$C_c^\infty(G)$ acts on $L^2(\Gamma \backslash G)$ by convolution operators: $$ \phi \in C_c^\infty(G): f \in L^2 \longmapsto \left( x \mapsto \int\limits_{G} \phi(g) f(xg) d g\right).$$ Call the is algebra representation $\pi$.

Restrict this algebra representation to $C_c^\infty(K\backslash G/K)$ and call it $\sigma$. It acts non-trivial only on the subspace $L^2(\Gamma \backslash G /K)$ by the Schur orthogonality relations.

Let $\hat{G}$ denote the unitary dual of $G$. Note that both algebra respresentation have a direct integral decomposition into irreducible subrepresentations $$ \pi = \int\limits_{\hat{G}} \tau \; d \mu_\pi(\tau), \qquad \pi = \int\limits_{\hat{G}} \tau \; d \mu_\sigma(\tau).$$

The support of $\mu_\sigma$ is then precisely the spherical representaion of $G$, i.e., those with a $K$-invariant vector. On this subset of $\hat{G}$ it coincides with $\mu_\pi$, whose support is larger.

The following works in the generality of a locally compact unimodular group $G$ and $\Gamma$ being a closed unimodular subgroup and $K$ being compact.

$C_c^\infty(G)$ acts on $L^2(\Gamma \backslash G)$ by convolution operators: $$ \phi \in C_c^\infty(G): f \in L^2 \longmapsto \left( x \mapsto \int\limits_{G} \phi(g) f(xg) d g\right).$$ Call this algebra representation $\pi$.

Restrict this algebra representation to $C_c^\infty(K\backslash G/K)$ and call it $\sigma$. It acts non-trivial only on the subspace $L^2(\Gamma \backslash G /K)$ by the Schur orthogonality relations.

Let $\hat{G}$ denote the unitary dual of $G$. Note that both algebra representations have a direct integral decomposition into irreducible representations $$ \pi = \int\limits_{\hat{G}} \tau \; d \mu_\pi(\tau), \qquad \pi = \int\limits_{\hat{G}} \tau \; d \mu_\sigma(\tau).$$

The support of $\mu_\sigma$ is then contained in the set of the spherical representations of $G$, i.e., those with a $K$-invariant vector. On its support, $\mu_\sigma$ coincides with $\mu_\pi$, whose support may be larger.