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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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