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Endoscopy for Hecke categories, character sheaves and representations

Abstract

For a split reductive group GG over a finite field, we show that the neutral block of its mixed Hecke category with a fixed monodromy under the torus action is monoidally equivalent to the mixed Hecke category of the corresponding endoscopic group HH with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on GG with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group HH, after passing to asymptotic versions. The other is a similar relationship between representations of G(Fq)G(\mathbb{F}_q) with a fixed semisimple parameter and unipotent representations of H(Fq)H(\mathbb{F}_{q}).Comment: 57 pages. A new section on application to representations added. A few small gaps fixed. To appear in Forum of Mathematics P

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