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Characteristic cycles for the loop Grassmannian and nilpotent orbits. (English) Zbl 1160.22306
From the introduction: We compute microlocal multiplicities for certain cases of interest in representation theory. Let \(G\) be a connected reductive group with loop group \(LG\), and let \(P\) be the subgroup of \(LG\) consisting of loops with positive Fourier coefficients. Then \(P\)-orbits \(\lambda\) on the loop Grassmannian \(LG /P = \mathcal G\) correspond to the irreducible representations \(L(\lambda)\) of the dual reductive group \(\hat G\). For dominant weights \(\mu\) and \(\lambda\) of a torus of \(\hat G\), let \(m_{\mu} (L(\lambda))\) denote the multiplicity of \(\mu\) in \(L(\lambda)\). Let us embed the orbit closure \(\overline{{\mathcal G}_{\lambda}}\) into a finite-dimensional manifold \(Z\).
Theorem 0.1. (a) Let \(i : {\mathcal G}_{\lambda}\to Z\) be the inclusion, and let \(i_! (\mathbb{C}_{\lambda})\) be the extension by zero of the constant sheaf on \(\mathcal O_\lambda\). Then \(CC(i_! (\mathbb{C}_{\lambda})[\dim {\mathcal G}_{\lambda} ]) = \overline{T^*_{{\mathcal G}_{\lambda}}} Z\).
(b) The characteristic cycle of the intersection cohomology sheaf of \(\overline{{\mathcal G}_{\lambda}}\) is given by the character of \(L(\lambda)\).
We also consider the nilpotent cone in a semisimple Lie algebra.

MSC:
22E67 Loop groups and related constructions, group-theoretic treatment
20G05 Representation theory for linear algebraic groups
32C38 Sheaves of differential operators and their modules, \(D\)-modules
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