subregular nilpotent representations of lie algebras in prime ...
subregular nilpotent representations of lie algebras in prime ...
subregular nilpotent representations of lie algebras in prime ...
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32<br />
0 +<br />
(b ) = 0 and 0<br />
(x, 0)=0. S<strong>in</strong>ce there exists only one <strong>subregular</strong> <strong>nilpotent</strong><br />
orbit <strong>in</strong> g , there exists g 2 G with g = 0 .<br />
Let us use notations from B.13 like C0 for g = 0 . Apply<strong>in</strong>g the results so<br />
far to 0 and 0 (<strong>in</strong>stead <strong>of</strong> and )wesee that there is a simple U 0(g){module<br />
L1 <strong>in</strong> C0 <strong>of</strong> dimension equal to (p ,h + ; _ 0 i)pN ,1 and satisfy<strong>in</strong>g (T , ) 0L1 6= 0.<br />
Twist<strong>in</strong>g L1 with g ,1 we get a simple module L2 <strong>in</strong> C <strong>of</strong> the same dimension with<br />
T , L2 6= 0(byB.13(2)). Now C.4.a imp<strong>lie</strong>s that L2 ' L; the claim follows.<br />
Remarks: 1) The pro<strong>of</strong> <strong>in</strong> a2) shows that the claim holds also <strong>in</strong> many cases where<br />
R <strong>of</strong> type E8, F4, and G2 and p h +1. Onemay hope that it holds always <strong>in</strong><br />
good characteristic.<br />
2) We have here deduced b) from a). Note conversely: If we know that b) holds<br />
(for a speci c ), then also a) holds (for that ): We have T , Z (w ; ) '<br />
Z (, )forw as <strong>in</strong> a) [by B.11]. Therefore (and by C.4) the socle <strong>of</strong> Z ( )is<br />
a composition factor <strong>of</strong> Z (w ; ). Now a dimension comparison shows that<br />
Z (w ; ) is isomorphic to that socle, hence simple. The rema<strong>in</strong>der <strong>of</strong> a) follows<br />
now asabove.<br />
D.5. Given 2 C 0 0 write J( ) for the set <strong>of</strong> simple roots with 2 F (J( )). Note<br />
that each , + $ with 2 J( ) is <strong>in</strong> the closure <strong>of</strong> the facet <strong>of</strong> .Let 2 g be<br />
<strong>nilpotent</strong>. We associate to each module M <strong>in</strong> C an <strong>in</strong>variant by sett<strong>in</strong>g<br />
(M) =f 2 J( ) [f0g jT $ , (M) 6= 0g (1)<br />
where we use the convention $0 =0.<br />
This <strong>in</strong>variant will turn out to be useful <strong>in</strong> the case where is <strong>subregular</strong>. In<br />
the general case one should have a ner <strong>in</strong>variant that keeps track <strong>of</strong> the behaviour<br />
under all translations to the boundary <strong>of</strong> a given facet.<br />
There may exist weights ; 0 2 C 0 0 with 6= 0 and C = C 0. If we replace<br />
by 0 <strong>in</strong> (1), then we will <strong>in</strong> general get di erent results. So depends not just<br />
on the category C , but also on the choice <strong>of</strong> . It might therefore be better to<br />
denote this map by . However, usually we x and then no problems should<br />
arise.<br />
Suppose that (b + ) = 0. Then C.9(1) and C.10(1) show that<br />
sbm(w; )=J( ) [f0g for all w 2 W . (2)<br />
In particular, the socle L = sbm(w0; )<strong>of</strong>Z ( ) satis es (L) = J( ) [f0g.<br />
Proposition C.4 says that this is the only simple module <strong>in</strong> C with 0 2 (L).<br />
Let g 2 G. Given M <strong>in</strong> C we have g M <strong>in</strong> C(g ) , see B.13. We get now<br />
from B.13(2).<br />
( g M)= (M) (3)