Induction of nilpotent orbits for real reductive groups and associated varieties of standard representations
This paper deals with the compatibility of induction of representations and nilpotent orbits with respect to the associated variety. The work generalizes results of Lusztig and Spaltenstein (1979) for complex reductive groups and relies heavily on work of Adams and Vogan (1992). In a little more detail, let \(G\) be a complex connected reductive algebraic group with a real form defined by the complex conjugation \(\tau\). Let \(\theta\) be a commuting Cartan involution and \(\mathfrak g =\mathfrak k +\mathfrak s\) be the corresponding Cartan decomposition. The paper begins by parameterizing the \(\mathfrak g\)-principal nilpotent \(K\)-orbits in \(\mathfrak s\). Then for \(\theta\)-stable parabolic subgroups \(Q=LU\) with \(\theta\)-stable and \(\tau\)-stable Levi factor \(L\), an induction of nilpotent orbits is defined from nilpotent \(L\cap K\)-orbits in \(\mathfrak l \cap \mathfrak s\) to nilpotent \(K\)-orbits in \(\mathfrak s\). The nilpotent \(K\)-orbits induced from \(\mathfrak l\)-principal nilpotent \(K\)-orbits are described. Similarly, for \(\tau\)-stable parabolic subgroups \(P=MN\) with \(\theta\)-stable and \(\tau\)-stable \(M\), an induction of nilpotent orbits (via Kostant-Sekiguchi) is defined from nilpotent \(M\cap K\)-orbits in \(\mathfrak m \cap \mathfrak s\) to nilpotent \(K\)-orbits in \(\mathfrak s\). The nilpotent \(K\)-orbits induced from the zero orbit are described. These two forms of induction are used to describe the generic \(K\)-orbits in the associated varieties of certain standard \((\mathfrak g,K)\)-modules.
scientific article; zbMATH DE number 1347352