Remark on the distributions supported on the nilpotent cone

Abstract

Let F be a local field of characteristic zero. This note gives two examples for the vanishing of certain distributions supported on the nilpotent cone. Then we give a shorter proof to the multiplicity one theorem for the symmetric variety ${\mathrm{GL}}_n\left(D\right)/{\mathrm{GL}}_n\left(E\right)$ where D is a 4-dimensional quaternion algebra over F and E is a quadratic field extension of F.