An alternative construction of character sheaves on parahoric subgroups

2025/09/22 by Alexander B. Ivanov, Ivanov, Alexander B., Sian Nie +3
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Algebraic Geometry and Number Theory

paper · pdf · doi:10.48550/arxiv.2509.18442

Abstract

Inspired by the foundational work of Bezrukavnikov and Chan \citeBC24 on character sheaves for parahoric subgroups and an alternative interpretation of deep level Deligne-Lusztig characters in \citeNie₂4, we present a parallel but closed (non-iterated) construction of character sheaves within the framework of J.--K. Yu's types. We show that our construction yields perverse sheaves, which coincide with those produced in \citeBC24 in an iterated way. In the regular case we establish the compatibility of their Frobenius traces with deep level Deligne-Lusztig characters. As an application, we prove the positive-depth Springer's hypothesis for arbitrary characters, thereby generalizing the generic case result of Chan and Oi \citeCO25. The proofs of our results make critical use of the strategies and results from \citeBC24 and \citeNie₂4.

Citations

• Character sheaves on loop Lie algebras: polar partition
• Green functions for positive-depth Deligne--Lusztig induction
• Convex elements and cohomology of deep level Deligne-Lusztig varieties
• Decomposition of higher Deligne-Lusztig representations
• The scalar product formula for parahoric Deligne--Lusztig induction
• Generic character sheaves on parahoric subgroups
• The algebraisation of higher level Deligne--Lusztig representations II: odd levels
• Geometric L-packets of Howe-unramified toral supercuspidal representations
• The cohomology of semi-infinite Deligne--Lusztig varieties
• Generic character sheaves on reductive groups over a finite ring
• The algebraisation of higher Deligne--Lusztig representations
• A Comparison of Two Complexes
• Regular supercuspidal representations
• The orbit method for profinite groups and a p-adic analogue of Brown's theorem
• On endoscopic decomposition of certain depth zero representations

Related

• Notes on character sheaves
• Character sheaves on the semi-stable locus of a group compactification
• Character sheaves in characteristic p have nilpotent singular support
• A Comparison of Two Complexes
• Character sheaves on unipotent groups in characteristic p>0