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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 $${\mathfrak g}_{1}={\mathfrak so} (n+1,n-1)$$.

##### MSC:
 17B45 Lie algebras of linear algebraic groups 22E46 Semisimple Lie groups and their representations
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##### References:
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