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So I asked this on maths SE because I don't truly consider it to be a research level question. This question mostly arises out of my completely limited understanding of perverse sheaves. However I do think it is much more likely to receive an answer here, so I am posting it here as well.

Assume $\mathbf{G}$ is a connected reductive algebraic group over an algebraic closure $\overline{\mathbb{F}_p}$ for some prime $p>0$. Let $\mathscr{M}\mathbf{G}$ be the category of all $\overline{\mathbb{Q}_{\ell}}$-constructible perverse sheaves on $\mathbf{G}$ where $\ell>0$ is a prime different from $p$. Now assume $K \in \mathscr{M}\mathbf{G}$ is a semisimple object. Recently I saw the following way to decompose $K$. Let $\mathcal{A}$ be the endomorphism algebra $\mathrm{End}_{\mathscr{M}\mathbf{G}}(K)$ of $K$ then for any finite dimensional $\mathcal{A}$-module $E$ we define $K_E = \mathrm{Hom}_{\mathcal{A}}(E,K)$. We then have $K_E \in \mathscr{M}\mathbf{G}$ and is simple if and only if $E$ is simple. We then have a canonical isomorphism (in $\mathscr{M}\mathbf{G}$)

$$K \cong \bigoplus_E (E\otimes_{\mathcal{A}} K_E)$$

where the sum runs over a set of representatives from the isomorphism classes of simple $\mathcal{A}$-modules.

Now this idea is not unfamiliar. In the representation theory of finite groups one frequently uses the same idea to decompose induced representations. However here I am slightly confused as to why $K_E \in \mathscr{M}\mathbf{G}$? My initial thought was that $\mathscr{M}\mathbf{G}$ is an abelian category so using the Freyd-Mitchell embedding theorem one could transport this question to a question in the category of modules for some ring, where it is clearly true. However I don't think this will really work. Is this simply a fact of perverse sheaves?

Does anyone know of a good reference to read about the above decomposition? Is the canonical isomorphism complicated or is it just a fitting's lemma style argument?

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Pick a presentation $\mathcal{A}^n\to \mathcal{A}^m\to E\to 0$. Now you can realize $K_{E}$ in your category as the kernel of $K^m\to K^n$ where the map is the transpose of the original matrix. –  Donu Arapura Mar 8 '13 at 19:04
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1 Answer

up vote 0 down vote accepted

$\operatorname{Hom}_{\mathcal A}(E,K)$, as a functor from the category of $\mathcal A$-modules to the category of groups, can be expressed in terms of generators of relations. Specifically, you sum one copy of $K$ for each generator of $E$, then you sum one copy of $K$ for each relation of $E$, and you form a map from the first to the second, and take the kernel.

All you're being asked to do is apply the same construction in the category of perverse sheaves.

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Thanks, once again, for answering my question. This is so simple, I feel like I really should have come across it before. –  Jay Taylor Mar 8 '13 at 20:40
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