Semi-Invariant Distribution Vectors for P-Adic Unipotent Groups

Let [Formula: see text] be a unipotent algebraic group defined over a [Formula: see text]-adic field of characteristic zero. We denote by [Formula: see text] the set of rational points of [Formula: see text]. It is a [Formula: see text]-adic Lie group with Lie algebra denoted by [Formula: see text]. Let [Formula: see text] be an irreducible unitary representation of [Formula: see text] in a Hilbert space [Formula: see text], [Formula: see text] be a linear form on [Formula: see text] and [Formula: see text] be a polarization at [Formula: see text]. We denote by [Formula: see text] a character of [Formula: see text] related to [Formula: see text]. The aim of this study is to give a precise description of the space of semi-invariant distribution vectors [Formula: see text] of [Formula: see text].