Orbits of spherical representations and Pyasetskii duality

Dual representations $V$ and $V^{\ast }$ of a complex connected algebraic group $G$ simultaneously have either infinitely or finitely many orbits. Whenever the latter holds, the orbits in $V$ and $V^{\ast }$ are in a bijective correspondence called Pyasetskii duality. We obtain a complete description of this duality in the case of spherical representations.