A compact symmetric space, for purposes of this article, is a quotient
G
/
K
G/K
, where
G
G
is a compact connected Lie group and
K
K
is the identity component of the subgroup of fixed points of an involution. A branching theorem describes how an irreducible representation decomposes upon restriction to a subgroup. The article deals with branching theorems for the passage from
G
G
to
K
2
×
K
1
K_{2}\times K_{1}
, where
G
/
(
K
2
×
K
1
)
G/(K_{2}\times K_{1})
is any of
U
(
n
+
m
)
/
(
U
(
n
)
×
U
(
m
)
)
U(n+m)/(U(n)\times U(m))
,
S
O
(
n
+
m
)
/
(
S
O
(
n
)
×
S
O
(
m
)
)
SO(n+m)/(SO(n)\times SO(m))
, or
S
p
(
n
+
m
)
/
(
S
p
(
n
)
×
S
p
(
m
)
)
Sp(n+m)/(Sp(n)\times Sp(m))
, with
n
≤
m
n\leq m
. For each of these compact symmetric spaces, one associates another compact symmetric space
G
′
/
K
2
G’/K_{2}
with the following property: To each irreducible representation
(
σ
,
V
)
(\sigma ,V)
of
G
G
whose space
V
K
1
V^{K_{1}}
of
K
1
K_{1}
-fixed vectors is nonzero, there corresponds a canonical irreducible representation
(
σ
′
,
V
′
)
(\sigma ’,V’)
of
G
′
G’
such that the representations
(
σ
|
K
2
,
V
K
1
)
(\sigma |_{K_{2}},V^{K_{1}})
and
(
σ
′
,
V
′
)
(\sigma ’,V’)
are equivalent. For the situations under study,
G
′
/
K
2
G’/K_{2}
is equal respectively to
(
U
(
n
)
×
U
(
n
)
)
/
diag
(
U
(
n
)
)
(U(n)\times U(n))/\text {diag}(U(n))
,
U
(
n
)
/
S
O
(
n
)
U(n)/SO(n)
, and
U
(
2
n
)
/
S
p
(
n
)
U(2n)/Sp(n)
, independently of
m
m
. Hints of the kind of “duality” that is suggested by this result date back to a 1974 paper by S. Gelbart.