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K-motives, Springer Theory and the Local Langlands Correspondence (2401.13052v2)
Published 23 Jan 2024 in math.RT, math.AG, and math.KT
Abstract: 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.
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