arXiv subjects

Konrad Zou

Publications and source records attributed to Konrad Zou.

5 recordsLinked to original sources

arXiv2026

On the action of the center in the $\ell$-adic categorical Langlands program

We discuss the action of the center in the categorical local Langlands program in the form of [FS24]. We provide a spectral description of the convolution action of a central torus in this situation. We show that the veracity of the $\ell$-adic categorical Langlands conjecture passes to the quotient by a central torus $D$ satisfying a minor cohomological condition in characteristic 0 and we show that under restriction to Langlands-Shahidi type parameters the conjecture passes to such a quotient in all characterstics and even integrally. In that situation we also show that the splitting of the semi-orthogonal decomposition descends. Finally we use these results to extend [Zou25] to $\mathrm{PGL}_n$.

math.RT↗

arXiv2026

On the Schematic and Analytic Constructions of the Local Langlands Category

We prove a folklore conjecture identifying two categorical enhancements of the automorphic side of the local Langlands correspondence. Concretely, we construct an equivalence for torsion coefficients between the category considered by Zhu and the one considered by Fargues-Scholze. To achieve this, we revisit Scholze's analytification functor and apply the first author's theory of kimberlites. We discuss unconditional applications to the splitting of the semi-orthogonal decomposition on BunG, and the compatibility with Eisenstein functors. Finally, we formulate a linearity conjecture for our functor with which we can show new vanishing statements for the cohomology of local Shimura varieties, and perverse exactness statements for Hecke operators.

math.NT↗

arXiv2025

Categorical local Langlands for $\mathrm{GL}_n$ for parameters of Langlands-Shahidi type with integral coefficients

We prove the categorical form of Fargues' geometrization conjecture for $\mathrm{GL}_n$ along $L$-parameters of Langlands-Shahidi type for rational, torsion, and integral coefficients. Additionally, we prove that in this case the categorical equivalence is $t$-exact, which yields new torsion vanishing results in the cohomology of unitary Shimura varieties.

math.RT↗

arXiv2024

Geometric Casselman-Shalika in mixed characteristic

We establish a geometric analog of the Casselman-Shalika formula for a split connected reductive group over a mixed characteristic local field. In particular, we construct sheaves on the Witt vector affine Grassmannian which geometrize the Fourier coefficients of spherical Hecke operators, and compute their cohomology.

math.AG↗

arXiv2024

The categorical form of Fargues' conjecture for tori

We prove the main conjecture of arXiv:2102.13459 for integral coefficients in the case of tori and prove that it is t-exact. Along the way we prove that the spectral action as constructed in that manuscript is compatible with the action of the excursion algebra and preserves the grading by $π_1(G)_Q$ on both sides. We additionally develop a (non-solidified) version of condensed group (co)homology and show that many constructions from classical group (co)homology extend to that case.

math.RT↗