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Good index behaviour of \(\theta \)-representations. I. (English) Zbl 1259.14049

Let \(Q\) be an algebraic group over an algebraically closed field of characteristic zero. Let \(\mathcal{\mathfrak{q}}\) be its Lie algebra, and let \(V\) be a \(Q\)-module. Then the index of \(V\) is the non-negative integer \[ \text{ind}\left( \mathcal{\mathfrak{q}},V\right) :=\dim V-\max_{\xi\in V^{\ast} }\left( \dim\mathcal{\mathfrak{q}}\cdot\xi\right), \] where \(V^{\ast}\) is the linear dual of \(V.\) It is always that case that \[ \text{ ind}\left( \mathcal{\mathfrak{q}},V^{\ast}\right) \leq \text{ind}\left( \mathcal{\mathfrak{q} }_{v},\left( V/\mathcal{\mathfrak{q}}\cdot v\right) ^{\ast}\right) , \] where \[ \mathcal{\mathfrak{q}}_{v}=\left\{ x\in\mathcal{\mathfrak{q}\,}|\,x\cdot v=0\right\}. \] If we have equality, we say \(\left( Q,V\right) \) has good index behavior (GIB). In general, it is difficult to determine if a representation has GIB – to date, no general principle is known.
Let \(G\) be a connected reductive algebraic group and let \(\mathfrak{g} =\)Lie\(\left( G\right)\). An involution \(\theta\) of \(\mathfrak{g}\) of order \(m\) provides a \(\mathbb{Z}/m\mathbb{Z}\)-grading of \(\mathfrak{g}\) into \(\mathfrak{g}=\bigoplus_{i=0}^{m-1}\mathfrak{g}_{i},\) where \(\mathfrak{g}_{i}\) is the eigenspace of \(\theta\) corresponding to the eigenvalue \(\zeta^{i}\) for \(\zeta\) a fixed primitive \(m^{\text{th}}\) root of unity. For \(G_{0}\subset G\) a connected algebraic group with Lie algebra \(\mathfrak{g}_{0}\) one has a natural action of \(G_{0}\) on \(\mathfrak{g}_{1}.\)
This work is an investigation of GIB for \(\left( G_{0},\mathfrak{g} _{1}\right) ,\) and the author focuses on two different cases, both assuming that the rank of \(\left( G_{0},\mathfrak{g}_{1}\right) \) is nonzero. In the case where \(\mathfrak{g}\) is an exceptional Lie algebra, a series of tables is given describing the finite order involutions \(\theta\) such that \(\left( G_{0},\mathfrak{g}_{1}\right) \) has GIB; the information is displayed using Kac diagrams. The other case considered is where \(\mathfrak{g=gl}_{n}\) – in this case attention is restricted to the inner automorphisms of \(\mathfrak{g;} \) here \(\theta\) must be given by conjugation by certain diagonal matrices.

MSC:

14L30 Group actions on varieties or schemes (quotients)
17B20 Simple, semisimple, reductive (super)algebras
17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras

Software:

Magma; SLA; GAP
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Full Text: DOI arXiv

References:

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