What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T20:22:35Z http://mathoverflow.net/feeds/question/83419 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/83419/what-is-a-canonical-set-of-representatives-in-gln-f-for-the-vertices-in-the-b What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building? Marc Palm 2011-12-14T12:41:04Z 2011-12-15T00:00:30Z <p>$F$ is a non archimedean field here. To be more precise, I would actually prefer a set of representative in $B(F)$ for the discrete space $B(F) / B(o)Z(F)$?</p> <p>This can be phrased also as question about lattices in $F^n$, but I would prefer to stay on the group level.</p> http://mathoverflow.net/questions/83419/what-is-a-canonical-set-of-representatives-in-gln-f-for-the-vertices-in-the-b/83464#83464 Answer by Paul Broussous for What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building? Paul Broussous 2011-12-14T20:13:20Z 2011-12-14T20:13:20Z <p>You somehow want to parametrise the vertices of the building of $G={\rm GL}(n,F)$ : $$G/F^\times K = BK/F^\times K= B(F)/B({\mathfrak o})Z(F)$$ (by Iwasawa decomposition).</p> <p>For $n=2$ ou can easily find representatives, but for $n>2$, it's going to be tricky!</p> <p>I just give some hints. Write $N$ for the unipotent radical of $B$ and $T$ for the diagonal torus so that $B=T\ltimes N$.</p> <p>-- If $n,n'\in N$ and $t, t'\in T$, then if $nt\sim nt'$ mod $B(O)Z(F)$, one has $t\sim t'$ mod $Z(F)T(O)$. So you may assume that $t$ is of the form</p> <p>$$t= {\rm diag}(\varpi^{k_1}, ...,\varpi^{k_n})$$ where $(k_1 ,...,k_n )$ is well defined modulo the diagonal action of $\mathbb Z$ on ${\mathbb Z}^n$. </p> <p>-- You have $nt \sim n't$ mod $Z(F)B(O)$ iff $n\sim n'$ mod $tN(0)t^{-1}$.</p> <p>So for each $t$ as above, you need to find a system of representatives of $$N(F)/tN(O)t^{-1}$$ For $n=2$, this is easy. For $n>2$, this seems tricky. I've never tried ...</p> http://mathoverflow.net/questions/83419/what-is-a-canonical-set-of-representatives-in-gln-f-for-the-vertices-in-the-b/83481#83481 Answer by Jim Humphreys for What is a canonical set of representatives in $GL(n,F)$ for the vertices in the Bruhat Tits building? Jim Humphreys 2011-12-14T23:57:51Z 2011-12-14T23:57:51Z <p>The comments together with Paul's answer emphasize the importance of formulating the question more precisely and with some context (what you've read on the subject, for instance). Though I'm not at all a specialist in buildings, I know some of the complicated history of the subject as it evolved into long and highly sophisticated papers by Bruhat-Tits and others. But your question about general (or equally well special) linear groups, which are split over the prime field, goes back to the foundational papers such as the 1965 Publ. Math. IHES paper by Iwahori and Matsumoto, freely available by a quick author search <a href="http://www.numdam.org" rel="nofollow">here</a>, followed by the detailed exercises in Chapter IV of Bourbaki's 1968 treatise <em>Groupes et algebres de Lie</em> where the BN-pair structure (or Tits system) is developed into the basic theory of buildings. </p> <p>In this early work there is a treatment of the special subgroup structure present in a split (Chevalley) group over a standard <code>$p$</code>-adic field: fixing a Borel subgroup over the finite residue class field, one can lift it to the <code>$p$</code>-adic integers where it becomes an Iwahori subgroup. Such groups are determined up to conjugacy in the ambient algebraic group. Along with a copy of the affine Weyl group, an Iwahori subgroup determines a BN-pair structure and Bruhat decomposition. In turn there are finitely many maximal (proper) "parahoric" subgroups. Their cosets in the big group become the vertices for the resulting building. If you regard the original Borel subgroup as "canonical", this pathway should lead to canonical vertices of the building. As Paul observes, in the case <code>$n=2$</code> all of this is fairly easy to write down; here the building is just an infinite tree. </p> <p>Once you get beyond split groups and ordinary <code>$p$</code>-adic extensions of the rationals, a lot more machinery has to be developed in order to work effectively with buildings and subgroup actions on them. But the general linear group, especially in semisimple rank 1, is the natural starting point for combining group theory and combinatorial geometry in a visualizable way.</p>