Introduction to the First Edition

1. This monograph is mainly concerned with two types of cohomology spaces

pertaining to a reductive Lie group G (real, p-adic, or product of such groups) and

a discrete cocompact subgroup T of G. The first one is the Eilenberg-MacLane

cohomology space H*(T; E) of V with coefficients in a finite dimensional unitary V-

module (or a finite dimensional G-module if G is real). The second one is attached

to G, or its Lie algebra g and a maximal compact subgroup K if G is real, and a

representation V of G, usually infinite dimensional, and appears in various guises:

continuous, smooth, or also (for G real) relative Lie algebra cohomology. Our

initial interest was in the former one. However, its study may be reduced in part

to the latter one (see Chapters VII and XIII), where G is the ambient group and

V runs through the irreducible subspaces of

L2(T\G).

The determination of this

cohomology is then a first step towards the determination of H*(T; E). But, as this

work developed, we were led to emphasize it more and more, and to treat it as our

main topic rather than as an auxiliary one. In fact, ten out of thirteen chapters are

devoted to it, or directly motivated by it.

The material presented here divides naturally into two parts, one devoted

mainly to real Lie groups (Chapters I to IX), the other to locally compact to-

tally disconnected groups (for short, t.d. groups), in particular reductive p-adic

groups, or products of real Lie groups and t.d. groups (Chapters X to XIII). Each

part in turn contains roughly three main items: general results on the cohomology

used, specific ones for cohomology and representations of reductive groups, and

applications to discrete cocompact subgroups.

We now give some indications on the contents of the various chapters.

2. In Chapters I to VIII, G is a real Lie group with finitely many connected

components, and the underlying cohomology is the relative Lie algebra cohomology

H*(g,t',V)

o r

rather, to allow for non-connected G's, a slight modification of it

denoted i7*(g,i^; V). Chapter I is devoted to foundational material on that coho-

mology. In §§1 to 4, g is a finite dimensional Lie algebra over a field of characteristic

zero and t a subalgebra. §1 recalls the direct definition of i7*($,£; V), §2 discusses

more generally the derived functors of Hom0 in the category C%$ of (B, ^-modules,

i.e., g-modules which are locally finite and semi-simple with respect to t. This ap-

proach differs only in minor details from that of G. Hochschild, in the framework

of relative homological algebra. The translation in the formalism of Yoneda's long

extensions is briefly recalled in §3. In §4, we give two proofs of a useful vanishing

theorem of D. Wigner. From §5 on, F — R, 9 is the Lie algebra of G and t that

of a maximal compact subgroup K of G. In §5, we transpose the previous consid-

erations to the category of (9, if)-modules. In §6 we introduce a slightly different

category

CQJ,L,

solely as a tool to prove the existence of a Hochschild-Serre spec-

tral sequence for (g, K)-modules. Also included are two results of Casselman (5.5)

XI