板上有對Lie theory比較熟悉的人嗎? 問題是BN-pairs在real reductive Lie group中:
a BN-pair (also known as Tits system) is a pair B,N of subgroups of a group G
such that
1. B and N together generate G
2. T = B∩N is normal in N
3. W = N/T admits a set of generators S such that for all s∈S, w∈W,
(i) BsBwB ⊆ BwB∪BswB
(ii) sBs^{-1} is not contained in B
BN-pair常用在axiomise Bruhat decomposition,
它在finite group of Lie type以及algebraic group中都有用處.
只是, 在real reductive Lie group中找不到其用處?
real reductive Lie group的Bruhat decomposition最初是由Harish-Chandra證明,
比Chevalley提出algebraic group的版本, 以及Tits提出的BN-pair還要早,
但是, 我看到的教科書上都沒有用BN-pair的方法, 而是用Harish-Chandra的方法證.
請問, 是因為這在real analytic的情況不適用嗎?
此外, 我在Kenneth S. Brown, Buildings這本書上的Appendix中找到這段話(k:field) :
To get a BN-pair, in general, one has to forget about Borel subgroups and
instead take B to be the group P(k) for some minimal parabolic subgroup
P of G, where now “parabolic” is defined by the property that G/P is a
projective variety. B is again unique up to conjugacy. We can choose B to
contain T(k) where T is now a maximal k-split torus in G, and we take N to be
the normalizer of T(k) in G(k).
在real reductive的情況下, 假設G是real reductive, Lie G = k + p是Cartan decomp,
G = KAN是Iwasawa decomp.
minimal parabolic subgroup好像是對應到MAN=Q,
這邊的M= Z_K(a)是K (compact part of G)裡面對a = Lie A的centraliser,
然後a是p(noncompact part of Lie G)的一個maximal abelian subalgebra, N是拿a對p作
weight space decomposition, 得到的root中取一個positive system,對應的subalgebra
Q=MAN是Q的Iwasawa decomposition.
然後在real的情況下, BN-pair中的N(和上面MAN的N不同)取作N_K(a), a的normaliser
請問這樣有辦法滿足BN-pair的axioms嗎?
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◆ From: 140.112.6.72
※ 編輯: willydp 來自: 220.132.161.204 (06/16 19:37)
※ 編輯: willydp 來自: 220.132.161.204 (06/16 19:39)