Let $G$ be a finite group, K a subgroup, and $X = \frac{G}{K}$ the space of cosets. Consider $L(X) = \bigoplus_{m=0}^N V_i$ the decomposition of $L(X)$ into irreducible sub-representations, in which each $(\rho_i,V_i)$ is unitary, and $N+1$ is equal to the dimension of the space $L(X)^K$ of K-invariant vectors.
How to show that $\frac{1}{|K|} \sum_{k \in K} \rho_i(k)$ is the orthogonal projection onto the one-dimensional subspace $V_i^k$, where $V_i^k = \{v \in V : \rho_i(k)v = v, \forall k \in K\}.$
information in the book: Harmonic Analysis on Finite Groups. TULLIO CECCHERINI-. SILBERSTEIN,. FABIO SCARABOTTI. AND FILIPPO TOLLI - Proposition 4.7.2