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Transformation Groups

, Volume 18, Issue 2, pp 507–537 | Cite as

Commuting involutions and degenerations of isotropy representations

  • Dmitri I. Panyushev
Article

Let σ 1 and σ 2 be commuting involutions of a semisimple algebraic group G. This yields a \( {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} \)-grading of \( \mathfrak{g} \) = Lie(G), \( \mathfrak{g}={\oplus_{i,j=0,1 }}{{\mathfrak{g}}_{ij }} \), and we study invariant-theoretic aspects of this decomposition. Let \( \mathfrak{g}\left\langle {{\sigma_1}} \right\rangle \) be the \( {{\mathbb{Z}}_2} \)-contraction of \( \mathfrak{g} \) determined by σ 1. Then both σ 2 and σ 3 := σ 1 σ 2 remain involutions of the non-reductive Lie algebra \( \mathfrak{g}\left\langle {{\sigma_1}} \right\rangle \). The isotropy representations related to \( \left( {\mathfrak{g}\left\langle {{\sigma_1}} \right\rangle, {\sigma_2}} \right) \) and \( \left( {\mathfrak{g}\left\langle {{\sigma_1}} \right\rangle, {\sigma_3}} \right) \) are degenerations of the isotropy representations related to \( \left( {\mathfrak{g},{\sigma_2}} \right) \) and \( \left( {\mathfrak{g},{\sigma_3}} \right) \), respectively. These degenerated isotropy representations retain many good properties. For instance, they always have a generic stabiliser and their algebras of invariants are often polynomial. We also develop some theory on Cartan subspaces for various \( {{\mathbb{Z}}_2} \)-gradings associated with the \( {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} \)-grading of \( \mathfrak{g} \) and study the special case in which σ 1 and σ 2 are conjugate.

Keywords

Symmetric Space Conjugacy Class Algebraic Group Maximal Rank Cartan Subalgebra 
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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institute for Information Transmission ProblemsRASMoscowRussia

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