An analogue of irreducible cuspidal representations for the group $PGL(2)$ over a two-dimensional local field

Abstract: Let $F$ be a local non-archimedian field of odd residue characteristic and let $G=PGL(2)$. In this paper we study an analog of irreducible cuspidal representations of the group $G(F)$ when $F$ is replaced by the field $K=F((t))$. The story turns out to be similar to the classical case, but also with some differences. We present a construction of such representations essentially (up to a small subtlety) starting from a quadratic extension $L$ of $K$ and a character $θ:L^{<em>/K^</em>\to} \mathbb C<sup>*$ which is not Galois invariant. We also show that the restriction of the representations we construct to the group $P(K)$ (here $P$ is a Borel subgroup of $PGL(2)$) is irreducible. However, contrary to the classical case it turns out that these restrictions are not isomorphic to the "standard" irreducible cuspidal representation of $P(K)$. In the Appendix we propose a notion of cuspidality for smooth representations of the group $H(K)$ for an arbitrary split reductive group $H$.

• The paper introduces an analogue of irreducible cuspidal representations for PGL(2) over two-dimensional local fields by extending classical smooth representation theory.
• It constructs special representations via induced modules from almost-unipotent subgroups, leveraging character data from quadratic extensions.
• The work classifies induced cuspidal representations and reveals new phenomena in irreducibility upon restriction to Borel subgroups, impacting the broader Langlands framework.

Analogue of Irreducible Cuspidal Representations for $PGL(2)$ over a Two-Dimensional Local Field

Introduction and Motivation

The paper investigates the representation theory of the group $G = PGL(2)$ over the two-dimensional local field $K = F((t))$, with $F$ a local non-archimedean field of odd residue characteristic, focusing on analogues of irreducible cuspidal representations. The principal motivation is to generalize the well-understood correspondence between irreducible cuspidal (a.k.a. special) representations and (non-Galois invariant) characters of tori attached to quadratic extensions in the one-dimensional case $G(F)$. The two-dimensional context introduces both structural analogies and nontrivial deviations from the classical scenario.

Crucially, in the two-dimensional setting, the notion of "special" must be formulated in terms of smooth representations for the group $G(K)$ and its "bounded" analogues $G(O)$ and $G'(O)$ (modular and twisted counterparts), fundamentally requiring actions on pro-vector spaces rather than just vector spaces. This analysis extends naturally to inquiries regarding the categorical definition of smooth representations, as well as the structure of the corresponding tori and congruence subgroups.

Special Representations: Definitions and Construction

A nontrivial, smooth irreducible representation $\pi$ of $G(K)$, $G = PGL(2)$0, or $G = PGL(2)$1 is declared special if its restriction to the Borel subgroup $G = PGL(2)$2 or its analogues is irreducible (Def. 1.1). This generalizes the classical equivalence of cuspidality and irreducibility upon restriction to $G = PGL(2)$3 for $G = PGL(2)$4.

A pivotal insight is that, for $G = PGL(2)$5 and $G = PGL(2)$6, the property of being special coincides with ellipticity: namely, the representation's restriction to the congruence subgroup's unipotent radical must support an elliptic orbit (i.e., one supported on a non-split torus corresponding to a quadratic field extension). This enables complete classification of special/elliptic irreducible representations of $G = PGL(2)$7 and $G = PGL(2)$8 via data $G = PGL(2)$9, with $K = F((t))$0 a quadratic extension and $K = F((t))$1 a character as follows:

• For ramified quadratic extensions, isomorphism classes correspond bijectively with admissible characters $K = F((t))$2 not fixed under Galois conjugation, up to Galois action.
• For unramified cases, there is a subtle $K = F((t))$3-torsor valued obstruction, interpreted via the possible quadratic character twist ambiguity on $K = F((t))$4.

The explicit representations are constructed as induced modules. For $K = F((t))$5 representations, the construction proceeds by induction from almost-unipotent subgroups with specific character data imposed on their pro-unipotent part, realized geometrically via the affine and twisted affine Grassmannians. The paper provides a careful analysis of the underlying congruence subgroups, the identification of tori in these groups, and the relevant supports in the Lie algebra representation.

Induction to $K = F((t))$6 and Properties

The central result of the paper is that, for each special/elliptic representation $K = F((t))$7 of $K = F((t))$8 or $K = F((t))$9, the compact induction to $F$0 (in the sense of Gaitsgory-Kazhdan; $F$1 or $F$2) yields a cuspidal and irreducible representation $F$3 of $F$4. Furthermore, its restriction to $F$5 remains irreducible. Hence, for every admissible pair $F$6 as above, there is an associated irreducible cuspidal representation of $F$7, denoted $F$8.

A notable deviation from the one-dimensional situation is clarified: while all irreducible cuspidal representations of $F$9 are isomorphic in the classical setup, for $G(F)$0 there exist many non-isomorphic irreducible cuspidal representations, and the straightforward extension of the standard cuspidal $G(F)$1-representation to $G(F)$2 fails. Nevertheless, the restriction to $G(F)$3 of each $G(F)$4, for special $G(F)$5, is always irreducible and shares a common associated graded (with respect to a canonical filtration) with a "universal" cuspidal $G(F)$6-module.

Classification Results

The classification of special (cuspidal irreducible) representations of $G(F)$7 and $G(F)$8 as well as the induced cuspidal irreducible representations of $G(F)$9 is accomplished explicitly in terms of quadratic extension and character data. The classification exploits:

• Mackey theory for analyzing restriction and induction,
• Fine structure of congruence subgroups,
• Coadjoint orbits for canonical supports,
• Quasi-Heisenberg and Heisenberg group structures present in the congruence quotients,
• Compatibility with the structure of the affine Grassmannian, relating orbits of $G(K)$0 to $G(K)$1.

In the ramified case, the non-canonical isomorphism classes arising from quadratic twisting are demonstrated, with subtle torsorial ambiguity in the unramified scenario. The ramifications for the identification of the full set of isomorphism classes and their parametric dependence (e.g., on "depth") are explicitly discussed.

Notions of Cuspidality and Theoretical Implications

A category-theoretic definition of cuspidality for smooth representations of an arbitrary split reductive group $G(K)$2 over $G(K)$3 is introduced in the appendix, generalizing the presented construction from $G(K)$4. This reflects a systematic strategy for relating cusp forms and associated modules in higher dimensions and more general settings.

The implications of these findings are notable:

• They provide new explicit families of irreducible, cuspidal representations for the "double affine" group $G(K)$5 over $G(K)$6, systematically extending the classical theory.
• The non-equivalence of all irreducible restrictions to $G(K)$7 signals a deeper richness in the representation theory for two-dimensional local fields.
• The work lays foundational stones towards understanding the harmonic analysis and the conjectural Langlands correspondence in higher-dimensional local field settings.

Open Questions and Prospects

A number of structural and classification questions remain open, as discussed in the concluding section of the paper. These include:

• The problem of characterizing all special representations of $G(K)$8 as induced from $G(K)$9 or $G(O)$0,
• The isomorphism criteria for induced representations given different depth or extension data,
• The extension to higher rank, specifically $G(O)$1, and the formulation of higher-level Hecke or double affine Hecke algebra connections.

These questions indicate fertile ground for future research, particularly in relation to two-dimensional adelic harmonic analysis, the structure of double affine Hecke algebras, and their potential interplay with arithmetic and geometric Langlands paradigms.

Conclusion

This work develops a comprehensive and technically detailed analogue of the classification and construction of irreducible cuspidal representations for $G(O)$2 over a two-dimensional local field. By systematically extending classical methods to the higher-dimensional context, it provides a precise correspondence between such representations and certain Galois-unstable characters of tori attached to quadratic extensions, realizes these explicitly via induction from congruence subgroups, and identifies new phenomena distinguishing the two-dimensional case. These findings establish foundational results for the emerging representation theory of reductive groups over two-dimensional local fields and invite further inquiry into their structure and applications.