CHAPTER 1
Introduction
A representation π of a group G defines a representation of a subgroup G by
restriction. In general irreducibility is not preserved by the restriction. If G is
compact then the restriction π|G is isomorphic to a direct sum of irreducible rep-
resentations π of G with multiplicities m(π, π ). These multiplicities are studied
by using combinatorial techniques. If G is not compact and the representation π
is infinite-dimensional, then generically the restriction π|G is not a direct sum of
irreducible representations [16] and we have to consider another notion of multi-
plicity.
For a continuous representation π of G on a complete, locally convex topological
vector space Hπ, the space

of
C∞-vectors
of is naturally endowed with a
Fr´ echet topology, and (π, Hπ) gives rise to a continuous representation
π∞
of G on

∞.
If is a Banach space, then the Fr´ echet representation
(π∞,

∞)
depends
only on the underlying (g,K)-module (Hπ)K. Given another continuous represen-
tation π of the subgroup G , we consider the space of continuous G -intertwining
operators (symmetry breaking operators)
HomG
(π∞|G
,
)∞).
The dimension m(π, π ) of this space yields important information of the restriction
of π to G and is called the multiplicity of π occurring in the restriction π|G . Notice
that the multiplicity m(π, π ) makes sense for non-unitary representations π and
π , too. In general, m(π, π ) may be infinite. For detailed analysis on symmetry
breaking operators, we are interested in the case where m(π, π ) is finite. The
criterion in [25] asserts that the multiplicity m(π, π ) is finite for all irreducible
representations π of G and all irreducible representations π of G if and only if
the minimal parabolic subgroup P of G has an open orbit on the real flag variety
G/P , and that the multiplicity is uniformly bounded with respect to π and π if
and only if a Borel subgroup of GC has an open orbit on the complex flag variety
of GC.
The classification of reductive symmetric pairs (g, g ) satisfying the former con-
dition was recently accomplished in [20]. On the other hand, the latter condition
depends only on the complexified pairs (gC, gC), for which the classification is much
simpler and was already known in 1970s by Kr¨ amer [28]. In particular, the multi-
plicity m(π, π
) is uniformly bounded if the Lie algebras (g, g ) of (G, G ) are real
forms of (sl(N + 1, C), gl(N, C)) or (o(N + 1, C), o(N, C)).
In this article we confine ourselves to the case
(1.1) (G, G ) = (O(n + 1,1),O(n,1)),
1
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