From the representation category of a Lie group and the representation on a homogeneous space, can we reconstruct the stabiliser subgroup reps?

Question

Given a Lie group $G$ and a transitive action $- \triangleright - : G \times X \to X$ on a homogeneous space, we can recover the stabiliser subgroup $H_x$ of a point $x \in X$. It is the subgroup of $G$ that leaves $x$ invariant. We then have $G/H \cong X$ as $G$-spaces.

Also, given the representation category of a compact Lie group and the forgetful functor of that category to vector spaces, it is possible to reconstruct the Lie group as natural transformations from this functor to itself. This is known as Tannaka-Krein duality.

I would like to combine these two reconstruction theorems. This is a follow-up on this question, where I learnt that the representation of $G$ on $\mathrm{L}^2(G/H)$ is: $$\bigoplus_{[\rho] \in \Lambda} \rho^* \otimes \operatorname{Inv}_H \rho$$ Here, $\Lambda$ is the set of irreducible unitary representations of $G$ and $\operatorname{Inv}_H$ is the $H$-invariant subspace.

My question is now, given the representation category of a compact Lie group and the representation of $G$ on $\mathrm{L}^2(G/H)$, is it possible to reconstruct the representation category of $H$?

In other words, can the reconstruction of homogeneous spaces be done in the "dual", Tannakian picture? If no, what further information is needed?

Edit: I'd like to recover the representations of $H$ as a monoidal category, that is, with tensor products. I'm happy about hints in the finite case as well.

Edit 2: Some further background: One can understand a group $G$ as a one-object category $BG$. The group inclusion is then a faithful functor $i: BH \hookrightarrow BG$. A representation is a functor $\rho: BG \to \operatorname{Vect}$. The restriction functor $\operatorname{Res}_i: \operatorname{Rep}G \to \operatorname{Rep}H$ then is composition with $i$. The forgetful functor $U$ that is important in Tannaka-Krein duality is restriction to the trivial group, or equivalently composition with the inclusion of the trivial group. (Following the slogan "Linear algebra is representation theory of the trivial group".) So I'm looking for factorisation $\operatorname{Rep}G \stackrel{U_G}{\to} \operatorname{Vect} = \operatorname{Rep}G \stackrel{F}{\to} \operatorname{Rep}H \stackrel{U_H}{\to} \operatorname{Vect}$, where I'm only given the data $\operatorname{Rep}G$, $\mathrm{L}^2(G/H)$ and $U_G$. Maybe this helps some more categorically minded people.

Answer 1

I understand it for finite groups now, and I suspect that it's similar for semisimple Lie groups.

These statements are equivalent, and true for finite groups:

• The adjunction $\operatorname{Res} \dashv \operatorname{CoInd}$ is monadic (for compact $G/H$), its monad is $\mathrm{L}^2(G/H) \otimes -$.
• $H$-representations are $L^2(G/H)$-representations internal to the category of $G$-representations.
• $H$-representations are $G$-equivariant maps $\mathrm{L}^2(G/H) \otimes \rho \to \rho$, where $\rho$ is a $G$-representation, satisfying certain associativity and unit laws.