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Let $k$ be a global field of positive characteristic, $\mathbb{A}$ its adele ring. Let $U$ be a unipotent algebraic group over $k$, of dimension sufficiently small relative to ${\rm char} (k)$. Is there a reference for the proof of a conjecture along the following lines? The space $L^2 (U(\mathbb{A}) / U(k))$, viewed as a $U(\mathbb{A})$-representation, decomposes into a direct sum of irreducibles, each with multiplicity one, parametrized by $\mathfrak{u} (k)^{*} / U(k)$ (in the spirit of the orbit method). Here $\mathfrak{u}$ is the Lie algebra of $U$.

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    $\begingroup$ So you are basically interested in a reference for Kirillov theory for positive char? $\endgroup$
    – Asaf
    Commented Nov 28, 2022 at 15:24
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    $\begingroup$ @Asaf I knew there is Kirillov theory for irreps of unipotent groups of local fields, say; here I ask about "Kirillov theory for the automorphic quotient". But if it included in what is usually called "Kirillov theory" then yes! I am interested in a reference for that. $\endgroup$
    – Sasha
    Commented Nov 28, 2022 at 17:03

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