K-motives, Springer Theory and the Local Langlands Correspondence
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911766096642048 |
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| author | Eberhardt, Jens Niklas |
| author_facet | Eberhardt, Jens Niklas |
| contents | We construct a geometric realization of categories of representations of affine Hecke algebras and split reductive $p$-adic groups via a $K$-motivic Springer theory. We suggest a connection to the coherent Springer theory of Ben-Zvi, Chen, Helm, and Nadler through a categorical Chern character and outline results and conjectures on $K$-motives within the Langlands program.
To achieve our results, we introduce a six functor formalism for reduced $K$-motives applicable to linearly reductive stacks and establish formality for categories of Springer $K$-motives. We work within a broader framework of Hecke algebras derived from Springer data. This makes the results applicable, for example, to the ($K$-theoretic) quiver Hecke and Schur algebra. Moreover, we relate our constructions to prior geometric realizations for graded Hecke algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13052 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | K-motives, Springer Theory and the Local Langlands Correspondence Eberhardt, Jens Niklas Representation Theory Algebraic Geometry K-Theory and Homology We construct a geometric realization of categories of representations of affine Hecke algebras and split reductive $p$-adic groups via a $K$-motivic Springer theory. We suggest a connection to the coherent Springer theory of Ben-Zvi, Chen, Helm, and Nadler through a categorical Chern character and outline results and conjectures on $K$-motives within the Langlands program. To achieve our results, we introduce a six functor formalism for reduced $K$-motives applicable to linearly reductive stacks and establish formality for categories of Springer $K$-motives. We work within a broader framework of Hecke algebras derived from Springer data. This makes the results applicable, for example, to the ($K$-theoretic) quiver Hecke and Schur algebra. Moreover, we relate our constructions to prior geometric realizations for graded Hecke algebras. |
| title | K-motives, Springer Theory and the Local Langlands Correspondence |
| topic | Representation Theory Algebraic Geometry K-Theory and Homology |
| url | https://arxiv.org/abs/2401.13052 |