Visible actions on flag varieties of exceptional groups and a generalization of the Cartan decomposition. (English) Zbl 1305.22013
Let \( G \) be a compact semisimple group and \( \sigma \) be a Chevalley-Weyl involution (this terminology seems not so popular; it is an involution which acts as \( (-1) \) on a Cartan subalgebra of \( \mathfrak{g} = \mathrm{Lie}(G) \)). The author classifies a pair \( (L, H) \) of Levi subgroups for which \( G = L G^{\sigma} H \) holds when \( G \) is of exceptional type. The author has already classified such pairs for classical groups of types B, C and D [Y. Tanaka, Bull. Aust. Math. Soc. 88, No. 1, 81–97 (2013; Zbl 1347.22012)]; [J. Math. Soc. Japan 65, No. 3, 931–965 (2013; Zbl 1296.22015)]; [Tohoku Math. J. (2) 65, No. 2, 281–295 (2013; Zbl 1277.22014)]. For type A, Kobayashi classified such pairs [T. Kobayashi, Transform. Groups 12, No. 4, 671–694 (2007; Zbl 1147.53041)], and this completes the classification. The author, as well as Kobayashi, is interested in such pairs because of the investigation of visible actions.
If \( L, H \) are both symmetric subgroups (i.e., its connected component is fixed by a certain involution), such classification goes down to a result of Matsuki, which generalizes the classical Cartan decomposition [T. Matsuki, J. Algebra 197, No. 1, 49–91, Art. No. JA977123 (1997; Zbl 0887.22009)]. In the exceptional case, there are a few cases which do not fall into Matsuki’s setting.
According to the classification given in Theorem 1.1, there are no such pairs except for the types \( E_6 \) and \( E_7 \) among the simple exceptional groups. When \( G \) is of type \( E_6 \) or \( E_7 \), one of the Levi subgroups \( L = G^{\tau} \) is always a symmetric subgroup (up to a switch of the position) and \( H \) is a subgroup of the symmetric subgroup \( G^{\tau'} \), where \( \tau \) and \( \tau' \) are involutions. Notably, the semisimple part of the subgroup \( H \) seems always to be of maximal rank in the semisimple part of \( G^{\tau'} \).
The proof uses Matsuki’s result first for the pair \( (G^{\tau}, G^{\tau'}) \), and then turns to decompose \( G^{\tau'} \) further. To prove the exhaustion of the classification, the author uses representation theory; namely the classification of multiplicity free decompositions of tensor product representations by P. Littelmann [J. Algebra 166, No. 1, 142–157 (1994; Zbl 0823.20040)] and J. R. Stembridge [Represent. Theory 7, 404–439 (2003; Zbl 1060.17001)] is a key.
If \( L, H \) are both symmetric subgroups (i.e., its connected component is fixed by a certain involution), such classification goes down to a result of Matsuki, which generalizes the classical Cartan decomposition [T. Matsuki, J. Algebra 197, No. 1, 49–91, Art. No. JA977123 (1997; Zbl 0887.22009)]. In the exceptional case, there are a few cases which do not fall into Matsuki’s setting.
According to the classification given in Theorem 1.1, there are no such pairs except for the types \( E_6 \) and \( E_7 \) among the simple exceptional groups. When \( G \) is of type \( E_6 \) or \( E_7 \), one of the Levi subgroups \( L = G^{\tau} \) is always a symmetric subgroup (up to a switch of the position) and \( H \) is a subgroup of the symmetric subgroup \( G^{\tau'} \), where \( \tau \) and \( \tau' \) are involutions. Notably, the semisimple part of the subgroup \( H \) seems always to be of maximal rank in the semisimple part of \( G^{\tau'} \).
The proof uses Matsuki’s result first for the pair \( (G^{\tau}, G^{\tau'}) \), and then turns to decompose \( G^{\tau'} \) further. To prove the exhaustion of the classification, the author uses representation theory; namely the classification of multiplicity free decompositions of tensor product representations by P. Littelmann [J. Algebra 166, No. 1, 142–157 (1994; Zbl 0823.20040)] and J. R. Stembridge [Represent. Theory 7, 404–439 (2003; Zbl 1060.17001)] is a key.
Reviewer: Kyo Nishiyama (Aoyama)
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 53C30 | Differential geometry of homogeneous manifolds |
| 22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |
| 32A37 | Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) |
Keywords:
Cartan decomposition; visible action; spherical action; symmetric space; exceptional Lie groupCitations:
Zbl 1296.22015; Zbl 1277.22014; Zbl 1147.53041; Zbl 0887.22009; Zbl 0823.20040; Zbl 1060.17001; Zbl 1347.22012References:
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