Timeline for Tits Reductive Groups over Local Fields Example 1.15 (Quasi-split special unitary groups in odd dimension)
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Aug 4, 2020 at 2:22 | comment | added | Marc Besson | It was that very post that inspired me to try out Tits again... I seem to be gradually piecing things together as I cycle through a couple of these excellent discussions. Thanks again! | |
| Aug 4, 2020 at 2:19 | comment | added | LSpice | It's also probably worth your while (it was definitely worth mine!) to see @Marty's lovely discussion of this example from the point of view of parahoric subgroups. | |
| Aug 4, 2020 at 2:17 | comment | added | LSpice | Incidentally, this is "as bad as (quasi-split but) non-split gets", which is why Tits does this particular example rather than, say, something not of type $\mathsf A$; in all other (quasi-split) cases at least the relative root system is reduced, so you do once again get additive groups (and $d_\alpha$ is an appropriate residual degree). | |
| Aug 4, 2020 at 2:15 | comment | added | Marc Besson | Thank you so much for all your help! I'll take some time to digest (especially with your good point about the $U(K)$ not being nice additive groups as in the split case) | |
| Aug 4, 2020 at 2:13 | comment | added | LSpice | By the way, as you go on, it may help to know that the $\delta$ that occurs later in this section is $0$ unless $p = 2$. (I thought that that was remarked somewhere, but I can't find it now.) Of course, Tits's Corvallis article is much easier reading than BT; but probably more pleasant than either is Yu's lovely article "Bruhat–Tits theory and buildings" in the Ottawa lectures on admissible representations of reductive, $p$-adic groups. It used to be on J.-K.'s web page, but anyway you can find it on Semantic Scholar. | |
| Aug 4, 2020 at 2:10 | comment | added | LSpice | You should not write $U_{a_{ij}}(c, d)$. Perhaps you mean $U_{a_i}(K)$, which is a set of matrices $u_i(c, d)$? This is not an additive group; it is not even Abelian. | |
| Aug 4, 2020 at 2:09 | comment | added | Marc Besson | Also you make a very good comment about the fact that the matrices I wrote do not form a group; but nonetheless, we know $U_{a_{ij}}(c,d)$ are isomorphic to a 2-dimensional additive group over $K$ (is this correct?) corresponding to choices of $c$ and $d$, and that when we quotient as Tits asks, in this case I am checking the ramified degree of the field where $c$ lives? | |
| Aug 4, 2020 at 2:09 | comment | added | LSpice | You are right; I pointed to the wrong thing. Note $\nu(m(u_i(c, d))v = v$ means $-v_i = v_{-i} = v_i + \omega(d)$, i.e., $2v_i + \omega(d) = 0$. This can be expressed as the vanishing of $a_i + \frac1 2\omega(d)$ or, when $c = 0$ and so we are in $U_{2a_i}$, of $2a_i + \omega(d)$. | |
| Aug 4, 2020 at 2:01 | comment | added | Marc Besson | So I don't know how you can draw this 1/2 out of (4). In particular, (4) seems to be identical to the formula for example 1.14, at the bottom of page 39; we let the normalizer act on the apartment by the natural action on the vectors plus a term corresponding to the valuation. I am probably being slow here but I still don't see where the 1/2$\omega(d)$ comes from. I would simply expect $a_i+\omega(d)$. | |
| Aug 4, 2020 at 1:58 | comment | added | LSpice | Oh, $d$ like $d_\alpha$, not $d$ like $u_i(c, d)$. Yes, it is more or less as you say (although I'd argue we really care about the residual degree—but the two pieces of information are complementary). Note, by the way, that, though it's fine for casual discussion, your discussion of the quotient group is more like a discussion of a transversal of the collection of cosets: the set of matrices you describe isn't a group! | |
| Aug 4, 2020 at 1:58 | comment | added | Marc Besson | In section 1.6, Tits writes: "For every affine function $\alpha$ on $A$ whose vector part $a=v(\alpha)$ belongs to $\Phi$, one has an obvious inclusion $\overline{X}_{2\alpha} \rightarrow \overline{X}_{\alpha}$, and the quotient has a natural vector space structure over $\overline{K}$"..., the residue field. The dimension of this space is $d$. | |
| Aug 4, 2020 at 1:57 | comment | added | LSpice | The $\omega(d)$ terms come from the fact that, similarly, the action of the normaliser of the split torus on the apartment corresponding to that torus is not what you might expect: see (4) on p. 41. | |
| Aug 4, 2020 at 1:54 | comment | added | LSpice | What do you mean by "compute the function $d$"? | |
| Aug 4, 2020 at 1:54 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| Aug 4, 2020 at 1:50 | comment | added | LSpice | As to "why they are $K$-points", it's the same reason that the unitary group $\{g \in \operatorname{GL}_n(\mathbb C) \mathrel: g g^* = 1\}$ is the $\mathbb R$-points of an $\mathbb R$-group, not the $\mathbb C$-points. A very easy check: the Lie-algebra version of this gives a $K$-vector space, not an $L$-vector space. (Why does this seem to behave differently from more familiar situations? Because, unlike those more familiar situations, the Galois group doesn't simply act coördinatewise.) | |
| Aug 4, 2020 at 1:44 | history | asked | Marc Besson | CC BY-SA 4.0 |