ENDOSCOPY FOR HECKE CATEGORIES, CHARACTER SHEAVES AND REPRESENTATIONS

For a reductive group $G$ 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 $H$ with trivial monodromy. We also extend this equivalence to all blocks. We give two applications. One is a relationship between character sheaves on $G$ with a fixed semisimple parameter and unipotent character sheaves on the endoscopic group $H$ , after passing to asymptotic versions. The other is a similar relationship between representations of $G(\mathbb{F}_{q})$ with a fixed semisimple parameter and unipotent representations of $H(\mathbb{F}_{q})$ .

A computational approach to the Frobenius–Schur indicators of finite exceptional groups

We prove that the finite exceptional groups [Formula: see text], [Formula: see text], and [Formula: see text] have no irreducible complex characters with Frobenius–Schur indicator [Formula: see text], and we list exactly which irreducible characters of these groups are not real-valued. We also give a complete list of complex irreducible characters of the Ree groups [Formula: see text] which are not real-valued, and we show the only character of this group which has Frobenius–Schur indicator [Formula: see text] is the cuspidal unipotent character [Formula: see text] found by Geck.