Small representations for Kac–Moody groups

Let $E_n(\mathbb{R})$ be the real Kac–Moody group with $n\geq 9$, and let $P_1$ be the maximal parabolic subgroup obtained by deleting the first node of the $E_n$ Dynkin diagram, with semisimple Levi factor of type $D_{n-1}$. Consider the canonically associated Eisenstein series induced from $P_1$ with parameter $s$.

Small-representations conjecture. Kac–Moody groups possess a minimal unitary representation realizable automorphically. For $E_n(\mathbb{R})$, it is obtained by inducing from $P_1$; the Eisenstein series at $s=\tfrac32$, defined by analytic continuation, is the spherical vector in the minimal automorphic representation, whose wave-front set is of Bala–Carter type $A_1$. A similar next-to-minimal representation is obtained at $s=\tfrac52$, with wave-front set of Bala–Carter type $2A_1$.

These assertions formalize observations about degenerate Whittaker coefficients and maximal-parabolic Eisenstein series for $E_9$, $E_{10}$ and $E_{11}$. The source calls the surrounding Fourier-coefficient analysis an open problem and does not provide a resolution of this formalized claim.