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 Lμ → 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 Cμ 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