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In this paper, we extend the notion of Shintani descent to general (possiblydisconnected) algebraic groups defined over a finite field $\mathbb{F}_q$. Forthis, it is essential to treat all the pure inner $\mathbb{F}_q$-rational formsof the algebraic group at the same time. We prove that the notion of almostcharacters (introduced by T. Shoji using Shintani descent) is well defined forany neutrally unipotent algebraic group, i.e. an algebraic group whose neutralconnected component is a unipotent group. We also prove that these almostcharacters coincide with the "trace of Frobenius" functions associated withFrobenius-stable character sheaves on neutrally unipotent groups. In the courseof the proof, we also prove that the modular categories that arise fromBoyarchenko-Drinfeld's theory of character sheaves on neutrally unipotent arein fact positive integral confirming a conjecture due to Drinfeld.

Shintani descent for algebraic groups and almost characters of unipotent groups