In representation theory, it's nice to decompose a given representation into smaller ones. One technique is by utilizing **extra symmetries**.

To be more precise, let $R$ be an associative unital complex algebra and $M$ a finite dimensional left $R$-module. Let further $G$ be a finite group with a complex-linear action on $M$, such that both actions commute. Then any $G$-isotypic component $M_i$ of $M$ is a left $R$-module, yielding a decomposition of $R$-module

$$ M = M_1 \oplus \ldots \oplus M_l.$$

The proof of it lies in the fact that the projection to $\chi$'th $G$-isotypic component is realizable as an element in the group algebra $A = \mathbb{C}[G]$

$$ \frac{1}{|G|} \Sigma_{g\in G} \overline{\chi(g)} g.$$

Examples arise in the field of modular forms (diamond operator.. etc) [1] and geometric representation theory [2, theorem 8.1.16] (classification of the representations of the affine Hecke algebra, extra symmetries from the stabilizer of a chosen point in the nilpotent cone.. etc).

**However**, this seems to rely on the fact that the projection is realizable as an element. Moreover, the element in this case seems exclusive for topological compact groups.. therefore the questions:

### Questions

1. What else algebra $A$ has an element that serves as the projection to a given isotypic component for *any* of its left modules?

2. More such examples as in *extra compatible symmetries reducing the complexity of certain mathematical object*?

### Reference

+ [1] A First Course in Modular Forms-[Diamond and Shurman]
+ [2] Representation Theory and Complex Geometry-[Chriss and Ginzburg]