Chun-Ju Lai

與王偉強有約 – (6) Representations of Lie Algebras in Prime Characteristic

In Lie Algebra, Math, Representation on 2011/11/10 at 5:03 PM

Representations of Lie Algebras in Prime Characteristic,
Jens Carsten Janzten

7. Premet Theorem

現在 g:reductive, \chi:nilpotent, 要用 Premet’s Theorem. 若 m{\leq} g 為 unipotent subalgebra 滿足
a) m 與 centralizer {c_g(\chi)} 交集為空
b) {\chi([m,m])} = 0 c) {\chi(m^{[p]})} = 0
則每個 {U_\chi(g)}-mod 都 free over {U_\chi(m)}

來證明 [Kac-Weisfeiler Conjecture]

{U_\chi(g)}-mod M 來說 {p^{\dim G.\chi /2}|\dim M}

(proof)

(A) 我們只要建構一個可套用 Premet Theorem 的 subalgbra m, 滿足 dim(m) = dim {G.\chi/2}

便知道 M is free over {U_\chi(m)}, which is of dim. {p^{\dim(m)}}, 就得證.

現在 {\chi} is nilpoteny, 由 G is good, 能找到對應的 e in g 使得 {\chi(x)=(e,x)} for some G-invariant non-deg bilinear form (-,-)

(B) 我們宣稱可以找到一組 g 的 {\mathbb{Z}}-grading as restricted Lie algebra

{g=\bigoplus_{i\in\mathbb{Z}}g(i)} 滿足:

(1) e in g(2), {c_g(e)} in {\bigoplus_{i\geq0}g(i)}
(2) dim {C_G(e)} = dim g(0) + dim g(1)
(3) dim g(i) = dim g(-i)

若如此, 則定 m = {\left( \bigoplus_{i\geq-1} g(-i)\right)\oplus g^1(-1)} 其中 {g^1(-1)} 為 maximal isotropic subalgebra wrt <-,-> 注意 dim {g^1(-1)} = dim g(-1)/2 所以我們便有

{\dim G.e = \dim g - \dim C_G(e)} = {\sum_{i\neq0,1} \dim g(i) = \dim g(-1) + 2\sum_{i\geq2} \dim g(i) }

因此 dim m = dim G.e/2

(C) 回來看這個 grading, 在 p 夠好的時候 我們有 Jacobson-Morozov 定理, 即對 nilpotent elt e, 都找得到 f,h in g 使得 同構於 {sl_2}

便可定  { g(i) :=\{x\in g;[h,x]=ix\}}

(D) 在一般的 p, 這種 eigenspaces 無法將它們好好 “分開” 我們須藉由特別的 1-parameter group {\psi}: {G_m\rightarrow G} 和 adjoint action, 將 eignevalue 轉換成 多項式才行, 即

{g(i):= \{ x\in g;\text{Ad}(\psi(t))(x)=t^ix\text{ for all }t\in K\backslash0\}}

這即是原本版本的 “integration” 將 {\psi(t)} 視為 {t^h}, 將 LHS 對 t “微分”便得 {(t^h x-xt^h)'=t^{h-1}(hx -xh)} 將 RHS 對 t “微分”便得 {(t^i x)'=t^{-1}ix} 代 t = 1 便得 [h,x] = ix

(E) 而此 1-parameter group 便來自 Bala-Carter parametrization of nilpotent orbits

9. Centers

(A) 在 char 0 的情況, 我們證明 Z(g) 即是 G-invariant part {U(g)^G} 再證明在 W-dot action 底下的 invariant part {S(H)^W}{U(g)^G} 同構 便可得到 Harish-Chandra 定理

{Z(g)\rightarrow S(H)^W} acted by projection 是個 algebra isomorphism

在 modular case, Z(g) 不再等於 {U(g)^G}, Kac-Weilsfeiler 證明了推廣的版本:

{U(g)^G\rightarrow S(H)^W} acted by projection 是個 algebra isomorphism

(B) As a corollary, 我們可定 “central” character cen{\lambda}: {U(g)^G\rightarrow K} 記錄 elt u 在 {Z_\chi(\lambda)} 上作用的倍率 便可證

cen{\lambda} = cen{\mu} <=> {\lambda}{\mu} 落在同一個 W-dot orbit

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