Timeline for For a spherical pair $(G, H)$, which $G$-representations appear in $k[G/H]$?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 17, 2019 at 13:51 | comment | added | David Loeffler | Splendid, thanks! | |
| Dec 16, 2019 at 12:54 | vote | accept | David Loeffler | ||
| Dec 16, 2019 at 12:47 | comment | added | Friedrich Knop | The answer to this question is related to the factoriality of $G/H$ which is related to the character group of $H$. In particular, if $H$ is semisimple then the monoid is free. Otherwise, you can find counterexamples on p. 149 of Krämer's paper "Sphärische Untergruppen...", the easiest being $G=SL(5)$ and $H=\mathbf G_m\cdot Sp(4)$. There the cone is $3$-dimensional with $4$ extremal reays. | |
| Dec 16, 2019 at 8:45 | comment | added | David Loeffler | This is great, thanks! Apologies for my vagueness about "lattice cone": the definition I had in mind was the monoid spanned by a linearly independent set of lattice vectors, i.e. a set of the form $\{ n_1 v_1 + \dots + n_r v_r : n_i \in \mathbb{Z}_{\ge 0}\}$, where $v_i$ are vectors in $\Lambda(G)$ that are linearly independent in $\Lambda(G) \otimes \mathbb{Q}$. Does $\Lambda_+(G, H)$ necessarily have this form, at least if $H$ is reductive? | |
| Dec 15, 2019 at 20:28 | history | answered | Friedrich Knop | CC BY-SA 4.0 |