Modified Traces and the Non-Tempered Spectrum: A Non-Semisimple Completion of BZSV for Finite Groups of Lie Type

Abstract

The Ben-Zvi–Sakellaridis–Venkatesh (BZSV) relative Langlands duality framework provides a categorical architecture for period integrals of automorphic forms, but requires regularization for non-tempered representations. We show that the modified trace from non-semisimple modular representation theory provides this regularization for finite groups of Lie type. For a finite group of Lie type G = G(Fq) at cross-characteristic ℓ (ℓ ∤ q), define d˜(PIM(φ)) = dim(PIM(φ))/|Sylℓ(G)| for each indecomposable projective module. We prove that the nonsemisimple spectral sum decomposes as [G : Sylℓ(G)] = Sgen/|P| + Snongen/|P|, where Sgen and Snongen are the generic and non-generic spectral masses—sums of dim(π)2 over representations with and without Whittaker models. This decomposition is independent of ℓ and matches the BZSV tempered/non-tempered split. We compute the per-stratum spectral decomposition for GL(2,Fq), GL(3,Fq), Sp(4,Fq), and the exceptional group G2(Fq), verifying agreement with Arthur parameter predictions. For GL(n), we prove that every non-generic representation arising from parabolic induction of one-dimensional data satisfies dim(πU) = 1, identifying the modified trace with a degenerate Whittaker period (Theorem C). For Sp(4), we prove that the cuspidal representation θ10 satisfies dim( ) = 0, making its spectral contribution invisible to any degenerate Whittaker functional (Proposition D). This dichotomy demonstrates that the modified trace carries strictly more information than any period integral for groups beyond GL(n), identifying it as a categorical invariant that specializes to period integrals only in favorable cases.

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