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On the cleanness of cuspidal character sheaves. (English) Zbl 1263.20044
From the introduction: We prove the cleanness of cuspidal character sheaves in arbitrary characteristic in the few cases where it was previously unknown.
Let $$\Bbbk$$ be an algebraically closed field of characteristic exponent $$p\geq 1$$. Let $$G$$ be a connected reductive algebraic group over $$\Bbbk$$ with adjoint group $$G_{\text{ad}}$$. It is known that, if $$A$$ is a cuspidal character sheaf on $$G$$, then $$A=IC(\overline\Sigma,\mathcal E)[\dim\Sigma]$$, where $$\Sigma$$ is the inverse image under $$G\to G_{\text{ad}}$$ of a single conjugacy class in $$G_{\text{ad}}$$, $$\mathcal E$$ is an irreducible local system on $$\Sigma$$ equivariant under the conjugation $$G$$-action and $$IC$$ denotes the intersection cohomology complex. (For any subset $$\gamma$$ of $$G$$ we denote by $$\overline\gamma$$ the closure of $$\gamma$$ in $$G$$.) We say that $$A$$ is clean if $$A|_{\overline\Sigma-\Sigma}=0$$.
This paper is concerned with the following result. Theorem 1.2. Any cuspidal character sheaf of $$G$$ is clean.

##### MSC:
 20G05 Representation theory for linear algebraic groups 20G10 Cohomology theory for linear algebraic groups 14L30 Group actions on varieties or schemes (quotients)
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