Integral Categorical Local Langlands

• Integral categorical Local Langlands correspondence is a refined framework relating smooth representations of reductive groups to Galois or Weil-Deligne parameters, integrating geometric, spectral, and categorical methods.
• It employs a Λ-linear, t-exact equivalence between derived categories of automorphic sheaves and spectral stacks, utilizing integral and torsion coefficients.
• Key implications include powerful torsion vanishing theorems and precise cohomological results in Shimura varieties, with explicit illustrations in the GL₂ case.

The integral categorical Local Langlands correspondence provides a categorical and integral form of the expected relationship between smooth representations of reductive groups over local fields and Galois or Weil-Deligne parameters, refining the classical correspondence by fully incorporating integral structures and the architecture of triangulated, stable ∞-categories. This framework brings together geometric, spectral, and categorical structures—most notably via equivalences or fully faithful embeddings between derived categories of automorphic sheaves and categories of perfect complexes over stacks (often Artin or derived) of Langlands parameters, now considered with integral or torsion coefficients. When restricted to suitable classes of parameters (e.g., Langlands–Shahidi type), the integral categorical correspondence becomes $t$-exact, with profound implications for cohomological vanishing phenomena and integrality properties in the cohomology of Shimura varieties and moduli spaces.

1. Categorical Formulation and Main Theorems

The integral categorical Local Langlands correspondence for $G = \mathrm{GL}_n$ with coefficients in a $\mathbb{Z}_\ell$-algebra $\Lambda$ (for $\ell \neq p$) can be summarized by the following $\Lambda$-linear, $t$-exact equivalence: $$\mathcal{E}: D^b_{\mathrm{int}}\big(\mathrm{Perf}(\Phi^{\mathrm{LS}}(W_E, \mathrm{GL}_n)_{\mathbb{Z}_\ell})\big) \xrightarrow{\sim} D^b_{\mathrm{int}}\big(D_{\mathrm{lis}}(\mathrm{Bun}_{\mathrm{GL}_n}, \Lambda)\big),$$ where:

• $$\Phi^{\mathrm{LS}}(W_E, \mathrm{GL}_n)_{\mathbb{Z}_\ell} \subset \mathrm{LocSys}_{\widehat{\mathrm{GL}}_n}$$ is the open substack of Langlands–Shahidi-type $L$-parameters with integral coefficients.
• $D^b_{\mathrm{int}}(\mathrm{Perf}(-))$ denotes the bounded derived category of perfect complexes with integral coefficients.
• $$D^b_{\mathrm{int}}(D_{\mathrm{lis}}(\mathrm{Bun}_{\mathrm{GL}_n}, \Lambda))$$ is the bounded derived category of $\ell$-adic lisse sheaves (or shtukas) on the stack of rank-$n$ vector bundles, equipped with the normalized perverse $t$-structure.

The correspondence is equivariant under the excursion algebra (the spectral action) and sends the structure sheaf $\mathcal{O}_\Phi$ to the Whittaker sheaf $\mathcal{W}$ on $\mathrm{Bun}_{\mathrm{GL}_n}$ (Zou, 9 Apr 2025).

2. Spectral and Automorphic Sides: Stacks and Parameters

2.1 Spectral Stack and LS-Type Parameters

• The spectral side is represented by the moduli stack

$$\mathrm{LocSys}_{\widehat{\mathrm{GL}}_n} = [Z^1(W_E, \mathrm{GL}_n)/\mathrm{GL}_n],$$

where $Z^1(W_E, \mathrm{GL}_n)$ is the space of continuous cocycles.

• The LS-type locus, $\Phi^{\mathrm{LS}}$, consists of semisimple cocycles $\phi: W_E \to \mathrm{GL}_n(\Lambda')$ decomposing as $\phi \simeq \phi_1 \oplus \dots \oplus \phi_r$, with each $\phi_i$ irreducible and pairwise non-isomorphic (even after cyclotomic twist: $\phi_i \not \simeq \phi_j(1)$ for $i \neq j$), ensuring the normal bundle to the spectral stack inclusion has vanishing cohomology at $\phi$.

2.2 Automorphic Side

• The geometric side involves the stack of vector bundles $\mathrm{Bun}_{\mathrm{GL}_n}$ on the Fargues–Fontaine curve $X_E$.
• Sheaves are considered in the derived category $$D_{\mathrm{lis}}(\mathrm{Bun}_{\mathrm{GL}_n}, \Lambda)$$, with the perverse $t$-structure normalized such that the skyscraper at a bundle of Newton slope $\nu$ sits in degree $\langle 2\rho, \nu \rangle$.

3. Construction via Hecke Operators and Spectral Action

The equivalence is realized by identifying spectral and automorphic actions:

• To each finite-dimensional representation $V$ of the Langlands dual group $\widehat{\mathrm{GL}}_n$, the Scholze–Fargues Hecke functor

$$T_V: D_{\mathrm{lis}}(\mathrm{Bun}_{\mathrm{GL}_n}, \Lambda) \to D_{\mathrm{lis}}(\mathrm{Bun}_{\mathrm{GL}_n}, \Lambda)^{BW_E}$$

acts compatibly with the action of the perfect derived category $$\mathrm{Perf}(\mathrm{LocSys}_{\widehat{\mathrm{GL}}_n})$$.

• For complexes $A$ in $D(\mathrm{Bun})$ “supported” on the LS locus, $T_V$ acts by tensoring with the perfect complex $V \circ \phi$ on the spectral side.
• For the standard $n$-dimensional representation $\mathrm{std}$, this functor recovers the cohomology of the infinite-level Lubin–Tate tower:

$$R\Gamma_c(\mathrm{LT}_\infty, \overline{\mathbb{Q}}_\ell) \simeq i_O(1/n)^* T_{\mathrm{std}}(i_O^n)_! \Lambda[n-1]((n-1)/2).$$

4. $t$-Exactness and Vanishing Theorems

A central result is that the constructed equivalence is $t$-exact:

• On the spectral side, the connective $t$-structure is generated by pullbacks of connective complexes from $B\mathrm{GL}_n$.
• On the automorphic side, the perverse $t$-structure is generated from shifts of compact generators $i_{b!}\rho[\text{$b$-shift}]$ corresponding to irreducible smooth representations $\rho$ of Levi subgroups $G_b(E)$.
• For LS-type $\phi$, the stalks of Hecke functors $T_V$ at $\phi$ are concentrated in degree zero, whence the $t$-exactness.
• The proof proceeds by first establishing $t$-exactness for irreducible $\phi$ (using fundamental properties of classical Hecke operators in the supercuspidal case), then extending to general LS-type parameters using induction on modifications and Hodge–Newton reducibility arguments.

As an application, $t$-exactness yields powerful vanishing theorems:

• The cohomology of (Type A or EL) unitary Shimura varieties with LS-type local parameter at $p$ is concentrated in the middle degree even with torsion coefficients:

$$H_c^i(\mathrm{Sh}_{K^pK_p}, \mathbb{Z}/\ell^r)_{\phi_p} = 0 \quad \text{for } i \neq \dim \mathrm{Sh}.$$

• For $\mathrm{GL}_2$, the mod-$\ell$ cohomology of modular curves at nonscalar $L$-parameters is torsion-free.

5. Illustrative Example: The $\mathrm{GL}_2$ Case

The situation for $\mathrm{GL}_2$ is completely transparent:

• $\mathrm{Bun}_{\mathrm{GL}_2}$ decomposes into two strata: one for the trivial bundle (slope $0$), one for the basic non-split bundle (slope $1/2$).
• The spectral side consists of parameters $\phi = \phi_1 \oplus \phi_2$ with $\phi_1 \not\simeq \phi_2(\pm1)$, and $\mathrm{Perf}(\Phi^{\mathrm{LS}})$ is the coordinate ring for two points times $\mathbb{G}_m$ (line bundles $\mathcal{O}(\chi_1, \chi_2)$ for $\chi_i \in \mathbb{Z}$).
• On the geometric side, the Hecke operator for $\mathrm{std} = (1,0)$ yields an explicit functor

$$T_{\mathrm{std}}: D(G(E)) \to D(G(E)) \oplus D(D^\times)$$

whose components on the two strata can be computed explicitly, mirroring induction and restriction between $\mathrm{GL}_2(E)$ and the division algebra $D^\times$.

6. Implications and Applications

The integral categorical Local Langlands correspondence has powerful applications across arithmetic geometry and representation theory:

• It categorifies the classical correspondence in the presence of integral and torsion coefficients, maintaining full spectral and automorphic functoriality.
• $t$-exactness implies strong torsion vanishing results which are unattainable with $\mathbb{Q}_\ell$-coefficients, and is fundamental for the study of integral cohomology in Shimura varieties.
• The formalism is compatible with the action of excursion operators (spectral action), fitting into the conjectural frameworks of Fargues–Scholze and the categorical geometrization of Langlands correspondences.

7. Context and Further Directions

This integral categorical structure can be compared to several related approaches:

• The categorical Deligne–Langlands correspondence for Iwahori-spherical representations and the coherent Springer theory provides a related functorial embedding for characteristic zero coefficients (Ben-Zvi et al., 2020), but faces further technical obstacles for integral coefficients, such as torsion and formality issues in $K$-theory and Hochschild homology.
• Explicit integral categorical realizations for tori rely on Fourier–Mukai transforms between stacks of parameters and their Picard duals, yielding $t$-exact and monoidal equivalences of stable $\infty$-categories (Fu, 9 Nov 2025).
• The integral Bernstein center for smooth representations is canonically identified with the coordinate ring of the universal deformation spaces of Galois parameters via gamma factor theory (Helm et al., 2016).

The integral categorical approach clarifies the structure of moduli of representations, lifts geometric arguments to the integral setting, and provides new tools for the analysis of torsion phenomena in both local and global contexts. Potential future developments include generalizations beyond Langlands–Shahidi parameters, extensions to non-GL-type groups, and deeper compatibility with the six-functor formalism and $p$-adic geometrizations.