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If $G$ is a (say) compact group and $V=\bigoplus_{i\in I}V_i$ the isotypic (a.k.a. primary) decomposition of a $G$-module, then

any $G$-invariant subspace $W\subset V$ writes $W=\bigoplus_{i\in I}(W\cap V_i)$.

While this isn’t hard to prove (similar to Hoffman-Kunze 1971, §7.5 for a single operator), it seems silly to redo it in a paper. Unfortunately, the only reference I’m familiar with omits the proof (Kirillov 1976, §8.3), and when using it (e.g. Lie groups VIII.3.1) Bourbaki refers to such an abstrusely worded version (Algebra VII.2.2) that unpacking it takes as much work as a direct proof.

Q: What is a good reference to quote for this? Bonus points if the case of non-algebraically closed fields is spelled out.

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    $\begingroup$ A better Bourbaki reference is Algèbre VIII, §4, Proposition 4 (unfortunately not yet translated, as far as I know). It works for semi-simple modules, no fields involved. $\endgroup$
    – abx
    Commented Mar 29, 2020 at 12:44
  • $\begingroup$ @abx Would you like to make your comment an answer? It is the one I wish I could accept, so far. $\endgroup$
    – Francois Ziegler
    Commented Apr 2, 2020 at 11:58
  • $\begingroup$ OK, done. $\qquad$ $\endgroup$
    – abx
    Commented Apr 2, 2020 at 13:45

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As suggested by the OP, I am turning my comment into an answer: a better Bourbaki reference is Algèbre VIII (new edition), §4, Proposition 4 d) (unfortunately not yet translated, as far as I know). It works for semi-simple modules over an arbitrary ring.

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  • $\begingroup$ Seems perfect, thanks. The earlier edition has it as VIII, §3, Proposition 9 b): “Tout sous-module $\mathrm N$ de $\mathrm M$ est somme directe des $\mathrm N\cap\mathrm M_\lambda$”. $\endgroup$
    – Francois Ziegler
    Commented Apr 2, 2020 at 21:30
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For compact groups you can quote IV.2.7 in Naimark-Stern. There the $T$-isotypical component is described as the image of an operator $E^T$.

For general semisimple categories it may be better to give a short modern proof. Define the center of the category. It is a product of division rings. Each simple object gives an idempotent in the center. This idempotent gives compatible idempotents in $Hom(V,V)$ and $Hom(W,W)$, which you can split off.

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  • $\begingroup$ Thank you. Naimark and Stern’s long section 2.7 (pp. 199-205) has much useful material on the primary decomposition of a given $V$ (what they call $S$) and projectors onto a $V_i$ (what they call $\smash{E^T}$), but they don’t seem to state let alone prove results on restriction to arbitrary invariant $W\subset V$, do they? $\endgroup$
    – Francois Ziegler
    Commented Apr 1, 2020 at 23:27
  • $\begingroup$ They don't, you are right. They prove that $E^T$ is the projection $V\rightarrow V_{i}$, killing all other isotypical components. The operator does not depend on a representation. Since $V_i$ (or $W_i$) is the 1-eigenspace of $E^T$ on $V$ (or $W$), it is a basic Linear Algebra exercise to deduce the statement you need. $\endgroup$
    – Bugs Bunny
    Commented Apr 2, 2020 at 6:11

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