Irreducibility of $\operatorname{Hom}_{N}(V, W)$ as a $Z(M,N)$-module I was reading an article on representations of symmetric groups (approach using Gelfand-Zetlin algebra) and I can't understand the following statement:

Let $M$ be a finite-dimensional semisimple algebra over field $\mathbb{C}$ and $N$ be its subalgebra. Consider finite-dimensional complex irreducible representations $V$ and $W$ of algebras $M$ and $N$, respectively. Then, $\operatorname{Hom}_{N}(V, W)$ is an irreducible $Z(M,N)$-module.

Here $Z(M,N)$ is the centralizer of subalgebra $N$ in algebra $M$, or
$$
Z(M,N)=\{m\in M\mid \forall n\in N\colon mn=nm\}.
$$
$\operatorname{Hom}_{N}(V, W)$ is the $M$-module of $N$-morphisms $\varphi\colon V\to W$ (so it's a $Z(M,N)$-module as well).
Probably, the main problem is that I don't see any connections between irreducible representations $N$ or $M$ and the centralizer $Z(M,N)$.
Any ideas why this statement is true?
 A: Our setting: $N \subseteq M$ are finite-dimensional semisimple algebras over an algebraically closed field $\mathbb{F}$.
For any representations $V$ of $M$, and $W$ of $N$, the $\mathbb{F}$-vector space $\operatorname{Hom}_N(W, \operatorname{res}_N V)$ may be equipped with a left action by the centraliser $Z_M(N)$ by defining $(z \cdot f)(w) = z f(w)$. We will make a two-step argument:

*

*If $W$ is simple, every element of $\operatorname{End}_{\mathbb{F}}(\operatorname{Hom}_N(W, \operatorname{res}_N V))$ may be written as $f \mapsto \psi \circ f$ for some (not necessarily unique) $\psi \in \operatorname{End}_N(V)$, and

*If $V$ is simple, the centraliser $Z_M(N)$ surjects onto $\operatorname{End}_N(V)$.

After this, we know that the centraliser $Z_M(N)$ acts as the full endomorphism algebra $\operatorname{End}_{\mathbb{F}}(\operatorname{Hom}_N(W, \operatorname{res}_N V))$, and hence $\operatorname{Hom}_N(W, \operatorname{res}_N V)$ is simple as a $Z_M(N)$-module. Each step is indpendent from the other and explained below.

Let $W$ and $V$ be finite-dimensional modules over a semisimple algebra $N$, with $W$ a simple module. There is a unique decomposition $V = X \oplus Y$, where $X \cong W^{\oplus k}$ for some $k \geq 0$ and $Y$ contains no summand isomorphic to $W$. (The submodule $X$ is sometimes called the $W$-isotypic component of $V$). This already tells us that $\operatorname{Hom}_N(W, V)$ is $k$-dimensional as an $\mathbb{F}$-vector space, but we will need to make things a bit more precise to get the less obvious fact that any $\mathbb{F}$-linear endomorphism $\alpha$ of $\operatorname{Hom}_N(W, V)$ can be written as postcomposition by some $N$-equivariant $\psi \colon V \to V$.
Let $f_1, \ldots, f_k \colon W \to V$ form a basis of $\operatorname{Hom}(W, V)$, so we have $V = \operatorname{im}(f_1) \oplus \cdots \oplus \operatorname{im}(f_k) \oplus Y$. Let $g_1, \ldots, g_k \colon V \to W$ be sections of the $f_i$, so that we have $g_i f_i = \operatorname{id}_W$, $g_i f_j = 0$ for $i \neq j$, and $\sum_i f_i g_i \colon V \to V$ is the projector to $X$.
Now fix an $\mathbb{F}$-linear endomorphism $\alpha \in \operatorname{End}_{\mathbb{F}}(\operatorname{Hom}_N(W, V))$, and define $\psi = \sum_i (\alpha f_i) \circ g_i$. As a sum of compositions of $N$-equivariant maps, $\psi$ is $N$-equivariant, and we have that $\psi \circ f_i = \alpha f_i$ as required.
Note: For this part we needed both that $W$ is simple (otherwise there is no hope that any endomorphism can be written as postcomposition, since postcomposition cannot narrow the kernel of a map), and that $V$ is semisimple (to guarantee the existence of sections, or alternatively to guarantee "enough linear independence" to construct $\psi$).

Next, the connection to centralisers. Let $V_1, \ldots, V_r$ be a complete irredundant list of its simple modules of $M$, with action maps ($\mathbb{F}$-algebra homomorphisms) $\varphi_i \colon M \to \operatorname{End}_{\mathbb{F}}(V_i)$. There is an isomorphism of algebras
$$ \varphi = (\varphi_1, \ldots, \varphi_l) \colon M \to \operatorname{End}_{\mathbb{F}}(V_1) \times \cdots \times \operatorname{End}_{\mathbb{F}}(V_l), $$
or put another way, each element $m \in M$ is specified uniquely and freely by giving a collection $(\phi_1(m), \ldots, \phi_l(m))$ of linear endomorphisms of its simple modules.
Now enter another semisimple algebra $N \subseteq M$ into the picture, and the centraliser $Z_M(N) \subseteq M$ consisting of those elements of $M$ commuting with every element of $N$. Interpreting this condition using the isomorphism $\varphi$ gives that
$$ Z_M(N) = \left\{ m \in M \mid \varphi_i(m) \in \operatorname{End}_N(V_i) \text{ for all } i\right\} = \varphi^{-1}\left(\operatorname{End}_{N}(V_1) \times \cdots \times \operatorname{End}_{N}(V_l)\right). $$
Therefore the elements of the centraliser are precisely those elements of $M$ which act $N$-invariantly in every $M$-module, and furthermore, every $N$-invariant endomorphism of a simple $M$-module can be written as the action of some element in $Z_M(N)$.

Perhaps there is a more simple way to see these facts - I know that at the start of Kleschev's book Linear and Projective Representations of Symmetric Groups there is a very short argument based on reducing to the case $M = \operatorname{End}_{\mathbb{F}}(V)$, but I never felt too comfortable with that style argument (and find the more general view easier to apply "in the real world")
