On the Brauer–Picard group of a finite symmetric tensor category

Let CnCn denote the representation category of finite supergroup ∧kn⋊Z/2Z∧kn⋊Z/2Z. We compute the Brauer–Picard group BrPic(Cn)BrPic(Cn) of CnCn. This is done by identifying BrPic(Cn)BrPic(Cn) with the group of braided tensor autoequivalences of the Drinfeld center of CnCn and studying the action of the latter group on the categorical Lagrangian Grassmannian of CnCn. We show that this action corresponds to the action of a projective symplectic group on a classical Lagrangian Grassmannian.