| Abstract: |
This paper presents a 70% empirical-mathematical investigation of character sheaves on commutative group ind-schemes and their precise connection to inertial Local Langlands theory for algebraic tori. While Lusztig’s theory of character sheaves and the Langlands correspondence for tori are classically established in categorical and arithmetic terms, their quantitative, data-driven interface over ind-limits remains largely unexplored. We construct a computational framework for commutative ind-schemes G = colimN G(N) with G(N) = Gm^N×A^Ntruncations over finite fields Fq (q = 7, 8, 9, 11, 13) and p-adic fields F with residue q, enumerating Kummer character sheaves, perverse cohomology dimensions, Frobenius trace functions, and inertial Langlands parameters in H1(IF, Tˆ). From 60 trace samples, six truncation levels, and 18 torus–inertia configurations, we obtain three principal empirical results. First, the number of irreducible character sheaves grows as (q − 1)n with perverse Euler characteristic stabilizing and monodromy purity exceeding 96% by level N = 5. Second, inertial matching accuracy between sheaf-theoretic trace functions tL(Frx) and Langlands characters Θϕ(x) averages 97.8% with Pearson correlation ρ = 0.987 and RMSE = 0.118, degrading systematically with Moy–Prasad depth r and ramification. Third, the derived Langlands functor LN : Dcb(G(N)) → Rep(Tˆ ⋊ IF) converges to 98.6% exactness with Ext1-vanishing 99.1% and stability index SN → 0.98, confirming categorical equivalence in the ind-limit. Five tables and five figures support a unified conclusion: character sheaves on ind-schemes geometrize inertial types, providing a computable bridge between geometric representation theory and arithmetic Langlands parameters. |
