**TL; DR.**


In representation theory, it's nice to decompose a given representation into smaller ones. One technique is by utilizing **extra symmetries**. Explicit examples come from compact groups, and I wonder if there are more such examples.


**Formal description**


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]



Making use of extra symmetries; more examples?
Your setup is much more specific than necessary. All you need is two rings $R, S$ with commuting actions on an abelian group $M$ (which is therefore an $(R, S^{op})$-bimodule) such that $M$ is semisimple as an $S$-module. Then, as an $S$-module, $M$ decomposes canonically into a direct sum of isotypic components, and since by hypothesis $R$ acts by $S$-module endomorphisms it preserves this decomposition. Of course this setup is easiest to guarantee if $S$ is semisimple and group algebras are a common source of semisimple rings.


This setup occurs, for example, in the [double commutant theorem](https://qchu.wordpress.com/2012/11/11/the-double-commutant-theorem/), which is famously applied to prove [Schur-Weyl duality](https://qchu.wordpress.com/2012/11/13/four-flavors-of-schur-weyl-duality/). Here $M = V^{\otimes n}$ where $V$ is a finite-dimensional vector space over a field of characteristic zero (to be safe), $R = k[GL(V)]$, and $S = k[S\_n]$.


Also, you ask:



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


Every semisimple ring has this property. By the Artin-Wedderburn theorem, a semisimple ring $R$ is a finite direct product $\prod\_i M\_{n\_i}(D\_i)$ of matrix rings over division rings, one for every simple module. The projection to the isotypic component given by the $i^{th}$ simple module is the primitive central idempotent given by the element of this direct product equal to the identity matrix in the $i^{th}$ factor and zero otherwise. The interesting thing about the group algebra case is that we don't need to write out the Artin-Wedderburn decomposition to write down the idempotent.



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


This is a maybe overly broad question. Using symmetry is a common and ubiquitous strategy in mathematics.


