I am solving an assignment problem on the Hecke algebra of a finite group, and looking for an idea that might help find a right direction.
Given a pair of finite groups $G\geq K$, the Hecke algebra $\mathcal{H}_{G,K}$ can be defined as
$$ \mathcal{H}_{G,K} = \mathbb{C}[K\backslash G/K], $$
equipped with the convolution product s.t. $\delta_{KgK} \cdot \delta_{KhK} = \sum_{k\in K}\delta_{K(gkh)K}.$
Given a representation $\pi:G\to \mathrm{GL}(V)$, it is straightforward that $\mathcal{H}_{G,K}$ acts on the space $V^K$ of $K-$invariants
$$ V^K = \{v\in V : \pi(k) v = v, \; \forall k\in K\}. $$
The question is to figure out in what sense $\mathcal{H}_{G,K}$ acts "in a universal way" to $V^K$, given the hint that one may invoke Frobenius reciprocity. However, I have no idea to proceed on.
Am I supposed to formulate the answer in terms of the usual categorical notion of universal property? What aspects of $\mathcal{H}_{G,K}$ should I ponder on? Any kind of instruction will be greatly appreciated.
The assignment $F(V) = V^K$ is functorial in $V$, and one can ask for its endomorphisms as a functor. The answer is the following: $F(V)$ is representable by $\mathbb{C}[G/K]$ (this is a nice exercise, and is where you use Frobenius reciprocity here, thinking of $V^K$ as $\text{Hom}(1, \text{Res}_K^G(V))$), so it follows by the Yoneda lemma that
$$\text{End}(F) \cong \text{End}_G(\mathbb{C}[G/K]).$$
Now by the universal property this can be identified with $\mathbb{C}[G/K]^K$, which can further be identified as as a vector space with $\mathbb{C}[K \backslash G/K]$. It's another exercise to check that this identifies the above endomorphism algebra with the Hecke algebra, and identifies the action of the above endomorphism algebra on $F$ with the usual one. (You may or may not have to take a transpose somewhere, I'm not sure.)