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 mπ ∞. 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 Hπ 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

mπ 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 Hπ is replaced by

Hπ ⊗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,

- Cλ 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 : Hπ ⊗ Spin −→ Hπ ⊗ 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

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 mπ ∞. 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 Hπ 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

mπ 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 Hπ is replaced by

Hπ ⊗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,

- Cλ 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 : Hπ ⊗ Spin −→ Hπ ⊗ 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