Timeline for Difference between $\mathfrak{g}/\!/G$ and $G/\!/G$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 12 at 1:41 | vote | accept | lafes | ||
| Aug 4, 2022 at 12:24 | comment | added | Friedrich Knop | It is only fro simply connected simisimple groups that $G//G$ is smooth and even an affine space. As free generators of $\mathcal{O}(T)^W$ one can take the orbit sums of the fundamental weights. | |
| Aug 4, 2022 at 1:53 | comment | added | LSpice | Thanks, @DavidESpeyer! Does this all carry over to the positive-but-good-characteristic case? | |
| Aug 4, 2022 at 1:51 | comment | added | David E Speyer | The coordinate ring of the torus in $PSL_3$ is the subring of $k[x_1, x_2, x_3]/(x_1 x_2 x_3-1)$ consisting of polynomials which have total degree $0 \bmod 3$ in the $x$'s. Taking invariants by $S_3$, we get the subring of $k[e_1, e_2]$ generated by $e_1^3$, $e_1 e_2$ and $e_2^3$: an $A_2$-singularity. | |
| Aug 4, 2022 at 1:49 | comment | added | David E Speyer | @LSpice It will still be true that $G//G$ will be $T//W$, but $T//W$ need not be affine space, if $G$ is a non-simply connected reductive group. For example, let's compare the cases of $SL_3$ and $PSL_3$. The coordinate ring of the torus in $SL_3$ is $k[x_1, x_2, x_3]/(x_1 x_2 x_3-1)$, with $W = S_3$ acting by permuting the variables. The invariant ring is generated by $e_1 := x_1+x_2+x_3$ and $e_2 := x_1 x_2 + x_1 x_3 +x_2 x_3$ (we don't need $e_3$ because $x_1 x_2 x_3 = 1$). | |
| Aug 4, 2022 at 0:59 | comment | added | LSpice | I appreciate this answer, because I didn't know the name of the result in the group case. I can easily imagine that things go wrong, even on the Lie-algebra side, for bad primes (in the technical sense of Springer–Steinberg). I don't know much about the behaviour of geometric quotients, even in characteristic $0$. What, if anything, is known about the effect on $G//G$ of replacing $G$ by an isogenous reductive group? | |
| Aug 4, 2022 at 0:54 | comment | added | LSpice |
TeX note: It is usually better to use \operatorname rather than \mathrm for things that are semantically operators. Compare, for instance, $\mathrm{rank} G$ \mathrm{rank} G to $\operatorname{rank} G$ \operatorname{rank} G (although you used parentheses). MathJax note: Unlike in TeX, a preamble line $\newcommand\mmod{…}…$ followed by a newline will force a space in the text. Unfortunately, as ugly as it is in the source, the only way to get rid of the space is to put the closing $ directly next to the first line of the post, with no intervening whitespace. I have edited accordingly.
|
|
| Aug 4, 2022 at 0:51 | history | edited | LSpice | CC BY-SA 4.0 |
Name of reference; `\operatorname`
|
| Aug 3, 2022 at 23:50 | history | edited | skd | CC BY-SA 4.0 |
added 44 characters in body
|
| Aug 3, 2022 at 23:44 | history | answered | skd | CC BY-SA 4.0 |