2 DAN BARBASCH AND PAVLE
PANDˇ
ZI
´
C
Let Γ G be a discrete cocompact subgroup. A question of interest is the compu-
tation of H∗(Γ). Let X := Γ\G/K. Then Hi(Γ) := Htop(X, i C), where Htop(X, i C)
denotes the usual cohomology of the topological space X: The theory of automor-
phic forms provides insight into Htop(X, i C). A fundamental result of Gelfand and
Piatetski-Shapiro is that
L2(Γ\G)
= mπHπ
where π are irreducible unitary representations of G, and ∞. It implies that
Hi(Γ)
= Htop(X,
i
C) = mπHct(G,
i
Hπ) =
mπHi(g,K;
Hπ).
Here Hct(G, i Hπ) denotes the continuous cohomology groups (see [BW]), and the
groups
Hi(g,K;
Hπ) are the relative Lie algebra cohomology groups defined in
[BW], Chapter II, Section 6, or [VZ]. Here the unitary representation is
replaced by the corresponding (g,K)-module, denoted again by Hπ.
Thus to obtain information about
Hi(Γ)
one needs to have information about
and
Hi(g,K,π).
It is very difficult to obtain information about the multiplic-
ities mπ. On the other hand, knowledge about the vanishing of
Hi(g,K)
for all
unitary representations translates into vanishing of
Hi(Γ).
This approach leads one
to consider the following problem.
Problem. Classify all irreducible admissible unitary modules with nonzero
(g,K) cohomology.
A more general problem where the trivial representation C of Γ is replaced by
an arbitrary finite dimensional representation was solved by Enright [E] for complex
groups. Introducing more general coefficients has the effect that is replaced by
⊗F for some finite-dimensional representation F . The results were generalized
later by Vogan-Zuckermann [VZ] to real groups as follows. The λ appearing below
is such that the infinitesimal character of Rq(Cλ) s equals the infinitesimal character
of F . The answer is that π = Rq(Cλ), s where
- q = l + u g is a θ stable parabolic subalgebra,
- is a unitary character of l,
- Rq s is cohomological induction, and s = dim(u k).
The starting point for the proof is the fact
Hi(g,K;
π)

= HomK [
i
s, π]. The
reference [BW] gives consequences of these results. For a survey of related more
recent results, the reader may consult [LS].
A major role in providing an answer to the above problem is played by the
Dirac Inequality of Parthasarathy [P2]. The adjoint representation of K on s lifts
to Ad : K −→ Spin(s), where K is the spin double cover of K. The Dirac operator
D : Spin −→ Spin
is defined as
D =
i
bi di U(g) C(s),
where C(s) denotes the Clifford algebra of s with respect to the Killing form, bi is a
basis of s and di is the dual basis with respect to the Killing form, and Spin is a spin
module for C(s). D is independent of the choice of the basis bi and K-invariant. It
satisfies
D2
= −(Casg ⊗1 + ρg
2)
+ (Δ(Cask) + ρk
2).
2
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