# Invariant regular cones in Lie group representation

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.