A general theory of multiplicity-free actions of $G\times H$?

Question

There seem to be a bunch of different results of the form "This nice representation $V$ of $G\times H$ breaks up as $\bigoplus_\lambda V_\lambda \otimes W_\lambda$, where the $(V_\lambda)$ are distinct irreps of $G$ and the $(W_\lambda)$ are distinct irreps of $H$."

Examples I know (described algebraically):

1. $G = S_n$, $H = GL(k)$, $V = ({\mathbb C}^k)^{\otimes n}$.

2. $G = H$, $V = Fun(G)$ (the Peter-Weyl theorem)

3. $G = GL(m)$, $H = GL(n)$, $V = Sym({\mathbb C}^m \otimes {\mathbb C}^n)$.

I know there should be some analogous one with $O(m)$ and $Sp(n)$...

My questions:

1. What are other examples?

2. Is there a general theory of these, that in particular would encompass Peter-Weyl? (So not just about actions on symmetric algebras.)

Answer 1

When $K$ is a complex reductive linear algebraic group, an affine variety $X$ equipped with an action of $K$ is called multiplicity-free if its ring of functions $\mathbb{C}[X]$ is multiplicity-free as a $K$-module. When $K$ is connected, this means that a Borel subgroup has a dense orbit on $X$ (if $X$ is also normal, such a variety is called spherical).

In the question's example (2), $K = G \times G$ and $X = G$ (with $G \times G$ acting by left and right multiplication) and $V= \mathbb{C}[G]$. In example (3), $K=GL(m)\times GL(n)$ and $X = (\mathbb{C}^m \otimes \mathbb{C}^n)^*$.

There is a general theory of spherical varieties, and there are classifications. For example, Kac[MR575790], Leahy[MR1650378] and Benson-Ratcliff[MR1382030] did the case where $X$ is itself a representation of $K$ (see also [Howe and Umeda, MR1116239] and [Knop, MR1653036]). Homogeneous spherical varieties were classified in [Kramer, MR0528837] and [Brion, MR0822838].

Edit: As pointed out by Allen in the comments, the condition that $V$ be multiplicity-free as a $(G \times H)$-module is weaker than the condition on $V$ in the original question. Let's call a $V$ as in the question strongly MF. As Allen points out in another comment, $\mathbb{C}[X]$ is strongly MF if and only if $B_G \times U_H$ and $U_G \times B_H$ have open orbits on $X$ (here $B_G$ is a Borel subgroup of $G$, and $U_G$ a maximal unipotent subgroup of $G$).

Answer 2

Well, one "general" theorem is

If A and B are semi-simple algebras acting faithfully on a finite dimensional vector space $V$ such that $\mathrm{End}_A(V)=B$ and vice versa, then tensoring with $V$ is a Morita equivalence between $A$-modules and $B$-modules.

You mostly have infinite examples up there, but they are direct limits of situations like this. Presumably with a little more care, one could make this work for something "graded finite dimensional" or a Hilbert space.

If you think for a moment about what it means to be a Morita equivalence between semi-simple algebras, it means must have exactly this multiplicity free form. Of course, the question of how to find situations like this is a little more fraught.