Minimal representations: spherical vectors and automorphic functionals

TL;DR

This paper provides explicit formulas for spherical vectors in minimal representations of certain split groups over local fields and describes automorphic functionals over global fields, advancing understanding of these representations.

Contribution

It offers the first explicit formulas for spherical vectors in minimal representations of split groups of type D_k and E_k over local non-archimedean fields and analyzes automorphic functionals globally.

Findings

Explicit formulas for spherical vectors in local minimal representations

Structural insights into smooth vectors in minimal representations

Description of automorphic functionals over global fields

Abstract

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.