PGL_n(ℂ)-character stacks and Langlands duality over finite fields

2024/12/04 by Emmanuel Letellier, Tommaso Scognamiglio, Letellier, Emmanuel +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2412.03234

openalex publication_date 2024/12/04 · openalex created_date 2024/12/06 · openalex updated_date 2026/07/28

Abstract

In this paper we study the mixed Poincaré polynomial of generic PGL_n(ℂ)-character stacks with coefficients in some local systems arising from the conjugacy classes of PGL_n(ℂ) which have non-connected stabiliser. We give a conjectural formula that we prove to be true under the Euler specialisation. We then prove that this conjectured formula interpolates the structure coefficients of the two based rings (C(PGL_n(\mathbbF_q)),Loc(PGL_n),*) and (C(SL_n(\mathbbF_q)), CS(SL_n),⋅) where for a group H, C(H) denotes the space of complex valued class functions on H, Loc(PGL_n) denotes the basis of characteristic functions of intermediate extensions of equivariant local systems on conjugacy classes of PGL_n and CS(SL_n) the basis of characteristic functions of Lusztig's character-sheaves on SL_n. Our result reminds us of a non-abelian Fourier transform.

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