Matrix coefficient

Remarks

If $\dim V = n$ and $e_1, \dots, e_n$ is a basis with dual basis $e_1^*, \dots, e_n^*$ in the dual space, then $\pi_{ij}(g) = e_i^*(\pi(g) e_j)$ is the $(i, j)$ entry of the matrix of $\pi(g)$. Every matrix coefficient of $\pi$ is a linear combination of the $\pi_{ij}$, because $c_{v, \ell}$ is linear in $v$ and in $\ell$. The left and right translates of a matrix coefficient of $\pi$ are again matrix coefficients of $\pi$: $c_{v, \ell}(h g) = c_{\pi(g) v, \ell}(h)$.

Example: let $G = \mathrm{SO}(2)$ act on $\R^2 \subset \C^2$ by rotations $r_\theta = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix}$. With $v = e_1$ and $\ell = e_2^*$, the matrix coefficient is $\theta \mapsto \sin \theta$.

Non-example: on the circle group $\R / 2\pi\Z$, the matrix coefficients of the finite-dimensional continuous representations are exactly the trigonometric polynomials. The function $\theta \mapsto \abs{\sin \theta}$ has infinitely many nonzero Fourier coefficients, so it is not a matrix coefficient of any finite-dimensional continuous representation.