Spherical representations of Lie supergroups

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)(G,K) of even type. Along the way, we compute the Harish-Chandra c  -function of the symmetric superspace G/KG/K. By way of an application, we show that in type AIII|AIIIAIII|AIII, all spherical representations are self-dual.