Stability of certain elements in the Bernstein center

Source abstract

Let $G$ be a split connected reductive algebraic group over a non-archimedean local field $F$ of characteristic zero. The stable center conjecture asserts that the subspace of stable elements in the Bernstein center of $G(F)$ is a sub-algebra, and this subring is conjecturally isomorphic to the regular functions on the moduli of semi-simple Langlands parameters. Bezrukavnikov-Kazhdan-Varshavsky suggested an approach towards the stable center conjecture by explicitly constructing elements and proving their stability directly without assuming LLC, and proved the stability of the depth-$r$ projectors. Following the idea, the first author and Tsao-Hsien Chen produced a family of elements in the center which are conjecturally stable using Deligne-Lusztig parameters attached to irreducible representations. This paper proves the stability of the projectors corresponding to strongly regular Deligne-Lusztig parameters in depth-zero, and for all the positive depth elements with some restrictions on the residue characteristic.