Simple Iwasawa modules with large canonical dimension of quaternion algebra over $\mathbb{Q}_p$
Weibo Fu
TL;DR
The paper constructs explicit absolutely irreducible Banach representations of the quaternion division algebra $D$ over ${\mathbb Q}_p$ with canonical (Gelfand–Kirillov) dimension $2$ and uses them to produce counterexamples to the Dospinescu–Schraen conjecture on the existence of infinitesimal characters and finite length for locally analytic vectors. By analyzing a uniform pro-$p$ subgroup of $D^\times$, the associated Lie algebra $\mathfrak{g}_D$, and finite-length weight modules over the completed enveloping algebra, the author builds representations whose locally analytic vectors have infinite length and lack an infinitesimal character, while the Banach representation itself can be decomposed into finite-length semisimple components. The construction leverages microlocalization between $K[[G]]$, $D(G,K)$, and completed enveloping algebras, and yields GK-dimension $2$ examples that exceed half the largest coadjoint orbit dimension, illuminating limitations in the $p$-adic Langlands correspondence for non-split groups. These results highlight the nuanced interaction between Banach and analytic structures in $p$-adic representation theory and provide a new class of pathological yet explicit counterexamples to existing conjectures.
Abstract
We construct certain absolutely irreducible Banach representations of the quaternion algebra of large canonical (Gelfand-Kirillov) dimension which yield counterexamples to a natural conjecture of Dospinescu-Schraen on the existence of an infinitesimal character and topological finite length for the locally analytic vectors of an absolutely irreducible Banach representation.