14 INTRODUCTION
In section IV.D we state and begin the proof of the result, Theorem (IV.D.3),
concerning the field of definition of cuspidal arithmetic automorphic classes eval-
uated in the fibres of Hodge bundles at CM points. The two main points in the
argument are: (i) that using the [EGW] formalism automorphic cohomology classes
of higher degree may be evaluated at points of the correspondence space, and (ii)
the comparison between the HT and AG arithmetic structures in the classical case.
In the appendix to this section we discuss the Penrose transform and arithmeticity
of automorphic cohomology classes in H1(X, Lμ) where X = Γ\D and μ + ρ is in
the anti-dominant Weyl chamber. In some sense the anti-dominant one is the most
important Weyl chamber for representation theory but it is not the Weyl chamber
where the Penrose transform of Picard and Siegel modular forms ends up.25 One
point of note here is that there are many Penrose transforms, a topic that will be
discussed elsewhere when the general definitions and properties of correspondence
spaces will be treated.
In section IV.E we complete the proof of theorem (IV.D.3). The argument
involves a somewhat intricate analysis of compatible pairs of CM points in corre-
spondence spaces.
Finally, in section IV.F we will present an exposition of the main result in [C1]
and [C2] concerning the relation of the cup-products of the Penrose transform of
Picard automorphic forms and their conjugates to the automorphic cohomology
group
H2(X,
L−ρ). It is this group that appears in the automorphic representation
of the adele group U(2, 1; A) whose infinite component is a totally degenerate limit
of discrete series (TDLDS); i.e., one whose Harish-Chandra infinitesimal character
is zero (cf. [CK]). It is known that such a representation cannot arise from the
cohomology, either l-adic or coherent, associated to a Shimura variety of Hodge
type.26
To be able to define an arithmetic structure on
H2(X,
L−ρ) was a moti-
vating question for Carayol, and his result is given in theorem (IV.F.1) with his
proof presented in the context of this
work.27
In Carayol’s work he used the explicit
construction of the TDLDS represented by functions on the closed SU(2, 1)-orbit,
which is the 3-sphere as depicted by the third of the above pictures of non-open
orbits acted on by SU(2, 1) through linear functional transformations. In the ap-
pendix to section IV.F we shall discuss a different geometric realization of a part
of the TDLDS and relate this to n-cohomology considerations. This construction,
interpreted in the context of Beilinson-Bernstein localization [BB] and the duality
theorem in [HMSW] and coupled with the general construction of correspondence
spaces should allow the methods of this work to be extended to further interesting
geometric examples. In particular, we note that the analogue of (IV.F.1) for Sp(4)
has been carried out in the forthcoming paper [Ke1].
25In
the appendix to section IV.D we give an alternate method for evaluating cohomology
classes in the case μ + ρ is anti-dominant and pose an interesting question that arises from this
construction.
26A
necessary condition for (0, C) to give a TDLDS is that no compact root in C is simple
[CK]. This implies that the complex structure given by C on GR/T is non-classical. It also enters,
in an essential way, in the subtle issues concerning the Hochschild-Serre spectral sequence.
27Although Carayol’s result does not yet give the sought for arithmetic structure on
H2(X, L−ρ), and by duality one on H1(X, L−ρ), we feel that his work is extremely interesting
and the arguments bring new and deep insight into automorphic cohomology.
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