Cuspidal endo-support and strong beta extensions

Abstract: Let $G$ be an inner form of a general linear group or classical group over a non-archimedean local field of residual characteristic $p$, assumed odd in the classical case. We prove that every smooth representation of $G$ over an algebraically closed field $R$ of characteristic $\ell\neq p$ contains a maximal semisimple character, i.e., one for which the point in the building of the corresponding centralizer is a vertex. Further, for every endo-parameter adapted to $G$, we define its support, which leads also to the notion of cuspidal endo-support of an irreducible representation, and we relate this to its cuspidal support. We also introduce beta extensions for strong facets in the building of a centralizer, and show these are sufficient for the construction of types. These results are used in a subsequent paper to decompose the category of smooth $R$-representations of $G$.

• The paper advances p-adic representation theory by introducing a refined notion of cuspidal endo-support via m-realizations.
• It establishes finiteness results and an exhaustive parametrization of smooth representations in each Bernstein block.
• Strong beta extensions are constructed to ensure a uniform and compatible framework for building types across general linear and classical groups.

Summary of "Cuspidal endo-support and strong beta extensions" (2604.01781)

The paper develops a refined categorical and character-theoretic analysis of smooth representations of $p$-adic classical groups and their inner forms, particularly over fields of characteristic distinct from the residual characteristic. It establishes structural results regarding the containment of maximal semisimple characters, introduces the notion of cuspidal endo-support for irreducible representations, and provides an enhanced approach to the construction of types via strong beta extensions for the associated Bernstein blocks.

Semisimple Characters, Endo-parameters, and Representation Categories

The work builds upon the machinery of semisimple characters, endo-classes, and endo-parameters, unifying previous approaches (Bushnell-Kutzko, Bushnell-Henniart, Stevens, Kurinczuk-Skodlerack-Stevens) for both general linear groups and classical groups, including their inner forms. Central to the paper is the analysis of the category $\mathrm{Rep}_R(G)$ of smooth representations of a $p$-adic group $G$ over an algebraically closed field $R$ of characteristic $\ell \neq p$.

Given an endo-parameter $t$, defined as a formal sum of simple endo-classes subject to degree and divisibility constraints, the authors isolate the subcategory $\mathrm{Rep}_R(t)$ generated by semisimple characters of a fixed intertwining class ("endo-factor"). The principal result demonstrates that:

• Every smooth representation of $G$ (over $R$) contains a maximal semisimple character: That is, the associated point in the building of the centralizer is a vertex, corresponding to the notion of "m-realization".
• Finiteness of essential $\mathrm{Rep}_R(G)$0-conjugacy classes: For each endo-parameter $\mathrm{Rep}_R(G)$1, there are only finitely many essential $\mathrm{Rep}_R(G)$2-conjugacy classes of m-realizations, crucial for the construction of projective generators and for categorical decomposition.

These results culminate in a categorical exhaustion theorem: The sum over isotypic components attached to m-realizations of $\mathrm{Rep}_R(G)$3 suffices to exhaust $\mathrm{Rep}_R(G)$4. The passage from arbitrary semisimple realizations (as in Dat's theorem) to m-realizations is not only more precise but also yields crucial finiteness properties required for further block decomposition results.

Cuspidal Endo-support and its Relation to Classical Supports

A further innovation is the notion of cuspidal endo-support, a parameterization of irreducible representations by conjugacy classes of pairs $\mathrm{Rep}_R(G)$5, where $\mathrm{Rep}_R(G)$6 is a Levi and $\mathrm{Rep}_R(G)$7 an endo-parameter, with $\mathrm{Rep}_R(G)$8 realized as the restriction of a semisimple character to $\mathrm{Rep}_R(G)$9. Cuspidal endo-support quantities are shown to control the location of cuspidal supports within the block structure of $p$0, and are maximal among cuspidal sub-endo-parameters.

The authors demonstrate a strong compatibility between this categorification and classical inertial cuspidal support: For any irreducible representation $p$1, the endo-parameter of its inertial cuspidal support is always dominated by its maximal cuspidal endo-support.

Theoretical Implications

This categorical and character-theoretic framework enables a universal description of Bernstein block decompositions via explicit invariants (endo-parameters and their supports), and establishes a tight connection between the representation theory of $p$2-adic groups, the combinatorics of their Bruhat-Tits buildings, and the structure theory of associated division algebras.

Beta Extensions and Strong Simplicial Structures

The construction of types for Bernstein blocks and, more generally, the explicit parameterization of irreducible representations, relies on the existence of beta extensions of Heisenberg representations associated to semisimple characters. The paper advances the theory by:

• Defining and constructing beta extensions attached to maximal parahoric subgroups in the strong simplicial structure of the centralizer building (rather than just the weak structure).
• Demonstrating that these beta extensions suffice for the construction of all types required for Bernstein block theory, aligning the depth-zero and positive-depth cases.
• Introducing the notion of compatible families of beta extensions, conjecturing their existence with "full intertwining" (i.e., maximal possible centralizer action), and proving this for inner forms of general linear groups.

Practical Implications

The results translate into explicit algorithms for constructing types and Hecke algebras associated with Bernstein blocks, and thus for decomposing categories of smooth representations in a manner compatible with the local Langlands correspondence, Jacquet-Langlands correspondences, and distinction problems under Galois involutions.

Structural Finiteness and Block Decomposition

By demonstrating that only finitely many essential or cuspidal conjugacy classes of m-realizations exist for each endo-parameter, the authors obtain strong control over the structure and size of endo-factors. This paves the way, as indicated for a subsequent paper, for explicit block decomposition of the category of smooth representations, potentially over arbitrary bases such as $p$3-algebras, unifying $p$4-modular and complex representation theory.

Numerical Results and Contrasts

While the paper's emphasis is conceptual and categorical rather than computational, a strong finiteness result is established: for each endo-parameter, the set of essential $p$5-conjugacy classes of m-realizations is finite, and these suffice to generate the representation category $p$6. These statements represent an advance over prior results wherein only the sum over all semisimple characters was known to suffice, with no parallel finiteness property.

Contrasts arise relative to previous work on simple character theory for general linear groups, as the present framework unifies this with classical groups and their inner forms and extends the utility of building-theoretic invariants.

Prospects for Future Research

The theory developed in this paper is foundational for ongoing efforts to:

• Explicitly decompose the category of smooth $p$7-representations of $p$8-adic groups into endo-factors and, via compatible beta extensions and types, to realize Bernstein blocks functorially.
• Generalize block decompositions beyond complex representations to $p$9-modular and integral coefficients, facilitating new advances in the context of mod-$G$0 Langlands correspondences.
• Further clarify the relationship between categorical block theory, the geometry of the Bruhat-Tits building, and the construction of types and Hecke algebras for both classical and exceptional groups.

The introduction of cuspidal endo-support and strong beta extensions provides a new technical framework for analyzing distinction problems and for transferring character-theoretic data between inner forms, laying groundwork for advances in automorphic descent and explicit correspondences.

Conclusion

This paper establishes categorical finiteness and new invariants for the block decomposition of smooth representations of $G$1-adic classical and general linear groups, founded on the geometry of the building, semisimple character theory, and the arithmetic of division algebras. The introduction of cuspidal endo-support and strong beta extensions unifies prior approaches and provides precise machinery for future work on explicit decomposition theorems, type theory, and the local Langlands program for general coefficients.