A function canonically associated to an irreducible representation in L^2(M) for a Riemannian G-manifold M. Who has seen it?

Question

The following is my first question here on mathoverflow.

Let $M$ be a closed connected Riemannian manifold with an isometric effective action of a compact connected Lie group $G$. Consider the representation of $G$ on $L^2(M)$ given by $gf(x):=f(g^{-1}x)$, which decomposes into a direct sum of irreducible representations. Let $V\subset L^2(M)$ be such an irreducible $G$-representation. Choose a function $a\in V$ with $||a||_{L^2(M)}=1$ and define a new, $G$-invariant function in $L^1(M)$ by

$v(x):=\intop_G |a(g^{-1}x)|^2dg$, $\quad x\in M$,

where $dg$ is the Haar measure on $G$. Then it is easy to prove (using the Schur orthogonality relations) that the function $v$ is independent of the choice of $a$. Indeed, let $u_1,\ldots,u_{dim(V)}$ be some $L^2$-ONB of $V$. Then

$v(x)=\frac{1}{dim(V)}\sum_{i=1}^{dim(V)} |u_i(x)|^2$, $\quad x\in M$.

Does the function $v$ have a common name? Do you know references where such a construction has been studied? The map $v$ seems to be a non-trivial function associated to the irreducible representation $V$ in an elementary way, and surely it has been studied or at least mentioned somewhere in the literature.

Thanks in advance for any helpful remarks and references.

Edit I: To get a more concrete grasp of the function $v$, consider the following example. Let $SO(2)$ act on the two-sphere (with standard metric) by rotations around the axis through the north pole and the south pole. Then each single spherical harmonic spans a one-dimensional irreducible representation of $SO(2)$ in $L^2(S^2)$, and in this case the function $v$ associated to such an irreducible representation is just the squared absolut value of the spherical harmonic spanning that representation. The squared absolute value of most spherical harmonics is indeed a non-trivial (i.e. non-constant) $SO(2)$-invariant function on $S^2$. More generally, whenever $V$ is one-dimensional, spanned by some function $u$ with $L^2$-norm $1$, then $v=|u|^2$. Note that this implies that every function spanning a one-dimensional subrepresentation of the representation on $L^2(M)$ given above must have a $G$-invariant absolut value. At the moment even that last fact seems non-trivial to me.

Edit II: As a second, easier example, consider the case $M=G$ with $G$-action given by left-multiplication. Then $v\equiv 1$, so it holds

$\frac{1}{dim(V)}\sum_{i=1}^{dim(V)} |u_i|^2 \equiv 1$

for every $L^2$-ONB $u_1,\ldots,u_{dim(V)}$ of an irreducible $G$-rep. $V\subset L^2(G)$.