Affine Springer fibers and depth zero L-packets
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866915433903292416 |
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| author | Bezrukavnikov, Roman Varshavsky, Yakov |
| author_facet | Bezrukavnikov, Roman Varshavsky, Yakov |
| contents | Let $G$ be a connected reductive group over a field $F=\mathbb{F}_q((t))$ splitting over $\overline{\mathbb{F}}_q((t))$. Following [KV,DR], a tamely unramified Langlands parameter $λ:W_F\to{}^L G(\overline{\mathbb{Q}}_{\ell})$ in general position gives rise to a finite set $Π_λ$ of irreducible admissible representations of $G(F)$, called the $L$-packet.
The main goal of this work is to provide a geometric description of characters $χ_π$ of $π\inΠ_λ$ and of their endoscopic linear combinations $χ_λ^κ$ in terms of homology of affine Springer fibers, thus establishing an analog of Lusztig conjectures in this case. Furthermore, each $χ_λ^κ$ can be described as the trace of Frobenius function of a conjugation equivariant perverse sheaf on the loop group by the sheaf-function correspondence.
As another application, we prove that the sum $χ_λ^{st}:=\sum_{π\inΠ_λ}χ_π$ is stable and show that the $χ_λ^{st}$'s are compatible with inner twistings. More generally, we prove that each $χ_λ^κ$ is $\mathcal{E}_{λ,κ}$-stable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_13123 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Affine Springer fibers and depth zero L-packets Bezrukavnikov, Roman Varshavsky, Yakov Representation Theory Algebraic Geometry 22E50, 22E57 Let $G$ be a connected reductive group over a field $F=\mathbb{F}_q((t))$ splitting over $\overline{\mathbb{F}}_q((t))$. Following [KV,DR], a tamely unramified Langlands parameter $λ:W_F\to{}^L G(\overline{\mathbb{Q}}_{\ell})$ in general position gives rise to a finite set $Π_λ$ of irreducible admissible representations of $G(F)$, called the $L$-packet. The main goal of this work is to provide a geometric description of characters $χ_π$ of $π\inΠ_λ$ and of their endoscopic linear combinations $χ_λ^κ$ in terms of homology of affine Springer fibers, thus establishing an analog of Lusztig conjectures in this case. Furthermore, each $χ_λ^κ$ can be described as the trace of Frobenius function of a conjugation equivariant perverse sheaf on the loop group by the sheaf-function correspondence. As another application, we prove that the sum $χ_λ^{st}:=\sum_{π\inΠ_λ}χ_π$ is stable and show that the $χ_λ^{st}$'s are compatible with inner twistings. More generally, we prove that each $χ_λ^κ$ is $\mathcal{E}_{λ,κ}$-stable. |
| title | Affine Springer fibers and depth zero L-packets |
| topic | Representation Theory Algebraic Geometry 22E50, 22E57 |
| url | https://arxiv.org/abs/2104.13123 |