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On semiregular nilpotent orbits in simple Lie algebras of type \(D\). (English) Zbl 1023.17011
Let \(G\) be a connected almost simple Lie group over \(\mathbb{C}\) of rank \(n\) with Lie algebra \({\mathfrak g}.\) A nilpotent element \(x\in {\mathfrak g}\) is called semiregular if \(G\cdot x\cap {\mathfrak a}=\emptyset \) for all proper regular subalgebras \({\mathfrak a}\) of \({\mathfrak g.}\) Now suppose \( G=SO_{2n}(\mathbb{C}) \). Then the semiregular nilpotent \(G\)-orbits are denoted by \(D_{n}(a_{k}) \) with \(k\) a nonnegative integer and \(k\leq n/2-1.\)
In this paper, the characteristics \(H^{(k) }\) of such \( D_{n}(a_{k}) \) are considered. If \(H_{\alpha }\) is the coroot corresponding to the root \(\alpha ,\) and \(\{ \alpha _{1},\ldots ,\alpha _{n}\} \) if a base of the root system of \(({\mathfrak g,h}) \) for \({\mathfrak h}\) a Cartan subalgebra of \({\mathfrak g}\), then \(H^{(k) }\) is realized as a linear combination of the \(H_{\alpha _{i}},\) and explicit formulas for the coefficients to the \(H_{\alpha _{i}}\) are given, depending on \(i,k,\) and \(n.\)
Given a nontrivial nilpotent \(G\)-orbit \(\mathcal{O}\subset G\) it is well-known that there exists a standard triple \((E,H,F) \) with \( E,F\in \mathcal{O}, H\in {\mathfrak h},\) and \(0\leq \alpha _{i}(H) \leq 2\) for all \(\alpha _{i}.\) Here for each \(k\) two standard triples \((E,H,F) \) are constructed where \(H=H^{(k) }\) and \(E,F\in D_{n}(a_{k}) .\) One of the types of standard triples constructed have \(E\) and \(F\) integral linear combinations of Chevalley root vectors. The other type satisfies \(F=\theta (E) ,\) where \(\theta :{\mathfrak g\to g}\) is the involution \(\theta (x)=- ^{t}x.\)
Let \({\mathfrak g=so}(n,n) \) be the split real form of \({\mathfrak g}\) defined by the Chevalley system of \({\mathfrak g}\) and \(\theta _{0}\) (i.e. \(\theta \) on \({\mathfrak g}_{0}\)). If \(G_{0}\) is the Lie subgroup corresponding to \({\mathfrak g}_{0},\) then \({\mathfrak g}_{0}\cap D_{n}(a_{k})\) is a union of two or four \(G_{0}\)-orbits. The authors conclude by constructing representatives \(E\) for each \(G_{0}\)-orbit and give explicit \( {\mathfrak g}_{0}\)-standard triples \((E,H,F)\). Finally, a similar result is given for the real form \({\mathfrak g}_{1}={\mathfrak so} (n+1,n-1)\).

17B45 Lie algebras of linear algebraic groups
22E46 Semisimple Lie groups and their representations
Full Text: DOI
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