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Hi Vinoth, here are my thoughts, hopefully they're correct and what you're after: You can think of $\mathfrak{\tilde{g}}$ as the set of pairs $$\{(X,L)\in \mathfrak{g}\times \mathbb{P}^{1}\;|\; X(L …

I have been thinking about quotients lately and pondered the following: Let $G$ be a connected linear algebraic group and $X$ a $G$-variety where the action is the morphism $\sigma:G\times X\rightarr …

The dimension of the nilpotent orbits of $\mathfrak{gl}_{6}$ can be described using the corresponding partition $\pi: d_{1}+d_{2}+\ldots + d_{k} =6$ associated to a nilpotent orbit - so $\pi$ is the p …

Peterson varieties (in type A) can be described as the subvarieties of the full flag variety $$\{(F_{i})\;|\; F_{i}\subset \mathbb{C}^{n}, \; \dim F_{i} =i,\; N(F_{i})\subset F_{i+1}\}$$ where $N$ …

Bezrukavnikov and Ginzburg have unpublished notes, 'Hilbert Schemes and Reductive Groups' (referenced here, for example): does anyone know what became of these notes? Did Bezrukavnikov-Ginzburg publis …

I'm struggling to see what the actual question is but here is an example of a non-trivial use of geometry to prove a simple statement in representation theory; moreover, it is the only known way to o …

Yes: this is discussed in Chriss & Ginzburg 'Representation Theory and Complex Geometry', p.161 Remark 3.3.26. In short, the nilpotent cone is normal and we can apply Zariski's Main Theorem to deduce …

A nice example is given by Borho-Macpherson's construction of Weyl group representations (Representations des groupes de Weyl et homologie d'intersection pour les varietes nilpotentes, Comptes rendus. …