1 st order differential operator P : E F between the corresponding sheaves with solution sheaf Θ, construct a canonical resolution P 0 Θ E 0 P 0 1 E1, E0 = E.

41 He was able to do this under the assumption that the PDE system ( ) Pu = v is involutive. This initiated a whole new chapter in formal PDE theory.

42 The most interesting case is when the PDE ( ) is involutive and overdetermined; otherwise, at least formally P(E) = F. Naturally occurring systems, other than what is essentially the case of holonomic D-modules, 3 are relatively rare. It is interesting that representation theory gives a whole host of examples, and additionally raises interesting new issues in PDE theory. 4 3 One may think of these as ODE s made categorical. 4 In the finite dimensional case, the BGG (for Bernstein-Gelfand-Gelfand) resolution gives the symbol sequence for the holonomic system whose solutions are an irreducible G C -module (some poetic license taken here).

43 To explain this, the bottom line of which is that for µ + ρ anti-dominant and µ 0, the localization of the above spectral sequence at u 0 has the property of involutivity, and is in fact the Spencer sequence associated to the localization along Z of the Harish-Chandra module V µ. The terms to be explained are

44 involutive (this is the most subtle), symbol, tableau and its prolongations; Spencer cohomology, characteristic variety Ξ PT M, symbol module and characterisic module, the characteristic sheaf M µ on PT M with supp M µ = Ξ. We shall work locally and in the holomorphic category.

45 Involutive This arises as the condition to be able to solve ( ) by a sequence of Cauchy problems along a generic flag of submanifolds M 0 M 1 M n.

46 Example Trying to solve the determined system 2 u 3 3 u 2 = u 1 + v 1 3 u 1 1 u 3 = u 2 + v 2 1 u 2 2 u 1 = u 3 + v 3 ( ) as a sequence of Cauchy problems does not work, because for any solution we have 1 u u u 3 = 0 and so this equation must be added to the system ( ). If we do this, then the system becomes overdetermined and there are integrability conditions on v 1, v 2, v 3 to be able to solve.

47 Note The symbol matrix of ( ) is 0 ξ 3 ξ 2 σ(ξ) = ξ 3 0 ξ 1 det σ(ξ) = 0 ξ 2 ξ 1 0 which suggests that something is funny.

48 For overdetermined systems, such as N i=1 y i (x) y i (x) = g x α x β αβ (x) α, β = 1,..., n for locally embedding a Riemannian manifold M n R N for N < n(n + 1)/2 the situation is more subtle. There are 2 nd order integrability conditions (Gauss equations) on the g αβ, and higher order ones beyond these (Codazzi, etc.). For determined systems involutivity is a generic condition but for overdetermined ones it is highly non-generic. We will give the definition below.

49 Symbol If the system is (using summation convention) then the symbol matrix is P λ i α (x) uα x i (x) = v λ (x) P λ i α (x)ξ i. If E, F, V are the fibres of E, F, TM at a reference point, then the symbol is σ : E V F.

50 Tableau This is given by A = ker σ E V. In the constant coefficient homogeneous case (P λ i α (x) = constant and v λ = 0), A = 1-jets of solutions.

51 For the first prolongation A (1) E S 2 V we have E S 2 V V E V A. A (1) Thinking of E V as E-valued linear forms in x 1,..., x n, A (1) = E-valued quadratic forms Q(x) such that all Q(x)/ x i A. Inductively one defines the A (k) E S k+1 V by A(k) x i A (k 1).

52 Spencer cohomology Set C k,q (A) = { A (k 1) q V k 1 E q V k = 0 δ : C k,q (A) C k 1,q+1 (A) (δ = d ) H k,q (A) = cohomology of the above. Definition The PDE system ( ) is involutive if H k,q (A) = 0, k 1 and q 0.

53 It is a deep and interesting story (E. Cartan, Spencer, Singer-Sternberg-Guillemin-Quillen,... ) that this definition implies that a solution to the sequence of Cauchy problems gives a solution to the PDE system (Cartan-Kähler theorem). 5 5 This is in the real analytic case (Cauchy-Kowalevski). In the C case it is false (Levy). In case ( ) is elliptic, Spencer s conjecture is that in the involutive case local solutions exist (OK for b (Kohn leading to L 2 methods for the -equation)).

54 Note The prolongation of ( ) is obtained by introducing new variables p i α and differentiating ( ) to obtain the system { u α (x) x i = p α i (x) / x i( P λ (x i, u α (x), p α i (x)) ) = v λ (x) x i. Then the Cartan-Kuranishi theorem is that a finite prolongation of ( ) either leads to incompatibilities or is involutive.

55 Characteristic variety Ξ PT M is defined by Ξ = {[ξ] : ker σ(ξ) 0}. We usually work pointwise and consider Ξ PV. In the involutive case the sequence of initial value problems is determined at M k where k = codim Ξ. k = 1 determined case (usual IVP) k 2 overdetermined case M n = R n ) (k = n for u = v in C n where k = n Ξ =, the holonomic case (dim Θ < ).

56 Symbol module B = A E V, B q = A (q) E S q+1 V = B = B q E S V is an S V -module. q 0 The characteristic module is defined by ( ) 0 B( 1) E S V M A 0. In the involutive case B has a minimal free resolution of a special type, meaning B = generators B 1 = relations among the generators B 2 = generators of the 1 st syzygies, which are relations among the relations.

57 and all of these syzygies are linear. In PDE terms this means that the compatibility or integrability relations to solve are the linear ones Pu = v P 1 v = 0, and similarly for P 1 u 1 = v 2 etc.

58 The dual of ( ) is the symbol sequence for a canonical exact sequence P 0 Θ E 1 P 0 1 E1 El 0 of locally free sheaves and linear, 1 st order PDE s whose fibres are the prolongations A (k) of A. This is the Spencer sequence in the involutive case. Finally, the characteristic sheaf M A is the localization, in the sense of algebraic geometry (FAC), of the characteristic module. Then ( ) translates into a resolution of the sheaf M A, all of whose higher cohomology vanishes. The characteristic variety Ξ A = supp M A.

59 We note that codim Ξ A = k measures the degree of overdeterminedness of the original PDE. It is also the length of the minimal resolution of B. Note A subvariety Y T M is integrable if f, g I Y = the Poisson bracket {f, g} I Y. Then at smooth points Y integrability is OK (Cartan). It is also true at reduced points (Gabber). It is not true for embedded components in Y (Bryant).

60 VI. The Spencer sequence of V µ Theorem For µ anti-dominant and µ 0, the localization at u 0 of the previous spectral sequence is the Spencer sequence. We shall describe, in terms of the complex geometry of Z D, the various objects from the PDE theory tableau: A = H d (Z, N Z/D L µ) (only needs µ + ρ anti-dominant) prolongations: A (k) = H d (Z, N (k+1) Z/D L µ ) (needs this plus µ 0)

61 Thus, the tableau and its prolongations give the K-type of V µ. It gives more, because if we write the K-type as ( ) V µ = Vµ k k 0 where Vµ k = H d (Z, N (k) Z/D L µ), the action of p on V µ is given by Vµ k p Vµ k+1 Vµ k 1 and the symbol maps give rise to the first piece. Since V µ is unitarizable and ( ) is an orthogonal direct sum, the adjoints of these maps give the second piece.

62 Characteristic variety Using p H 0 (Z, N Z/D ) we have f PN Z/D F L (c, 2c 1; p) π Pp Z e Gr L (c, p) where π is the map {F c F 2c 1 p} F 2c 1 Pp, and then Ξ = f (PN Z/D) Pp (we drop the subscript A on Ξ). It is the smallest Ad K-invariant subvariety of Pp containing Pp. Moreover, it is non-degenerate (i.e., does not lie in a linear subspace), and Ξ Q = quadric given by the Cartan-Killing-form.

63 In our examples, except in the SU(2, 1) case, codim Ξ 2 so that the PDE system associated to V µ is overdetermined. The differential of e is given by ( ) e : n + c Hom(p, p + ) X e (X )(Y ) = [X, Y ] + for Y p, and the differential of f may be obtained from this in the standard way using the above diagram.

64 Characteristic sheaf: M µ = f ν (L µ ). Roughly speaking, M µ = O Ker σ1. Examples We note that the characteristic variety Ξ depends only on the complex structure of D; the characteristic sheaf depends on µ. We will first use ( ) and root diagrams to illustrate some characteristic varieties.

65 SU(2, 1) = dim p = 4 and Ξ = quadric in P 3.

66 Sp(4) β f is injective except at [X β ] PN Z/D,x 0 = Pp ; Ξ Q P 5 and all inclusions are of codimension one (simplest overdetermined case).

67 SO(4, 1) + α β α = f is surjective with 1-dimensional fibres (simplest example when f is not an immersion).

68 G 2 There are three different choices for the non-classical complex structure, and there are qualitative differences in the behavior of f : PN Z/D Pp for each. Case 1: The root picture is e 2 e

69 Then ad e1 : p p + has image of dimension 2, while ad e2 : p p + is zero. Thus the generic fibres of f have dimension 1, dim Ξ = = 4 and we have Ξ Q Pp = P 7 where Ξ has codimension two in the quadric Q and codimension three in Pp.

70 Case 2: Then ad e1 : p p + has rank 2 and ad e2 : p p + has rank 2 but its image intersects that of ad e1 in dimension 1. It is easy to check that for generic X p, ad X : p p + is injective and so f : PNZ/D Ξ is equidimensional. In Ξ Q P 7 the codimensions are 2 and 1 respectively.

71 Case 3: Then ad e1 : p p + is zero and ad e2 : p p + is an isomorphism. Once again f : PNZ/D Ξ has generic fibre of dimension 1.

72 We note that in all cases the PDE system defining the Harish-Chandra module is overdetermined. We recall that the spectral sequence abutting to H q (D, L µ ) does not require that µ + ρ be anti-dominant, and we shall illustrate the symbol maps in a few of these cases. With the notations F p,q = H q (Z, p N Z/D L µ ), S = C[[p ]] = p (m) m 0 the symbol maps of d 1 F p,q p F p+1,q

73 give rise to the horizontal rows in S (c) F 0,d S (c 1) F 1,d S (c 2) F 2,d S F c,d S (c) F 0,d 1 S (c 1) F 1,d 1 S (c 2) F 2,d 1 S F c,d 1 S (c) F (0,0) S (c 1) F 1,0 S (c 2) F 2,0 S F c,0 Setting E p,q 2, = homology at the (p, q)th spot we obtain a S -linear and K C -linear complex E p 2,q+1 2, E p,q 2, E p+2,q 1 2, with morphisms σ(d 2 ). Continuing, one obtains the symbol spectral sequence (E p,q r,, σ(d r )).

74 Definition The q th characteristic variety Ξ q = { [ξ] : E 0,q 1,ξ = E } 1,q 1 2,ξ = = E c,q c+1 c,ξ = {[ξ] : d 1,ξ = = d c,ξ = 0}. SU(2, 1): We set W = standard U(2)-module; p = W W ; deg L Z µ = k = l + 2.

75 The following are the tables of the H q (Z, p N Z/D L µ ): 1. l > 0 W (l) 2 W (l 1) W (l 2) k = 2, l = 0 W (0) W (0) (TDLDS case) 3. k W (k) W (k+1) W (k+2)

76 Case (i) The symbol maps at the p = 0, p = 1 spots are ( ) 2 W (l) W 2 W (l 1) ( ) ( ) 2 W (l 1) 2 W W (l 2). These may be identified as follows: P (w w ) P w P w P W (l) ; w, w W, (P P ) (w w ) P w P w P, P W (l 1).

77 In P(C 2 C 2 ) the condition w w = 0 defines a quadric. This leads to the Conclusion For k 3, the characteristic variety is a quadric in P 3. For ξ non-characteristic the symbol sequence is exact.

78 Case (iii) The symbols are then maps ( ) 2 W (k) W 2 W (k+1) ( ) ( ) W (k+1) 2 W W (k+2), and using notations as above they may be identified as P (w w ) Pw Pw (P P ) (w w ) Pw P w leading to the

79 Conclusion For k 0 the characteristic variety Ξ =, and for any ξ 0 the symbol sequence is exact. This implies that dim H 0 (D, L k ) < for k 0.

80 Case (ii) This is the most interesting case corresponding to the TDLDS for SU(2, 1). The only non-trivial part of the symbol spectral sequence is σ(d 2 ) : W (0) p (2) W (0). Identifying W (0) = C one may show that For P p (2), σ(d 2 )(P) = (1/2) Ω, P where Ω g (2) Z(U(G)) is the Casimir operator. Thus the characteristic varieties in the spectral sequence are Ξ 1 = P 3 ; Ξ 2 = quadric in P 3.

81 Thus the symbol spectral sequence reduces to the symbol of d 2, which is defined on all of E 0,1 1 p, and this is the map H 1( Z, O Z ( 2) ) p (2) H 0 (Z, 2 N Z/D L ρ ) = = H 0 (Z, ω Z ) p (2) H 0 (Z, O Z ) that was described above. One may also work things out for Sp(4), including the two TDLDS s. So far no general pattern is suggested.

82 Conclusion Although non-classical D s and X s have perhaps non-familiar properties, 6 they have a rich geometry and many interesting aspects that open up intriguing open problems. 6 e.g., O(D) = C, and conjecturally C(D) = C(Ď). Also, X is not algebraic, and conjecturally C(X ) = C.