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Non-classical flag domains and Spencer resolutions Phillip Griffiths 1 Talk based on joint work with Mark Green
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
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 ◮ 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; � α ∈ Φ g α � ◮ g C = h ⊕ ⊕ is the root space decomposition of g C relative to the Cartan sub-algebra h = t C ;
◮ 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 � α ∈ Φ + α 2 p = p + ⊕ p − , p − = p + ; ◮ For a choice of Borel subgroup B ⊂ G C with T C ⊂ B ˇ D = G C / B is a flag variety;
◮ A flag domain is given equivalently by ◮ an open G R -orbit D in ˇ D , ◮ a choice of positive roots giving an integrable almost complex structure on G R / T = D with x 0 D ∼ T 1 , 0 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 ;
� � ◮ N Z / D → Z is the normal bundle of Z ⊂ D ; ◮ a weight µ ∈ i t ∗ 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 W W � � � � � ������� � ������� � � � � � � � � � � D ′ D U D where D , D ′ are open G R -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 [ 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 ;
◮ Hirzebruch’s proportionality principle gives χ ( X , L µ ) = ± vol X · χ (ˇ D , L µ ) , where χ (ˇ D , 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).
III. Equivalent forms of non-classical (assume g C is simple) p ⊂ H 0 ( Z , N Z / D ) , U ⊂ G C / K C ✟✟✟✟✟ Geometric ❍❍❍❍❍ D is Z -connected.
Notes g C 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 → Gr L ( c , p ) , dim p = 2 c whose geometry measures the “non-classicalness” of D . 2 2 For example, e is an immersion ⇐ ⇒ TDLDS.
Z -connectivity means that any two points x , x ′ ∈ D may be joined by a chain of Z u ’s, u ∈ U r x ′ x r (cf. Colleen Robles’ talk).
L ∂ D has special properties ✟✟✟✟✟ complex geometric ❍❍❍❍❍ d -pseudo-concavity
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
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 .
for any realization of D as a Mumford-Tate ✟✟✟✟ domain the IPR � = 0 Hodge theoretic ❍❍❍❍ 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 Im z →∞ exp( zN ) · F ∈ G R -orbit in ∂ D Φ ∞ ( W ( N ) , F ) = lim and explain the above.
H 0 ( X , L µ ) = 0 for µ regular (maybe just µ � = 0) ✟✟✟✟ — lots of H q ( X , L ⊗ k µ ) for µ regular, q � = 0 Cohomological ❍❍❍❍ for a Harish-Chandra module V the ( E 1 , d 1 ) term of the Serre-Hochschild spectral sequence is not a bi-complex
Note α ∈ Φ + g − α and n c = g − α . Then the HSSS is n = ⊕ ⊕ α ∈ Φ + c = H q ( n c , ∧ p p + ⊗ V ) ⇒ H p + q ( n , V ) . E p , q 1 The complex ( E 1 , d 1 ) is constructed from c ⊗ ∧ p p + ← ∧ q n ∗ → ( p , q ) .
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 out only in the is a sub-algebra classical case
It is the far right term that gives d 2 E p , q → E p +2 , q − 1 − . 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.
arithmetic automorphic representations whose infinite component is a TDLDS ✟✟✟ corresponding to H d ( X , L − ρ ) Representation ❍❍❍ theoretic { 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 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 } .
Then the theorem of Akheiser-Gindikin is U = G R exp( i ω 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 ⊂ ˇ D ; ◮ U is Stein; in fact, Γ \ U is Stein (Burns-Halverscheid-Hind).
The correspondence space W is defined by ⊂ G C / T C = enhanced flag variety W � � ⊂ U G C / K C
� � Then we get the earlier diagrams ⊂ G C / T C (due to universality) W W � � � � ������� � ������ � � � π D π u � � � � � � � D ′ D D U ∩ G C / B The fibres of π D are ∼ = B / T C , and are contractable.
Example: S U (2 , 1) ✟✟✟✟✟✟ D = flags in P 2 ⊂ P 2 × P 2 ∗ l ˇ r p
D ′′ D ′ D = ❄ ❄ non-classical B B
l Z ( P , L ) = { ( p , l ) } ⇒ U ∼ p ∼ = = B × B P P 1 L
p W = p ′ p ′′ p p p ′ P = p ′ p ′′ L = pp ′′
Definitions ◮ V µ = H q ( D , L µ ); ◮ F p , q → U has fibres H q ( Z u , ∧ p N Z u / D ⊗ L µ ). µ Theorem There exists a spectral sequence { E p , q , d r } with r ◮ E p , q = H 0 ( U , F p , q µ ) ; 1 ◮ V µ = ker d 1 ∩ ker 2 ∩ · · · on E 0 , q 1 .
Idea H q ( D , L µ ) ∼ = H q � Γ( W , Ω • π D ⊗ π ∗ � D L µ ; d π D ) (EGW theorem) DR ∼ = H q ( n , O G W ) − µ where O G W = Γ( W , O W ) . Now use the HSSS and result that U is Stein.
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 = 0 for q < d and the spectral sequence looks like µ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 . . . . . . . . . 0 0 − − − − 0
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