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Complements to ‘The Bruhat order on symmetric varieties’. (English) Zbl 0826.20045

Summary: We give several complements to the paper mentioned in the title [ibid. 35, No. 1-3, 389-436 (1990; Zbl 0704.20039)]. Our main result shows that the partial order on the set \(\mathcal I\) of twisted involutions in the Weyl group \(W\), which was introduced in the earlier paper, agrees with the partial order on \(\mathcal I\) induced by the usual Bruhat order on \(W\).

MSC:

20G15 Linear algebraic groups over arbitrary fields
20G05 Representation theory for linear algebraic groups
14L30 Group actions on varieties or schemes (quotients)

Citations:

Zbl 0704.20039
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References:

[1] Borel, A.,Linear Algebraic Groups (2nd edn) Springer-Verlag, New York, Berlin, Heidelberg, 1991. · Zbl 0726.20030
[2] Borel, A. and Tits, J., ’Groupes réductifs’,Inst. Hautes Études Sci. Publ. Math. 27 (1965), 55–150. · Zbl 0145.17402 · doi:10.1007/BF02684375
[3] Deodhar, V. V., ’Some characterizations of Bruhat ordering on a Coxeter group and determination of the relative Möbius function’,Invent. Math. 39 (1977), 187–198. · Zbl 0346.20032 · doi:10.1007/BF01390109
[4] Richardson, R. W. and Springer, T. A., ’The Bruhat order on symmetric varieties’,Geom. Dedicata 35 (1990), 389–436. · Zbl 0704.20039 · doi:10.1007/BF00147354
[5] Springer, T. A., ’Some results on algebraic groups with involutions’,Advanced Studies in Pure Math., Vol. 6,Algebraic Groups and Related Topics, Kinokuniya/North Holland, 1985, pp. 525–543.
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