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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.

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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