  In the first part of this paper we study minimal representations of simply
connected simple split groups of type $D_k$ or $E_k$ over local non-archimedian
fields. Our main result is an explicit formula for the spherical vectors in
these representations. In the case of real and complex groups such a formula
was obtained recently in hep-th/0107222. We also use our techniques to study
the structure of the space of smooth vectors in the minimal representation
recovering some results of Magaard and Savin. In the second part we consider
groups as above defined over a global field. In this situation we describe the
form of the automorphic functional on the minimal representation of the
corresponding adelic group.
