English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $\mathfrak g$ be a simple Lie algebra over $\mathbb C$ and let $e \in\mathfrak g$ be nilpotent. We consider the finite $W$-algebra $U(\mathfrakg,e)$ associated to $e$ and the problem of determining the variety $\mathcalE(\mathfrak g,e)$ of 1-dimensional representations of $U(\mathfrak g,e)$. For$\mathfrak g$ of low rank, we report on computer calculations that have beenused to determine the structure of $\mathcal E(\mathfrak g,e)$, and the actionof the component group $\Gamma_e$ of the centralizer of $e$ on $\mathcalE(\mathfrak g,e)$. As a consequence, we provide two examples where thenilpotent orbit of $e$ is induced, but there is a 1-dimensional$\Gamma_e$-stable $U(\mathfrak g,e)$-module which is not induced via Losev'sparabolic induction functor. In turn this gives examples where there is a"non-induced" multiplicity free primitive ideal of $U(\mathfrak g)$.

On the variety of 1-dimensional representations of finite $W$-algebras in low rank