1 Non-classical flag domains and Spencer resolutions Phillip - - PowerPoint PPT Presentation

1

Non-classical flag domains and Spencer resolutions

Phillip Griffiths

1Talk based on joint work with Mark Green

Outline

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

• 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;

◮ 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.

• II. Notations and terminology

◮ GC will be a semi-simple complex Lie group; ◮ GR ⊂ GC will be a connected real form containing a

compact maximal torus T;

◮ KR ⊂ GR is the unique maximal compact subgroup

containing T;

◮ KC and TC are the complexifications; ◮ gC = h ⊕

• ⊕

α∈Φ gα

is the root space decomposition of gC relative to the Cartan sub-algebra h = tC;

◮ gR = kR ⊕ pR is the Cartan-decomposition; ◮ Φ = Φc ∪ Φnc 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 = pC

• ρ = 1

2

• α∈Φ+ α

p = p+ ⊕ p−, p− = p+;

◮ For a choice of Borel subgroup B ⊂ GC with TC ⊂ B

ˇ D = GC/B is a flag variety;

◮ A flag domain is given equivalently by

◮ an open GR-orbit D in ˇ

D,

◮ a choice of positive roots giving an integrable almost

complex structure on GR/T = D with T 1,0

x0 D ∼

= ⊕

α∈Φ+ gα; ◮ Z = KR/T is a maximal compact subvariety of D and

U = {gZ : g ∈ GC, gZ ⊂ D} is the cycle space; note that Z = KC/BK where BK = B ∩ KC;

◮ NZ/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 Lµ =: GC ×B Cµ;

◮ Vµ = Hq(D, Lµ); we will usually take q = d where

d = dim Z;

◮ W ⊂ GC/TC will be the correspondence space, defined

below, and the basic diagrams are W

• D

U W

• D

D′ where D, D′ are open GR-orbits in ˇ D.

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 [kC, p+] ⊆ p+, and using the

Cartan-Killing form non-classical is equivalent to [p+, p+] = 0;

◮ for Γ ⊂ GR a co-compact, neat subgroup

X = Γ\D and the Hq(X, Lµ) are automorphic cohomology groups;

◮ Hirzebruch’s proportionality principle gives

χ(X, Lµ) = ± vol X · χ(ˇ D, Lµ), where χ(ˇ D, Lµ) is known by the Borel-Weil-Bott theorem;

◮ of particular interest are the automorphic cohomology

groups when µ + ρ is singular (⇒ χ(X, Lµ) = 0), and of very particular interest are the Hq(X, L−ρ) corresponding to totally degenerate limits of discrete series (TDLD’s) (Henri Carayol’s talk).

• III. Equivalent forms of non-classical

(assume gC is simple)

Geometric

✟✟✟✟✟ ❍❍❍❍❍

p ⊂ H0(Z, NZ/D), U ⊂ GC/KC D is Z-connected.

Notes

gC are holomorphic vector fields on ˇ D and the first property is that p injects into (fibre-generating) holomorphic normal vector fields. This gives a map e : Z → GrL(c, p), dim p = 2c whose geometry measures the “non-classicalness” of D.2

2For example, e is an immersion ⇐

⇒ TDLDS.

Z-connectivity means that any two points x, x′ ∈ D may be joined by a chain of Zu’s, u ∈ U

r r x′

x (cf. Colleen Robles’ talk).

complex geometric

✟✟✟✟✟ ❍❍❍❍❍

L∂D has special properties d-pseudo-concavity

Notes

The intrinsic Levi form L∂D will be discussed in Mark Green’s

• talk. Recall that pseduo-convexity means

D ∂D H = complex-analytic hypersurface

d-pseudo-concavity means D ∂D Z dim Z = d In fact, it seems likely that we will have Z Z ∩ D = spherical shell in Cd

?

= ⇒ dim Hq(D, L) < ∞ for 0 ≦ q < d.

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       

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) = lim

Im z→∞ exp(zN) · F ∈ GR-orbit in ∂D

and explain the above.

Cohomological

✟✟✟✟ ❍❍❍❍

           H0(X, Lµ) = 0 for µ regular (maybe just µ = 0) — lots of Hq(X, L⊗k

µ )

for µ regular, q = 0                       for a Harish-Chandra module V the (E1, d1) term of the Serre-Hochschild spectral sequence is not a bi-complex           

Note

n = ⊕

α∈Φ+ g−α and nc =

⊕

α∈Φ+

c

g−α. Then the HSSS is E p,q

1

= Hq(nc, ∧pp+ ⊗ V ) ⇒ Hp+q(n, V ). The complex (E1, d1) is constructed from ∧qn∗

c ⊗ ∧pp+ ←

→ (p, q).

A priori the coboundary is

δ : (p, q) → (p − 1, q + 2) + (p, q + 1) + (p + 1, q) + (p + 2, q − 1)

✟✟✟✟✟✟✟✟ ✟ ✟ ✟ ✟ ✟ ✟

• ut because nc

is a sub-algebra

✻

• ut only in the

classical case

✻

It is the far right term that gives E p,q

1 d2

− → 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.

Representation theoretic

✟✟✟ ❍❍❍

       arithmetic automorphic representations whose infinite component is a TDLDS corresponding to Hd(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.

• IV. Realization of Vµ 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 ⊂ GC/KC. Let AR ⊂ pR be a maximal abelian subalgebra with Σ ⊂ A∗

R

the restricted roots of ad AR acting on gR. Set ω0 = {Y ∈ pR : | λ, Y | < π/2 for λ ∈ Σ}.

Then the theorem of Akheiser-Gindikin is U = GR exp(iω0) · u0 , u0 = eKC ∈ GC/KC. This leads to the GR-orbit structure of U, which will be described in Mark’s talk. Also it gives

◮ Universality: U is independent of the non-classical

D ⊂ ˇ D;

◮ U is Stein; in fact, Γ\U is Stein

(Burns-Halverscheid-Hind).

The correspondence space W is defined by W ⊂ GC/TC = enhanced flag variety 

• 
• U

⊂ GC/KC

Then we get the earlier diagrams W

πD

• πu
• ⊂ GC/TC

D U W

• D

D′ (due to universality) ∩ GC/B The fibres of πD are ∼ = B/TC, and are contractable.

Example: SU(2, 1)

ˇ D = flags in P2 ⊂ P2 × P2∗

✟✟✟✟✟✟

l p

r

D′′

❄

B D = non-classical D′

❄

B

P L p l ⇒ U ∼ = B × B Z(P,L) = {(p, l)} ∼ = P1

W = p′ p P = p′ p′′ p L = pp′′ p′ p′′ p

Definitions

◮ Vµ = Hq(D, Lµ); ◮ Fp,q µ

→ U has fibres Hq(Zu, ∧pNZu/D ⊗ Lµ).

Theorem

There exists a spectral sequence {E p,q

r

, dr} with

◮ E p,q 1

= H0(U, Fp,q

µ ); ◮ Vµ = ker d1 ∩ ker2 ∩ · · · on E 0,q 1 .

Idea

Hq(D, Lµ) ∼ = Hq

DR

• Γ(W, Ω•

πD ⊗ π∗ DLµ; dπD)

• (EGW theorem)

∼ = Hq(n, OGW)−µ where OGW = Γ(W, OW). Now use the HSSS and result that U is Stein.

Notes

The theorem and proof work when we quotient by Γ. The dr are linear differential operators of order r. If µ + ρ is anti-dominant and |µ| ≫ 0 (very regular case), then the Fp,q

µ

= 0 for q < d and the spectral sequence looks like ∗ ∗ ∗ ∗ ∗ ∗ ∗ . . . . . . . . . − − − −

The fibres at u0 and with Z = Zu0 are Hd(Z, Lµ), Hd(Z, N∗

L/D ⊗ Lµ), . . . , Hd(Z, ∧cNZ/D ⊗ Lµ).

Identifying p ∼ = Tu0U and p ∼ = p∗ via the Cartan-Killing form the symbol map in the first spot is Hd(Z, Lµ) ⊗ p

Hd(Z, NZ/D ⊗ Lµ)

∩ Hd(Z, Lµ) ⊗ H0(Z, NZ/D)

• cup-product

❍ ❍ ❍ ❨

For Lµ → Z and p ⊂ H0(Z, NZ/D) a fibre generating subspace, there is a highly developed theory, due to Mark Green and others, involving the sheaves p(m) ⊗ ∧pNZ/D ⊗ Lµ and their cohomology groups. When Lµ → 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

• n a wall, including µ + ρ = 0 (TDLDS). This situation has

yet to be understood; there are some suggestive examples.

• 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 Cn.

The Lie algebra of a manifold with this structure is the sheaf ΘM of holomorphic vector fields. The subsequent Kodaira-Spencer-Kuranishi theory uses the representation of the (Zariski) tangent space H1(M, ΘM) to the deformations of M by Dolbeault cohomology H0,1

∂ (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:

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

P0

− → E1

P1

− → · · · , E0 = E.

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.

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

3One may think of these as “ODE’s made categorical.” 4In the finite dimensional case, the BGG (for

Bernstein-Gelfand-Gelfand) resolution gives the symbol sequence for the holonomic system whose solutions are an irreducible GC-module (some poetic license taken here).

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 u0 has the property of involutivity, and is in fact the Spencer sequence associated to the localization along Z of the Harish-Chandra module Vµ. The terms to be explained are

◮ 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 Mµ on PT ∗M with supp Mµ = Ξ.

We shall work locally and in the holomorphic category.

Involutive

This arises as the condition to be able to solve (∗) by a sequence of Cauchy problems along a generic flag of submanifolds M0 ⊂ M1 ⊂ · · · ⊂ Mn.

Example

Trying to solve the “determined” system ∂2u3 − ∂3u2 = u1 + v 1 ∂3u1 − ∂1u3 = u2 + v 2 (∗∗) ∂1u2 − ∂2u1 = u3 + v 3 as a sequence of Cauchy problems does not work, because for any solution we have ∂1u1 + ∂2u2 + ∂3u3 = 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 v1, v2, v3 to be able to solve.

Note

The symbol matrix of (∗∗) is σ(ξ) =    −ξ3 ξ2 ξ3 −ξ1 −ξ2 ξ1    ⇒ det σ(ξ) = 0 which suggests that something is funny.

For “overdetermined” systems, such as

N

• i=1

∂y i(x) ∂xα ∂y i(x) ∂xβ = gαβ(x) α, β = 1, . . . , n for locally embedding a Riemannian manifold Mn ⊂ RN for N < n(n + 1)/2 the situation is more subtle. There are 2nd

• rder integrability conditions (Gauss equations) on the gαβ,

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.

Symbol

If the system is (using summation convention) Pλi

α (x)∂uα

∂xi (x) = v λ(x) then the symbol matrix is Pλi

α (x)ξi.

If E, F, V are the fibres of E, F, TM at a reference point, then the symbol is σ : E ⊗ V ∗ → F.

Tableau

This is given by A = ker σ ⊂ E ⊗ V ∗. In the constant coefficient homogeneous case (Pλi

α (x) =

constant and v λ = 0), A = 1-jets of solutions.

For the first prolongation A(1) ⊂ E ⊗ S2V ∗ we have E ⊗ S2V ∗ ⊗ V

E ⊗ V ∗

∪ ∪ A(1)

• A.

Thinking of E ⊗ V ∗ as E-valued linear forms in x1, . . . , xn, A(1) = E-valued quadratic forms Q(x) such that all ∂Q(x)/∂xi ∈ A. Inductively one defines the A(k) ⊂ E ⊗ Sk+1V ∗ by ∂A(k)

∂xi ∈ A(k−1).

Spencer cohomology

Set

◮ C k,q(A) =

• A(k−1) ⊗ ∧qV ∗

k ≥ 1 E ⊗ ∧qV ∗ k = 0

◮ δ : C k,q(A) → C k−1,q+1(A)

(δ = “d”)

◮ Hk,q(A) = cohomology of the above.

Definition

The PDE system (∗) is involutive if Hk,q(A) = 0, k ≧ 1 and q ≧ 0.

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-K¨ ahler theorem).5

5This 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 L2 methods for the ∂-equation)).

Note

The prolongation of (∗) is obtained by introducing new variables pi

α and differentiating (∗) to obtain the system

∂uα(x)

∂xi

= pα

i (x)

∂/∂xi Pλ(xi, uα(x), pα

i (x))

• = ∂vλ(x)

∂xi .

Then the Cartan-Kuranishi theorem is that a finite prolongation of (∗) either leads to incompatibilities or is involutive.

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 Mk where k = codim Ξ. k = 1 — determined case (usual IVP) k ≥ 2 — overdetermined case (k = n for ∂u = v in Cn where Mn = Rn) k = n — Ξ = ∅, the holonomic case (dim Θ < ∞).

Symbol module

B = A⊥ ⊂ E ∗ ⊗ V , Bq = A(q)⊥ ⊂ E ⊗ Sq+1V = ⇒ B = ⊕

q≧0 Bq ⊂ E ⊗ S•V is an S•V -module.

The characteristic module is defined by (♮) 0 → B(−1) → E ⊗ S•V → MA → 0. In the involutive case B has a minimal free resolution of a special type, meaning B = generators B1 = relations among the generators B2 = generators of the 1st syzygies, which are relations among the relations . . .

and all of these syzygies are linear. In PDE terms this means that the compatibility or integrability relations to solve Pu = v are the linear ones P1v = 0, and similarly for P1u1 = v2 etc.

The dual of (♮) is the symbol sequence for a canonical exact sequence 0 → Θ → E0

P1

− → E1

P1

− → · · · → El → 0

• f locally free sheaves and linear, 1st 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 MA is the localization, in the sense of algebraic geometry (FAC), of the characteristic

• module. Then (♮) translates into a resolution of the sheaf MA,

all of whose higher cohomology vanishes. The characteristic variety ΞA = supp MA.

We note that codim ΞA = 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 ∈ IY = ⇒ the Poisson bracket {f , g} ∈ IY . 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).

• VI. The Spencer sequence of Vµ

Theorem

For µ anti-dominant and |µ| ≫ 0, the localization at u0 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 = Hd(Z, N∗

Z/D ⊗ Lµ)

(only needs µ + ρ anti-dominant) prolongations: A(k) = Hd(Z, N∗(k+1)

Z/D

⊗ Lµ) (needs this plus |µ| ≫ 0)

Thus, the tableau and its prolongations give the K-type of Vµ. It gives more, because if we write the K-type as (♮♮) Vµ = ⊕

k≧0 V k µ

where V k

µ = Hd(Z, N∗(k) Z/D ⊗ Lµ), the action of p on Vµ is given

by V k

µ ⊗ p → V k+1 µ

⊕ V k−1

µ

and the symbol maps give rise to the first piece. Since Vµ is unitarizable and (♮♮) is an orthogonal direct sum, the adjoints

• f these maps give the second piece.

Characteristic variety

Using p ⊂ H0(Z, NZ/D) we have PN∗

Z/D

• f
• FL(c, 2c − 1; p)
• π

Pp∗

Z

e

GrL(c, p)

where π is the map {Fc ⊂ F2c−1 ⊂ p} → F2c−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.

In our examples, except in the SU(2, 1) case, codim Ξ ≧ 2 so that the PDE system associated to Vµ 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.

Characteristic sheaf: Mµ = fν(Lµ).

Roughly speaking, Mµ = OKer 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.

SU(2, 1)

s s

• s

s

+ + + = ⇒ dim p = 4 and Ξ = quadric in P3.

Sp(4)

• +

+ + + β

◮ f∗ is injective except at [Xβ] ∈ PN∗ Z/D,x0 ∼

= Pp−;

◮ Ξ ⊂ Q ⊂ P5 and all inclusions are of codimension one

(simplest overdetermined case).

SO(4, 1)

• +

+ + +

• ✒

❅ ❅ ❅ ❅ ❘

β α

• ✒

α = ⇒ f∗ is surjective with 1-dimensional fibres (simplest example when f is not an immersion).

G2 There are three different choices for the non-classical complex structure, and there are qualitative differences in the behavior

• f f : PN∗

Z/D → Pp∗ for each.

Case 1: The root picture is

• +

+ + + + +

✻ ✲

e2 e1

Then ade1 : p− → p+ has image of dimension 2, while ade2 : p− → p+ is zero. Thus the generic fibres of f have dimension 1, dim Ξ = 2 + 3 − 1 = 4 and we have Ξ ⊂ Q ⊂ Pp∗ ∼ = P7 where Ξ has codimension two in the quadric Q and codimension three in Pp∗.

Case 2:

• +

+ + + + + Then ade1 : p− → p+ has rank 2 and ade2 : p− → p+ has rank 2 but its image intersects that of ade1 in dimension 1. It is easy to check that for generic X ∈ p−, adX : p− → p+ is injective and so f : PN∗

Z/D → Ξ is equidimensional. In

Ξ ⊂ Q ⊂ P7 the codimensions are 2 and 1 respectively.

Case 3:

• +

+ + + + + Then ade1 : p− → p+ is zero and ade2 : p−

∼

− → p+ is an

• isomorphism. Once again f : PN∗

Z/D → Ξ has generic fibre of

dimension 1.

We note that in all cases the PDE system defining the Harish-Chandra module is overdetermined. We recall that the spectral sequence abutting to Hq(D, Lµ) 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 = Hq(Z, ∧pNZ/D ⊗ Lµ), ◮ S• = C[[p∗]] = ⊕ m≧0 p∗(m)

the symbol maps of d1 F p,q ⊗ p∗ → F p+1,q

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 KC-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 )).

Definition

The qth characteristic variety Ξq =

• [ξ] : E 0,q

1,ξ = E 1,q−1 2,ξ

= · · · = E c,q−c+1

c,ξ

• = {[ξ] : d1,ξ = · · · = dc,ξ = 0} .

SU(2, 1): We set

◮ W = standard U(2)-module; ◮ p∗ ∼

= W ⊕ W ;

◮ deg Lµ

• Z = k = l + 2.

The following are the tables of the Hq(Z, ∧pNZ/D ⊗ Lµ): 1. l > 0 W (l)∗

2

⊕ W (l−1)∗ W (l−2)∗ 2. k = −2, l = 0 W (0) W (0) (TDLDS case) 3. k ≧ 0 W (k) W (k+1) W (k+2)

Case (i) The symbol maps at the p = 0, p = 1 spots are W (l)∗ ⊗

• 2

⊕ 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)∗.

In P(C2 ⊕ C2) the condition w ∧ w ′ = 0 defines a quadric. This leads to the

Conclusion

For k ≦ −3, the characteristic variety is a quadric in P3. For ξ non-characteristic the symbol sequence is exact.

Case (iii) The symbols are then maps W (k) ⊗

• 2

⊕ 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

Conclusion

For k ≧ 0 the characteristic variety Ξ = ∅, and for any ξ = 0 the symbol sequence is exact. This implies that dim H0(D, Lk) < ∞ for k ≧ 0.

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

◮ Ξ1 = P3; ◮ Ξ2 = quadric in P3.

Thus the symbol spectral sequence reduces to the symbol of d2, which is defined on all of E 0,1

1

⊗ p∗, and this is the map H1 Z, OZ(−2)

• ⊗ p∗(2)

H0(Z, ∧2NZ/D ⊗ L−ρ)

∼ = ∼ = H0(Z, ωZ) ⊗ p∗(2) H0(Z, OZ) 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.

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.

6e.g., O(D) ∼

= C, and conjecturally C(D) ∼ = C(ˇ D). Also, X is not algebraic, and conjecturally C(X) ∼ = C.