2. SYMMETRY BREAKING FOR SPHERICAL PRINCIPAL SERIES 13

Lemma 2.2. 1) A (g,K)-module ZK is irreducible as a g-module if every irre-

ducible K-module occurring in ZK is irreducible as a K0-module.

2) For G = O(n +1,1), let P0 := P ∩ G0. Then P0 is connected, and is a minimal

parabolic subgroup of G. Then we have a natural bijection:

G0/P0

∼

→ G/P (

Sn).

Proof of Proposition 2.1. Let ZK be the underlying (g,K)-module of Z.

It is suﬃcient to discuss the irreducibility of ZK as a (g,K0)-module.

1) Any irreducible representation of K O(n+1)×O(1) occurring in the spherical

principal series representation I(λ) is of the form

Hi(Sn)

1 for some i ∈ N,

which is still irreducible as a representation of K0 = SO(n + 1) if n ≥ 2. Here 1

denotes the trivial one-dimensional representation of O(1). Hence the assumption of

Lemma 2.2 (1) is fulfilled, and the first statement follows.

2) By Lemma 2.2 (2), the restriction of I(λ) to G0 is isomorphic to a spherical

principal series representation of G0 = SO0(2, 1). Comparing the aforementioned

composition series of representation I(λ) of O(2, 1) with a well-known result for

G0 = SO0(2, 1) SL(2, R)/{±1}, we see that T (i) is a direct sum of a holomor-

phic discrete series representation and an anti-holomorphic discrete series repre-

sentation of G0 and that other irreducible subquotients of G remain irreducible as

G0-modules. See also Remark 16.2 for geometric interpretations of this decompo-

sition.

Proposition 2.1 and [12] imply that the representation I(λ) is reducible if and

only if

λ = n + i or λ = −i for i ∈ N.

A reducible spherical principal series representation has two irreducible composition

factors. The Langlands subquotient of I(n+i) is a finite-dimensional representation

F (i). We have for i ∈ N non-splitting exact sequences as Fr´ echet G-modules:

(2.10) 0 → F (i) → I(−i) → T (i) → 0,

(2.11) 0 → T (i) → I(n + i) → F (i) → 0.

Inducing from the minimal parabolic subgroup P of G , we define the induced

representation J(ν) and the irreducible representations F (j) ≡ F G (j) and T (j) ≡

T G (j) of G as we did for G. We shall simply write F (j) for F G (j) and T (j) for

T G (j), respectively, if there is no confusion.

2.2. Finite-dimensional subquotients of disconnected groups. Since

the group G = O(n + 1,1) has four connected components, we need to be care-

ful to identify the finite-dimensional subquotient F (i) with some other (better-

understood) representations.

First, we consider the space of harmonic polynomials of degree i ∈ N by

Hi(Rn+1,1)

:= {ψ ∈ C[x0, · · · , xn+1] : ψ = 0, ψ is homogeneous of degree i},

where =

∂2

∂x0 2

+ · · · +

∂2

∂xn 2

−

∂2

∂x2

n+1

. Then G = O(n + 1,1) acts irreducibly on

Hi(Rn+1,1)

for any i ∈ N. The indefinite signature is not the main issue here,

because this representation extends to a holomorphic representation of the com-

plexified Lie group O(n + 2, C). Similarly, the group G = O(n, 1) acts irreducibly