I am following Analysis and Geometry on Complex Homogeneous Spaces by Faraut et al. I'll set up all of what I need and then ask my questions.

Let $G$ be a connected semi-simple non-compact real Lie group. Let $K \subset G$ be a chosen maximal compact subgroup. We say an irreducible representation $(\pi,V)$ is spherical the space of $K$-fixed vectors $V^K$ is exactly $1$-dimensional.

We know that $(\pi,V)$ is spherical if and only if $V$ contains a cone $C$ invariant under the action of $G$. Moreover if $(\pi,V)$ is spherical then we can let $u_0 \in V$ be a $K$-fixed vector and $v_0 \in V$ a highest weight vector. We then have minimal and maximal cones $C_{min} = \overline{\mathbb{R}^+ conv(Gu_0 \cup \{0\})} = conv(G v_0)$ and $C_{max} = C_{min}^*$, the dual cone.

Here are my questions:

(1) In the general case, when we're not looking at the adjoint representation, are the cones interesting? Can the properties of the cone (a semi-algebraic set) tell us anything about the representation?

(2) Occasionally one gets $C_{min}=C_{max}^*$. Why is this interesting? Or what is the significance of it not happening?

I've worked out that $C_{min} \neq C_{max}$ for spherical representations of $SL(2,\mathbb{R})$. Is there some significance to this?


Regarding your second question, the cases where $C_{min} = C_{max}$ for cases where $G$ is a real form a complex semisimple Lie group have been classified by Misyureva. Basically, you get that $V$ is a Euclidean Jordan algebra and the cone is the closure of the self dual symmetric cone in $V$. According to Hilgert and Neeb this question in its general form was open in 1998.

As for your first question, let me quote from the latter reference:

Problems related to invariant cones in representations occur in various contexts. They seem to have been studied first by Vinberg in the context of homogeneous cones (cf.[Vi63]). Apart from the study of the adjoint and coadjoint representations (cf. Example 1.2) most of the work done more recently is connected to isotropy representations of symmetric spaces, where invariant cones show up in connection with orderings, Wiener-Hopf operators and Hardy spaces (cf. Example 1.4, [KN96] [HO96], and the references given there).

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