Types and compact-induction constructions for reductive groups

The starting point of this programme is the construction of supercuspidal representations of reductive groups over nonarchimedean local fields.

Let $F$ be such a field and let $G=\mathbf G(F)$, where $\mathbf G$ is connected reductive. It is conjectured that every irreducible smooth complex supercuspidal representation of $G$ has the form

$\pi\simeq\operatorname{c-Ind}_{K}^{G}\rho,$

where $K$ is an open subgroup compact modulo the center and $\rho$ is a finite-dimensional irreducible smooth representation of $K$. This is known in important cases, but remains open in general; see [1].

Our principal question is how to describe the inducing data explicitly. What determines the subgroup $K$? How is $\rho$ constructed, and what properties ensure that its compact induction is irreducible? Conversely, how can these data be extracted from an arbitrary supercuspidal representation?

We would like to develop this question through four stages.

1. Supercuspidal types and compact induction

The first task is to construct types for supercuspidal inertial classes and obtain explicit compact-induction models for the individual supercuspidal representations. These are closely related tasks, but they are not identical: a compact open type detects an inertial class, whereas a compact-induction model realizes a particular representation.

The structure of the relevant subgroups is central. We seek a description of how they arise from filtration subgroups, character data and stabilizers, together with a construction of their representations and a proof of the required intertwining statements. The resulting construction should be exhaustive.

2. Types for general Bernstein components

Starting with supercuspidal data in Levi subgroups, the next task is to construct types for the corresponding Bernstein components of $G$.

For an inertial class $\mathfrak s$, we seek a compact open subgroup $J$ and an irreducible representation $\lambda$ such that, for every irreducible smooth representation $\pi$,

$\operatorname{Hom}_{J}(\lambda,\pi)\ne0\quad\Longleftrightarrow\quad\pi\in\operatorname{Rep}^{\mathfrak s}(G).$

This stage should explain how the supercuspidal constructions interact with parabolic induction and with the theory of covers in the sense of Bushnell–Kutzko [3].

3. Comparison of Hecke algebras

Once the types have been constructed, we want to determine and compare their Hecke algebras.

The objective is to construct explicit algebra isomorphisms, identifying the intertwining operators and their convolution relations, including the parameters and cocycles that occur. Such comparisons should explain how the positive-depth data affect the algebra and when those data can be separated from a simpler representation-theoretic problem.

4. Reduction to depth zero

The final goal is to use these Hecke algebra comparisons to reduce Bernstein components to suitable depth-zero components.

This means an equivalence with a depth-zero block of an appropriate group, rather than changing the depth of a representation within the original group. Part of the problem is to identify the group and the type on the depth-zero side, and to determine the scope in which such a reduction is possible.

Two established approaches guide this programme. For general linear groups, Bushnell–Kutzko construct types and supercuspidal representations using simple strata and simple characters [2]. Stevens develops the corresponding theory for classical groups in odd residue characteristic, including wildly ramified data [4, 5]. Together with our forthcoming joint work with Yu Wang on reduction to depth-zero blocks for Stevens data, this provides the full picture described above for that classical-group setting.

For general reductive groups in the tame setting, Yu's construction [6], the exhaustion results of Kim and Fintzen [7, 9], and the types of Kim–Yu [8] provide the starting data. The two papers of Adler–Fintzen–Mishra–Ohara [11, 12] establish explicit Hecke algebra comparisons and reduction to depth zero under their hypotheses.

We would like to understand the common structure behind these approaches and extend it beyond their present scope. The first step is the construction of supercuspidal types and compact-induction models for linear reductive groups in small residue characteristic where the existing theories do not apply. Genuine representations of central covering groups remain a further goal of the programme; [14, 15] provide relevant constructions in particular covering settings.

The First step discussion (opens in a new tab) develops the initial compact-induction question and the distinction between tame and wild data.

References

1. J. Fintzen, Supercuspidal representations: construction, classification, and characters (opens in a new tab). Survey of the constructions and the general compact-induction conjecture.

2. C. J. Bushnell and P. C. Kutzko, The Admissible Dual of GL(N) via Compact Open Subgroups, Annals of Mathematics Studies 129, Princeton University Press, 1993.

3. C. J. Bushnell and P. C. Kutzko, Smooth representations of reductive p-adic groups: structure theory via types (opens in a new tab), Proc. London Math. Soc. 77 (1998), 582–634.

4. S. Stevens, The supercuspidal representations of p-adic classical groups (opens in a new tab), Invent. Math. 172 (2008), 289–352.

5. M. Miyauchi and S. Stevens, Semisimple types for p-adic classical groups (opens in a new tab), Math. Ann. 358 (2014), 257–288. Includes the correction to the supercuspidal construction in its appendix.

6. J.-K. Yu, Construction of tame supercuspidal representations (opens in a new tab), J. Amer. Math. Soc. 14 (2001), 579–622.

7. J.-L. Kim, Supercuspidal representations: an exhaustion theorem (opens in a new tab), J. Amer. Math. Soc. 20 (2007), 273–320.

8. J.-L. Kim and J.-K. Yu, Construction of tame types (opens in a new tab) (2017).

9. J. Fintzen, Types for tame p-adic groups (opens in a new tab), Ann. Math. 193 (2021), 303–346.

10. J. Fintzen, On the construction of tame supercuspidal representations (opens in a new tab), Compos. Math. 157 (2021), 2733–2746. A corrected proof of supercuspidality for Yu's construction.

11. J. D. Adler, J. Fintzen, M. Mishra and K. Ohara, Structure of Hecke algebras arising from types (opens in a new tab).

12. J. D. Adler, J. Fintzen, M. Mishra and K. Ohara, Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms (opens in a new tab).

13. Our joint work with Yu Wang on reduction to depth-zero blocks for Stevens data — forthcoming.

14. T. K. Howard and M. H. Weissman, Depth Zero Representations of Nonlinear Covers of p-adic Groups (opens in a new tab).

15. J. Zou, Simple type theory for metaplectic covers of GL(r) over a non-archimedean local field (opens in a new tab), J. Inst. Math. Jussieu 24 (2025), 1263–1335.

16. M. H. Weissman, An induction theorem for groups acting on trees (opens in a new tab). Proves compact-induction existence when the derived group has relative rank one.