14 2. SYMMETRY BREAKING FOR SPHERICAL PRINCIPAL SERIES

on

Hj(Rn,1)

for j ∈ N. By the classical branching law, we have a G -irreducible

decomposition:

(2.12)

Hi(Rn+1,1)|G

i

j=0

Hj(Rn,1).

Second, we notice that there are three non-trivial one-dimensional represen-

tations of the disconnected group G. For our purpose, we consider the following

one-dimensional representation

(2.13) χ : O(n + 1,1) → {±1}

by the composition of the following maps

G → G/G0 O(n + 1) × O(1)/SO(n + 1) × SO(1) {±1} × {±1}

pr2

→ {±1},

where G0 = SO0(n+1, 1) is the identity component of G, and pr2 denotes the second

projection. Similarly, we define χ : O(n, 1) → {±1}. Then by inspecting the action

of the four disconnected components of G, we have the following isomorphisms as

representations of G and G

, respectively:

F (i)

χi

⊗

Hi(Rn+1,1),

(2.14)

F (j) (χ

)j

⊗

Hj(Rn,1).

(2.15)

Combining (2.12) with (2.14) and (2.15), we get the following branching law

for the restriction G ↓ G :

F (i)|G

i

j=0

(χ

)i−j

⊗ F (j).

Thus we have shown the following proposition.

Proposition 2.3 (branching law of F (i) for G ↓ G ). Suppose i, j ∈ N.

1) HomG (F(i),F(j)) = 0 if and only if 0 ≤ j ≤ i and i ≡ j mod 2.

2) HomG0 (F(j),F(i)) = 0 if and only if 0 ≤ j ≤ i.

2.3. Symmetry breaking operators and spherical principal series rep-

resentations. We refer to non-trivial homomorphisms in

H(λ, ν) := HomG (I(λ),J(ν))

as intertwining restriction maps or symmetric breaking operators. In the next chap-

ter general properties of symmetry breaking operators for principal series repre-

sentations are discussed. In this section we will illustrate the functional equations

satisfied by the continuous symmetry breaking operators (Theorem 8.5, see also

Theorem 12.6) by analyzing their behavior on I(λ) × J(ν) where both representa-

tions I(λ) and J(ν) are reducible, i.e., (λ, ν) are in

L = {(i, j) : i, j ∈ Z and (i, j) ∈ (0,n) × (0,n − 1)}.

The Weyl group S2 × S2 of G × G acts on L. The action is generated by the

action of the generators (λ, ν) → (−λ + n, ν) and (λ, ν) → (λ, −ν + n − 1). We

write Leven ⊂ L for the orbit of

L = {(i, j) : i, j non-positive integers, i = j mod 2}