  The classical Cartan-Helgason theorem characterises finite-dimensional
spherical representations of reductive Lie groups in terms of their highest
weights. We generalise the theorem to the case of a reductive symmetric
supergroup pair $(G,K)$ of even type. Along the way, we compute the
Harish-Chandra $c$-function of the symmetric superspace $G/K$. By way of an
application, we show that all spherical representations are self-dual in type
AIII|AIII.
