Introduction
The objective of this work is to study aspects of the automorphic cohomology
groups Hq(X, Lμ) on quotients X = Γ\D by an arithmetic group Γ acting on a class
of homogeneous complex manifolds D = GR/T . Here GR is the connected real Lie
group associated to a reductive Q-algebraic group G, T GR is a compact maximal
torus, and μ the weight associated to a character of T that gives a homogeneous
holomorphic line bundle D. These D’s may be realized as Mumford-Tate
domains that arise in Hodge theory, and in general we shall follow the terminology
and notations from the monograph
[GGK1].1
We shall say that D is classical if
it equivariantly fibres holomorphically or anti-holomorphically over an Hermitian
symmetric domain; otherwise it is non-classical, and this is the case of primary
interest in this paper.
In the non-classical case it has been known for a long time that, at least when
Γ is co-compact in GR,

H0(X,
Lμ) = 0 for any non-trivial μ;
when μ is sufficiently
non-singular,2
then
Hq(X,
Lμ) = 0, q = q(μ + ρ)
Hq(μ+ρ)(X, Lμ) = 0
where q(μ + ρ) will be defined in the text.
More precisely, for k k0 and any non-singular μ
dim
Hq(μ+ρ)(X,
Lkμ) = vol(X) · Pμ(k)
where Pμ(k) is a Hilbert polynomial with leading term
Cμkdim D
where 0
is independent of Γ. Thus, in the non-classical case there is a lot of automorphic
cohomology and it does not occur in degree zero. In the classical case, the intensive
study of the very rich geometric, Hodge theoretic, arithmetic and representation
theoretic properties of automorphic forms has a long and venerable history and
remains one of great current interest. In contrast, until recently in the non-classical
case the geometric and arithmetic properties of automorphic cohomology have re-
mained largely mysterious.3
For three reasons this situation has recently changed. One reason is the works
[Gi], [EGW] that give a general method for interpreting analytic coherent cohomol-
ogy on a complex manifold as holomorphic de Rham cohomology on an associated
1Cf. the Notations and Terminology section below.
2Non-singular,
or regular, means that μ is not on the wall of a Weyl chamber; sufficiently
non-singular means that μ is at a large enough distance |μ| from any wall.
3Important
exceptions are the works Schmid [Schm1], Williams [Wi1], [Wi2], [Wi3], [Wi4],
[Wi5], Wells and Wolf [WW1], [WW2], [WW3], and Wolf [Wo], some of which will be discussed
below. These deal primarily with the representation-theoretic aspects of automorphic cohomology.
1
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