A minimal even type of the 2-adic Weil representation

The Weil representation is used to construct a minimal type of the two-fold central extension of ${\mathrm{Sp}}_{2n}({\mathbb{Q}}_2)$. The corresponding Hecke algebra is shown to be isomorphic to the classical affine Hecke algebra of the split adjoint orthogonal group ${\mathrm{SO}}_{2n+1}({\mathbb{Q}}_2)$.