## Regular Functions on Slodowy Slices

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

I recently learned that there is a natural identification, given a simple Lie algebra,

Polynomial functions on the Slodowy slice of a regular nilpotent orbit $\simeq$ polynomial functions on the Cartan subalgebra invariant under the Weyl group.

I guess there's a generalization of this statement for subregular or other nilpotent orbits. Which book/article should I have a look at? For example, I guess there's a equivalence of the form

Polynomial functions on the Slodowy slice of a subregular nilpotent orbit $\simeq$ polynomial functions on the Levi subalgebra of the form $sl_2\times \mathbb{C}^{r-1}$ satisfying conditions *

What are the conditions * I need to fill in the statement above?

-

## 1 Answer

The Slodowy slice to a nilpotent $e$ is by definition the set of elements $x$ such that $x-e$ commutes with some fixed $f$ such that $e$ and $f$ generate an $\mathfrak{sl}_2$. Thus, it is an affine space modeled on the commutant of $f$, a Lie subalgebra of $\mathfrak{g}$.

In the regular case, this subalgebra isn't a torus (it's nilpotent), though it is a flat limit of tori (just as a regular nilpotent is a limit of semi-simples). In $\mathfrak{sl}_n$, if we take the principal nilpotent, its commutant is the space of matrices with 0's on or below the diagonal, and with entries constant on every shifted diagonal.

-

Thank you, I learned the definition of Slodowy slice recently. I'm now more interested in understanding the "Miura map", see Premet's eprints.ma.man.ac.uk/495/02/GGG.pdf , p. 28, in the case of sub regular orbit. This should give a homomorphism from $\mathbb{C}[S]$ to  $\mathbb{C}[sl_2\times \mathbb{C}^{r-1}]$... – Yuji Tachikawa Nov 27 2011 at 6:51
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 15, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8787974119186401, "perplexity_flag": "head"}