Categorical Local Langlands

• Categorical Local Langlands is a framework that replaces classical representations with abelian, triangulated, or ∞-categories to capture richer two-dimensional data.
• It constructs categorified principal series via 2-inductive limits and parabolic induction, integrating central extensions from K₂-theory to maintain smooth group actions.
• The approach establishes noncommutative reciprocity laws that glue local and global categorical data, aligning with higher-dimensional arithmetic and geometric Langlands philosophy.

Categorical Local Langlands refers to a programmatic enhancement of the classical local Langlands correspondence, in which correspondences between representations of reductive groups over local fields and "Langlands parameters" are lifted from sets of irreducible objects to equivalences between richly structured categories. This categorification replaces individual automorphic (or admissible) representations by abelian, triangulated, or ∞-categories—often equipped with group actions or Hecke symmetries—and relates them functorially to categories of sheaves (quasi-coherent, coherent, or constructible) on moduli stacks of Langlands parameters or local systems.

1. Categorical Analogue of Principal Series for Two-Dimensional Local Fields

In the unramified context for two-dimensional local fields, the classical principal series representation of $GL_n(K)$, constructed via induction from a Borel subgroup $B$, is categorified. Here, the role of the vector space is replaced by a $k$-linear abelian category (a "generalized 2-vector space") with a smooth action of $GL_{2n}(K)$. For a fixed unramified parameter—an $n$-tuple $$(a_1, \dots, a_n)\in\mathbb{C}^n/(2\pi i/\log q)\mathbb{Z}^n$$ corresponding to a geometric parameter of the Weil group—one constructs, for each integer $m\geq1$, a category $V_m$ whose objects are $GL_{2n}(K)/C_m$-equivariant functions into a fixed semisimple category (such as $\mathrm{Vect}_n$).

The parabolic group $P$ of block-upper-triangular matrices (each block in $GL_2(K)$) acts via a system of functors $T_{(a_1,\dots,a_n)}$, defined using one-dimensional vector spaces $A_{g_i}$ (arising from central extensions): $$T_{(a_1,\dots,a_n)}(g)(Y) = (A_{g_1})^{a_1} \boxtimes \cdots \boxtimes (A_{g_n})^{a_n} \boxtimes Y.$$ The categorical principal series is then defined via a two-inductive limit (the 2-direct limit)

$V_{(a_1,\dots,a_n)} = 2\!\lim_m (V_m)^P,$

which yields a generalized 2-vector space with a smooth $GL_{2n}(K)$-action, serving as the categorical analogue of the unramified principal series representation (Osipov, 2012).

2. Central Extensions and K₂-Phenomena

A fundamental structural role is played by central extensions, reflecting K₂-theoretic phenomena intrinsic to two-dimensional fields. Concretely, one constructs a central extension

$1 \to R^* \to GL_2(K)_{R^*} \to GL_2(K) \to 1$

where $A_g$ are $R^*$-torsors canonically attached to $g\in GL_2(K)$, defined via Haar measures and more generally the canonical $R^*$-torsors $p(\cdot)$ on locally compact abelian groups. The category is twisted by these $A_g$'s, with compatibility ensured by isomorphisms

$(A_g)^a \otimes (A_h)^a \to (A_{gh})^a,$

ensuring coherence (associativity) for the $P$-action in the categorical induction.

The central extension is constructed explicitly via the Steinberg group, using the symbol

$$K_2(K) \longrightarrow R^*, \quad (f, g) \mapsto q^{-v_K(f, g)}$$

(where $v_K(f, g)$ denotes the K₂-symbol in the Kato sense). Crucially, the central extension splits over compact subgroups like $GL_2(\mathcal{O}_K)$, ensuring the smoothness of the action and reflecting the unramified condition categorically.

3. Noncommutative Reciprocity Laws

The paper establishes noncommutative reciprocity laws as higher-dimensional analogues of Artin reciprocity. These results guarantee that for a normal proper surface $X$ (arithmetic or over a finite field), the constructed central extension for $GL_n(\mathbb{A})$,

$$1 \to R^* \to GL_n(\mathbb{A})_{R^*} \to GL_n(\mathbb{A}) \to 1$$

splits over the "local" subgroups $GL_n(K_x)$ (for a closed point $x$) and $GL_n(K_C)$ (for a codimension one point/subscheme $C$), where $\mathbb{A}$ is the global adelic ring of $X$. These reciprocity laws ensure the compatibility of local and global categorical data and enable a "gluing" procedure in the nonabelian, 2-categorical context, extending the classical reciprocity principles to arithmetic surfaces and their function field analogues.

4. Categorical Construction and Two-Dimensional Local Langlands

Within this framework, the categorical local Langlands correspondence for two-dimensional local fields is formulated as a mapping from $n$-dimensional unramified representations of the Weil group to smooth $GL_{2n}(K)$-actions on abelian $k$-linear categories constructed via categorified induction. These categories

$V_{(a_1,\dots,a_n)} = 2\!\lim_m (V_m)^P$

encapsulate the local data (through the action of $GL_{2n}(K)$ and its compact subgroups, twisted via central extensions) and are predicted to exhaust, up to subquotients, the class of smooth, spherical $GL_{2n}(K)$-actions on such categories.

The combination of categorical induction and central extension twisting is necessary for the smoothness and sphericality properties that mirror those of the classical unramified principal series. The noncommutative reciprocity laws then assemble the local categorical data compatibly over arithmetic surfaces, completing the higher-dimensional, categorical extension of the Langlands philosophy.

5. Structural Summary and Theoretical Implications

• Smooth actions and "2-vector spaces": The categorical principal series representations $V_{(a_1,\dots,a_n)}$ generalize the principal series by replacing the target vector space with an abelian $k$-linear category, encoding both the representation-theoretic and geometric data required in two-dimensional local fields.
• Central extensions as obstructions and corrections: The presence of K₂-induced central extensions and their splitting over compact subgroups is essential to ensure that categorical induction yields the correct analogues of unramified representations, compensating for obstacles to naively defining two-dimensional invariants.
• Reciprocity and gluing: The noncommutative reciprocity laws ensure that local and global 2-categorical data are glued in a manner consistent with the (generalized) higher-dimensional class field theory—nontrivially intertwining arithmetic, K-theory, and categorified representation theory.
• Categorical local Langlands conjectural picture: The constructed categorical representations provide candidates for a categorical form of the unramified part of the two-dimensional local Langlands correspondence—any sufficiently smooth, irreducible, spherical categorical action must arise (up to geometric subquotients) via the procedure outlined above.

6. Outlook and Integration with Higher-Dimensional and Geometric Langlands

This work positions the categorical local Langlands correspondence for two-dimensional local fields within the broader landscape of higher Langlands correspondences, leveraging categorical induction, K₂-theory, and reciprocity laws. It anticipates a full categorical correspondence in which actions of $GL_{2n}(K)$ on abelian categories correspond, under precise functorial constructions, to higher local Galois parameters and their arithmetic counterparts. The methods harmonize classical aspects (e.g., parabolic induction, smoothness, central extensions) with the modern categorical techniques necessary for the "Langlands program in dimension two."

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Key Construction Table

Concept	Classical Local Langlands	Categorical Local Langlands (2D)
Representation	$\mathrm{Ind}_B^{G} \chi$	$V_{(a_1,\dots,a_n)} = 2\!\lim_m(V_m)^P$
Induced from	Character on $B \subset G$	Parabolic $P$ with categorical twisting
Action	$G=GL_n(K)$	$G=GL_{2n}(K)$ acts on abelian $k$-linear category
Central extension	Trivial or finite cover	Arising from $K_2(K) \to R^*$, may be nontrivial
Reciprocity law	Artin reciprocity	Noncommutative splitting over local/global subgps