On porcupine varieties in Lie algebras

405

In particular, every reductively embedded abelian subalgebra of q is contained in a Cartan subalgebra of q

1 4 Remark Suppose that qi is a subalgebra of a Lie algebra g We shall say that a subalgebra a of qi is q-reductwe if it is reductively embedded in q (not merely m

qi )

( i ) Every q-reductive abelian subalgebra a of qj is contained in a maximal q-reductive abelian subalgebra of qi

( u ) If a IS a maximal q-reductive abelian subalgebra of qi then so is (p{a) for every automorphism of q leaving qi invariant

( ill ) Every q-reductive abelian subalgebra of qj is contained in the center of a Cartan subalgebra of qi Conversely, every Cartan subalgebra of qi contains a unique maximal q-reductive abelian subalgebra, which lies in its center

( iv ) The intersection of all maximal q-reductive abelian subalgebras of q is the center of q

( v ) The set of all maximal q-reductive abelian subalgebras of qi is a finite union of conjugacy classes If all Cartan subalgebras of qi are conjugate then so are all maximal q-reductive abelian subalgebras

( vi ) All maximal q-reductive abelian subalgebras of qi are isomorphic

Proof (i) and (u) are straightforward

( ill ) Let a be a q-reductive subalgebra in qj Then a is reductive also m qi, hence by 1 3 is contained in the center of a Cartan subalgebra of qi The second assertion follows from the fact that two commuting reductively embedded subalgebras of \) have a reductively embedded sum

( iv ) The center of q is reductively embedded, hence contained in every maximal abelian q-reductive subalgebra Conversely let x be an element which is contained in every maximal q-reductive subalgebra Then x commutes with every element of each Cartan subalgebra by (iii) However, the Cartan subalgebras of g generate g

( v ) Follows from (iii) and the fact that Щцг) is a union of finitely many conjugacy classes

( vi ) The assertion is true if the ground field is С Let о be a maximal q-reductive abehan subalgebra of qi Then t)<cÇ(gi)(i: is gcc-reductive and abelian and is, therefore, contained in a maximal q^-reductive abelian subalgebra m of (gi)^ We claim that \)(£ = m once this claim is established then dim^o = dimc^m Since all maximal q(c-reductive abelian subalgebras of (qi)(n are conjugate by (v) (since in a complex Lie algebra all Cartan subalgebras are conjugate), it follows that they have the same dimension and the assertion follows

Since the abelian algebra о is reductively embedded in g i it is contained in the center of a Cartan algebra \) of gi Then l)^^(o. Qi) and thus ï)(i:^5(t>c^ (9ik) Also mc^(t)^^ q^j,) since m is abelian Since m is abehan and q^-reductively embedded, there is a Cartan algebra f of (qi)c containing m Since f and l)^^ are conjugate in (qi)(C, then a conjugate n of m in {c\i)(^ contains v^^ and is contained in i)(^ We may cissume without losing generality that n = m

Now let к q(c-> q^ be the real involution given by к(х + ï-y) = x — /«у for x,yGg Then ?c(l)(c) = Ьс whence к{^(1)^)) = ^(i)^) Thus mÇ5(l)(c) implies к(т) -5(Ьс) Therefore m -f к(т) is an abehan subalgebra of (q^c containing oc Moreover, fc(m) -h m is also qc-reductive for mem and m ек{т) the operators ad m and adm' are semisimple and commute, so their sum ad(m + m) = ad m + ad m' is semisimple By the maximality of m we conclude m + к(т) Ç m