Character sheaves on the semi-stable locus of a group compactification

Xuhua He

Algebraic Geometry mathscidoc:1912.43232

Advances in Mathematics, 225, (6), 3258-3290, 2009.12
We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the ordinary restriction of a character sheaf on the compactification to a semi-stable stratum is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer (2006) [21]. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture Lusztig (2004) [16
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  title={Character sheaves on the semi-stable locus of a group compactification},
  author={Xuhua He},
  booktitle={Advances in Mathematics},
Xuhua He. Character sheaves on the semi-stable locus of a group compactification. 2009. Vol. 225. In Advances in Mathematics. pp.3258-3290.
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