I've culled together a slick proof of $\Bbb C[G]\cong\bigoplus_{V\in\widehat{G}}{\rm End}(V)$ (Peter-Weyl decomposition) for finite groups using the fact that the fiber functor (that is, the forgetful functor ${\sf Rep}(G)\to{\sf Vect}_{\Bbb C}$) is representable (namely represented by $\Bbb C[G]$, that is $\hom_G(\Bbb C[G],V)\cong V$ canonically for all representations $V$), and from this noted that Schur's orthogonality relations are simply the statement that the isotypical identity maps in the direct sum are orthogonal idempotents. I've also seen the representability of the fiber functor used in the proof of Tannaka duality for finite groups.

I asked in MO's homotopy chatroom if the fiber functor is still representable if $G$ is compact, and if so what it should be represented by. One might consider the regular representation $L^2(G)$ again, but this has no identity element to evaluate a function $L^2(G)\to V$ at. One chatter suggested the dual of $L^2(G)$ so that we could evaluate at the Dirac delta functional, which is essentially what is being done for finite groups. Another commenter suggested that $L^2(G)$ should still be the correct representing object, which is because of its PW decomposition (which is unfortunate since ideally I'd want representability logically prior to knowing PW). I assume the natural isomorphism would be evaluating at some kind of approximate identity? Also, how would we leverage PW for proving representability?

I know for finite $G$ I used the distributivity of $\hom(-,-)$ in both arguments. I know that hom is finitely distributive in both arguments in the category of Hilbert spaces, but I don't know if distributivity extends far enough to work with $L^2(G)$. I can't even determine if one of $\bigoplus\hom(V_i,W)$ and $\hom(\bigoplus V_i,W)$ or if one of $\bigoplus\hom(V,W_i)$ and $\hom(V,\bigoplus W_i)$ canonically inject into the other. Comparing the first two spaces, I would need to connect the following two conditions somehow:

$$\sum_{i\in I}\left(\sup_{\|v_i\|_{V_i}=1}\|\varphi_i(v_i)\|_W\right)^2<\infty \tag{1}$$

$$\sup_{\sum_{i} \|v_i\|_{V_i}^2=1}\left\|\sum_{i\in I}\varphi_i(v_i)\right\|_W<\infty \tag{2}$$

I know I can square $(2)$, write it as a double summation of inner products $\langle\varphi_i(v_i),\varphi_j(v_j))\rangle_W$ and then employ the Cauchy-Schwarz identity but then I get $(\sum_i\|\varphi_i(v_i)\|_W)^2$ instead of $\sum_i\|\varphi_i(v_i)\|_W^2$. So:

  • Is the fiber functor $F:{\sf Rep}(G)\to{\sf Hilb}$ representable? If so, what is the representing object $U$, what is the natural isomorphism $\hom_G(U,V)\cong V$, and how do we prove it?
  • To what extent is $\hom(-,-)$ distributive in ${\sf Hilb}$, the category of Hilbert spaces with bounded linear operators for morphisms? Is it countably distributive? Are there canonical injections if not actual isomorphisms, and if so which direction are they?

With regards to hom-distributive for Hilbert spaces, Google has been surprisingly unhepful.

  • $\begingroup$ Am I the second commenter you refer to? If so, that is not what I said; I said that the correct representing object should be the algebraic direct sum $\bigoplus V \boxtimes V^{\ast}$, which is not a Hilbert space, so the appropriate category to work in is slightly different. The category one should work in is the ind-completion of the category of finite-dimensional representations (not unitary) of $G$, and then this clearly works for formal reasons but it doesn't tell you very much. $\endgroup$ – Qiaochu Yuan Mar 16 '15 at 4:39
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    $\begingroup$ I also don't recommend using the category of Hilbert spaces for these kinds of formal arguments; for example, what do you think a morphism of Hilbert spaces is? If the answer is a bounded operator, then you aren't requiring representations to be unitary. If the answer is a unitary operator, then homs are no longer vector spaces! You need to make this choice in order to check whether the Hilbert space direct sum is the categorical direct sum. $\endgroup$ – Qiaochu Yuan Mar 16 '15 at 4:40
  • $\begingroup$ @QiaochuYuan since representability of $F$ strikes me as so interesting/powerful when working with finite groups, I wanted to know to what extent that significance can still be true working with compact groups. That is my underlying motivation. Working with ind-completions might carry over the arguments from the case of finite groups naturally, so if that's all I want to accomplish I recognize that's what I should do, but as you say this might not tell us much - we tend to do representation theory of (locally) compact groups with Hilbert spaces anyway, for instance when we're discussing ... $\endgroup$ – whacka Mar 16 '15 at 17:09
  • $\begingroup$ ... Pontryagin duality, the Peter-Weyl decomposition of $L^2(G)$, or quantum mechanics, so $\sf Hilb$ is where I'm interested in being. I do want the homs to be spaces, so bounded linear operators are my morphisms. I believe I already know the Hilbert space direct sums are not categorical co/products, but I'm still interested if homs distribute over them. Also sorry for misunderstanding your comment in chat. I had a suspicious you might mean the algebraic direct sum (not the Hilbert space one) but didn't ask. Still curious if hom-distributivity is how you went from PW to representability. $\endgroup$ – whacka Mar 16 '15 at 17:10
  • $\begingroup$ Actually, in the case of semisimple Lie groups, Harish-Chandra introduced a purely algebraic setting for doing representation theory which in the compact case reduces precisely to the ind-completion of the category of finite-dimensional representations (en.wikipedia.org/wiki/%28g,K%29-module). It's a very natural setting for ignoring issues like "if morphisms between Hilbert spaces are bounded operators then what's the significance of the inner product in the first place?" $\endgroup$ – Qiaochu Yuan Mar 16 '15 at 17:13

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