𝐙/𝐦-graded Lie algebras and perverse sheaves, IV

Lusztig, George; Yun, Zhiwei

doi:10.1090/ert/546

$\mathbf {Z}/m$-graded Lie algebras and perverse sheaves, IV
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Represent. Theory 24 (2020), 360-396 Request permission

Abstract:

Let $G$ be a reductive group over $\mathbf {C}$. Assume that the Lie algebra $\frak g$ of $G$ has a given grading $(\frak g_j)$ indexed by a cyclic group $\mathbf {Z}/m$ such that $\frak g_0$ contains a Cartan subalgebra of $\frak g$. The subgroup $G_0$ of $G$ corresponding to $\frak g_0$ acts on the variety of nilpotent elements in $\frak g_1$ with finitely many orbits. We are interested in computing the local intersection cohomology of closures of these orbits with coefficients in irreducible $G_0$-equivariant local systems in the case of the principal block. We show that these can be computed by a purely combinatorial algorithm.

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