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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.

Hyeongmuk LIM
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1 Answers1

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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.)

Qiaochu Yuan
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  • Do you mean the universality in question was that the action of Hecke algebra on $V^K$ is natural in $V$? Can I say more, e.g. a $\mathcal{H}_{G,K}-$linear morphism $V^K\to W^K$ uniquely extends to a $\mathbb{C}[G]-$linear morphism $V\to W$? – Hyeongmuk LIM Sep 20 '23 at 11:04
  • @Hyeongmuk: no, that doesn't follow. I mean the universality in question is that the Hecke algebra is the universal thing which acts functorially on $V^K$. Said another way, consider the category of algebras $A$ over the base field equipped with a functorial action $A \otimes V^K \to V^K$; then the Hecke algebra is the terminal object in this category. This is just another way of describing the endomorphisms of the functor. – Qiaochu Yuan Sep 20 '23 at 16:52
  • Thank you for the reply. I will think about it. – Hyeongmuk LIM Sep 21 '23 at 03:35