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Register a new userOn Q-factorial terminalizations of nilpotent orbits
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Authors
Baohua Fu
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In a recent preprint, Y. Namikawa proposed a conjecture on Q-factorial terminalizations and their birational geometry of nilpotent orbits. He proved his conjecture for classical simple Lie algebras. In this note, we prove his conjecture for exceptional simple Lie algebras. For the birational geometry, contrary to the classical case, two new types of Mukai flops appear.
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We develop an algorithm for computing the closure of a given nilpotent $G_0$-orbit in $g_1$, where $g_1$ and $G_0$ are coming from a $Z$ or a $Z/mZ$-grading $g= bigoplus g_i$ of a simple complex Lie algebra $g$.
We study wave-front sets of representations of reductive groups over global or non-archimedean local fields.
Let G be a simple, simply-connected algebraic group over the complex numbers with Lie algebra $mathfrak g$. The main result of this article is a proof that each irreducible representation of the fundamental group of the orbit O through a nilpotent element $e in mathfrak g$ lifts to a representation of a Jacobson-Morozov parabolic subgroup of G associated to e. This result was shown in some cases by Barbasch and Vogan in their study of unipotent representations for complex groups and, in general, in an unpublished part of the authors doctoral thesis. In the last section of the article, we state two applications of this result, whose details will appear elsewhere: to answering a question of Lusztig regarding special pieces in the exceptional groups (joint work with Fu, Juteau, and Levy); and to computing the G-module structure of the sections of an irreducible local system on O. A key aspect of the latter application is some new cohomological statements that generalize those in earlier work of the author.
The description of nilpotent Chernikov $p$-groups with elementary tops is reduced to the study of tuples of skew-symmetric bilinear forms over the residue field $mathbb{F}_p$. If $p e2$ and the bottom of the group only consists of $2$ quasi-cyclic summands, a complete classification is given. We use the technique of quivers with relations.
In this paper, we study the metric geometric mean introduced by Pusz and Woronowicz and the spectral geometric mean introduced by Fiedler and Ptak, originally for positive definite matrices. The relation between $t$-metric geometric mean and $t$-spectral geometric mean is established via log majorization. The result is then extended in the context of symmetric space associated with a noncompact semisimple Lie group. For any Hermitian matrices $X$ and $Y$, Sos matrix exponential formula asserts that there are unitary matrices $U$ and $V$ such that $$e^{X/2}e^Ye^{X/2} = e^{UXU^*+VYV^*}.$$ In other words, the Hermitian matrix $log (e^{X/2}e^Ye^{X/2})$ lies in the sum of the unitary orbits of $X$ and $Y$. Sos result is also extended to a formula for adjoint orbits associated with a noncompact semisimple Lie group.
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