An analogue of irreducible cuspidal representations for the group $PGL(2)$ over a two-dimensional local field
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| Format: | Preprint |
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2026
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| author | Braverman, Alexander Kazhdan, David |
| author_facet | Braverman, Alexander Kazhdan, David |
| contents | 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 $θ:L^*/K^*\to \mathbb C^*$ 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 $H$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_11735 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An analogue of irreducible cuspidal representations for the group $PGL(2)$ over a two-dimensional local field Braverman, Alexander Kazhdan, David Representation Theory Algebraic Geometry Number Theory 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 $θ:L^*/K^*\to \mathbb C^*$ 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 $H$. |
| title | An analogue of irreducible cuspidal representations for the group $PGL(2)$ over a two-dimensional local field |
| topic | Representation Theory Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2604.11735 |