Automorphic functionals for the minimal representations of groups of type $D_n$ and $E_n$

TL;DR

This work constructs an explicit $G(F)$-invariant automorphic functional on the minimal representation model $\mathcal S(X(\mathbb A))$ for split groups of type $D_n$ and $E_n$, realized via abelian or Heisenberg unipotent radicals. The authors develop comprehensive local and global models, define main and boundary terms, and prove a main decomposition theorem: $\theta_{G,L}=\theta_X+p_X+|\Delta_F|^{1/2}\,\theta_{G_1,M_1}\circ\mathcal B$ (for Siegel-type), with a complementary boundary-augmented expansion in the Heisenberg-type case, supplemented by transition maps between models to handle non-admissible pairs. The paper then applies these results to explicit expansions for $D_n$, including a detailed analysis of the pair $(D_3,D_2)$, and extends the methodology to the exceptional groups $E_6,E_7,E_8$ via sequential transitions and Fourier-coefficient computations of degenerate Eisenstein series. Overall, the work provides a structured, computable framework for automorphic functionals on minimal representations, enabling concrete Fourier-analytic descriptions and connections to residual Eisenstein data for a broad class of Lie types.

Abstract

Let $G$ be a split simply-connected group of type $D$ or $E$. The minimal automorphic representation $Π$ of $G(\mathbb A)$ admits a realization on a space of functions $\mathcal S(X(\mathbb A))$ for a variety $X$. In this paper we write explicitly an automorphic, i.e. $G(F)$-invariant, functional on $\mathcal S(X(\mathbb A)).$

Theorems & Definitions (64)

• Definition 1
• Theorem 1.1
• proof
• Proposition 1.1
• Corollary 1.1
• Remark 2
• Theorem 1.2
• Remark 6
• Proposition 2.1
• proof