In this post, I mainly focus on introducing Gelfand pairs of finite groups.

### Representations of finite groups

Let’s recall some concepts. Let \(G\) be a finite group. A (complex) *representation* of \(G\) is a pair \((\rho, V)\) where \(V\) is a vector space over \(\C\) and \(\rho: G \to GL(V)\) is a group homomorphism. Let \((\rho_1, V_1)\) and \((\rho_2, V_2)\) be representations of \(G\), a *\(G\)-morphism*, or simply a *morphism*, between these two representations is a linear map \(\varphi: V_1 \to V_2\) such that for any \(g \in G\) and \(v \in V_1\),
\[\varphi(\rho_1(g)v) = \rho_2(g)\varphi(v).\]
Then \(\mathrm{Hom}_G(\rho_1, \rho_2)\), the set of morphisms between \(\rho_1\) and \(\rho_2\), is a vector space over \(\C\). A representation \((\rho, V)\) is *irreducible*, if there is no proper subspace \(W \subset V\), such that \(\rho(g)w \in W\) for any \(g \in G\) and \(w \in W\). Between irreducible representations, we have Schur’s lemma.

**Theorem 1 (Schur’s lemma).**
Let

\(\rho_1\) and

\(\rho_2\) be irreducible representations of

\(G\). Then

\[\begin{equation*}
\mathrm{Hom}_G(\rho_1, \rho_2) \cong \left\{
\begin{array}{ll}
\C, & \text{ if } \rho_1 \cong \rho_2; \\
0, & \text{ otherwise.}
\end{array}
\right.
\end{equation*}\]
It’s well known that \(G\) is semisimple, i.e., any representation of \(G\) decomposes into a direct sum of irreducible representations. A representation \((\rho, V)\) of \(G\) is called *multiplicity free*, if any irreducible representation appearing in the decomposition of \((\rho, V)\) appears exactly once, up to isomorphism.

Let \(H\) be a subgroup of \(G\). If \((\rho, V)\) is a representation of \(G\), we can certainly restrict the group homomorphism \(\rho: G \to GL(V)\) to \(H\) so that we can regard \((\rho, V)\) as a representation of \(H\). This is called the *restriction of representations* from \(G\) to \(H\), and the restricted representation of \(\rho\) will be denoted as \(r^G_H(\rho)\). Conversely, if \((\pi, W)\) is a representation of \(H\), we can build a representation \((\rho, V)\) of \(G\) out of \(\pi\), which is the *induced representation*, as follow.