Let
G
G
be a compact connected semisimple Lie group, let
K
K
be a closed subgroup of
G
G
, let
Γ
\Gamma
be a finite subgroup of
G
G
, and let
τ
\tau
be a finite-dimensional representation of
K
K
. For
π
\pi
in the unitary dual
G
^
\widehat G
of
G
G
, denote by
n
Γ
(
π
)
n_\Gamma (\pi )
its multiplicity in
L
2
(
Γ
∖
G
)
L^2(\Gamma \backslash G)
.
We prove a strong multiplicity one theorem in the spirit of Bhagwat and Rajan, for the
n
Γ
(
π
)
n_\Gamma (\pi )
for
π
\pi
in the set
G
^
τ
\widehat G_\tau
of irreducible
τ
\tau
-spherical representations of
G
G
. More precisely, for
Γ
\Gamma
and
Γ
′
\Gamma ’
finite subgroups of
G
G
, we prove that if
n
Γ
(
π
)
=
n
Γ
′
(
π
)
n_{\Gamma }(\pi )= n_{\Gamma ’}(\pi )
for all but finitely many
π
∈
G
^
τ
\pi \in \widehat G_\tau
, then
Γ
\Gamma
and
Γ
′
\Gamma ’
are
τ
\tau
-representation equivalent, that is,
n
Γ
(
π
)
=
n
Γ
′
(
π
)
n_{\Gamma }(\pi )=n_{\Gamma ’}(\pi )
for all
π
∈
G
^
τ
\pi \in \widehat G_\tau
.
Moreover, when
G
^
τ
\widehat G_\tau
can be written as a finite union of strings of representations, we prove a finite version of the above result. For any finite subset
F
^
τ
\widehat {F}_{\tau }
of
G
^
τ
\widehat G_{\tau }
verifying some mild conditions, the values of the
n
Γ
(
π
)
n_\Gamma (\pi )
for
π
∈
F
^
τ
\pi \in \widehat F_{\tau }
determine the
n
Γ
(
π
)
n_\Gamma (\pi )
’s for all
π
∈
G
^
τ
\pi \in \widehat G_\tau
. In particular, for two finite subgroups
Γ
\Gamma
and
Γ
′
\Gamma ’
of
G
G
, if
n
Γ
(
π
)
=
n
Γ
′
(
π
)
n_\Gamma (\pi ) = n_{\Gamma ’}(\pi )
for all
π
∈
F
^
τ
\pi \in \widehat F_{\tau }
, then the equality holds for every
π
∈
G
^
τ
\pi \in \widehat G_\tau
. We use algebraic methods involving generating functions and some facts from the representation theory of
G
G
.