CHAPTER 2

Symmetry breaking for the spherical principal

series representations

Before we start with the construction of the G -intertwining operators between

spherical principal series representations of G = O(n + 1,1) and G = O(n, 1) we

want to prove the main results (Theorems 1.2 and 1.3, see Theorems 2.5 and 2.6

below) about G -intertwining operators between irreducible composition factors of

spherical principal series representations. This is intended for the convenience of

the readers who are more interested in representation theoretic results rather than

geometric analysis arising from branching problems in representation theory. In the

proof we use the results about symmetry breaking operators for spherical principal

series representations of G and G that will be proved later in the article.

2.1. Notation and review of previous results. Consider the quadratic

form

(2.1) x0

2

+ x1

2

+ · · · + xn

2

−

xn+12

of signature (n+1,1). We define G to be the indefinite orthogonal group O(n+1, 1)

that preserves the quadratic form (2.1). Let G be the stabilizer of the vector

en =

t(0,

0, · · · , 0,1,0). Then G O(n, 1). We set

K :=O(n + 1) × O(1), (2.2)

K :=K ∩ G =

⎧⎛

⎨

⎩

⎝

A

1

ε

⎞

⎠

: A ∈ O(n),ε = ±1

⎫

⎬

⎭

O(n) × O(1). (2.3)

Then K and K are maximal compact subgroups of G and G , respectively.

Let g = o(n + 1,1) and g = o(n, 1) be the Lie algebras of G = O(n + 1,1) and

G = O(n, 1), respectively. We take a hyperbolic element H as

(2.4) H := E0,n+1 + En+1,0 ∈ g .

Then H is also a hyperbolic element in g, and the eigenvalues of ad(H) ∈ End(g)

are ±1 and 0. For 1 ≤ j ≤ n, we define nilpotent elements in g by

Nj

+

:= − E0,j + Ej,0 − Ej,n+1 − En+1,j, (2.5)

Nj

−

:= − E0,j + Ej,0 + Ej,n+1 + En+1,j. (2.6)

Then we have maximal nilpotent subalgebras of g:

n+ := Ker(ad(H) − 1) =

n

j=1

RNj

+,

n− := Ker(ad(H) + 1) =

n

j=1

RNj

−.

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