I'm given the following in a homework question:
Let $G$ be a group and $k$ an algebraically closed field.
(a) Show that the action of $G \times G$ on $C_k (G)$ defined by $ (g_1, g_2) \varphi (x) = \varphi(g_1^{-1} x g_2) \hspace{1cm} g_1, g_2 , x \in G , \varphi \in C_k(G)$ defines a representation $\pi$ of $G \times G$
Let $\rho : G \rightarrow GL(E) $ be a finite dimensional irreducible representation. Let $ M(\rho) = \{$ span of the matrix coefficients of $ \rho \} \subset C_k(G)$
(b) Show that $M(\rho)$ is a subrepresentation of $\pi$.
My question(s):
1) what is "span of the matrix coefficients"?
2) I need to show that for $m \in M(\rho)$: $\pi (g_1, g_2, m) \in M(\rho)$ $\forall g_1, g_2 \in G$. Can I write "let $M := \rho$", the matrix representation of $\rho$ and then $M(\rho) = \{cM | c \in k\}$?
3) And am I right in assuming that $E$ has to be a vector space over $k$?
Many thanks for your help.