Variation of Weyl modules in $p$-adic families

TL;DR

This paper investigates the stability of Weyl modules' structures within p-adic families, demonstrating their rigidity under certain conditions related to Weil-Deligne representations and pure specializations.

Contribution

It establishes the rigidity of Frobenius-semisimplified Weyl modules across p-adic families and clarifies conditions for their structural stability.

Findings

Rigidity of Frobenius-semisimplifications in pure specializations

Structural stability of Weyl modules in p-adic families

Conditions for lifting pure representations to Weil-Deligne representations

Abstract

Given a Weil-Deligne representation with coefficients in a domain, we prove the rigidity of the structures of the Frobenius-semisimplifications of the Weyl modules associated to its pure specializations. Moreover, we show that the structures of the Frobenius-semisimplifications of the Weyl modules attached to a collection of pure representations are rigid if these pure representations lift to Weil-Deligne representations over domains containing a domain $\mathscr{O}$ and a pseudorepresentation over $\mathscr{O}$ parametrizes the traces of these lifts.