On the E-polynomial of parabolic Sp_2n-character varieties

2017/08/01 by Vincenzo Cambò, Cambò, Vincenzo
Mathematics · Physics and Astronomy · #05E10 (Secondary) #20C33 (Primary) 22E10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1708.00393

openalex publication_date 2017/08/01 · openalex created_date 2017/08/08 · openalex updated_date 2026/07/28

Abstract

We find the E-polynomials of a family of parabolic Sp_2n-character varieties M^ξ_n of Riemann surfaces by constructing a stratification, proving that each stratum has polynomial count, applying a result of Katz regarding the counting functions, and finally adding up the resulting E-polynomials of the strata. To count the number of \mathbbF_q-points of the strata, we invoke a formula due to Frobenius. Our calculation make use of a formula for the evaluation of characters on semisimple elements coming from Deligne-Lusztig theory, applied to the character theory of Sp(2n,\mathbbF_q), and Möbius inversion on the poset of set-partitions. We compute the Euler characteristic of the M^ξ_n with these polynomials, and show they are connected.

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