Standard monomials for wonderful group compactifications. (English) Zbl 1120.20051
The author extends standard monomial theory to the wonderful compactification \(X\) of a semisimple group \(G\) of adjoint type. Recall that R. Chirivì and A. Maffei have already constructed standard monomials for the more general situation of a wonderful compactification of a symmetric space [J. Algebra 261, No. 2, 310-326 (2003; Zbl 1055.14052)]. We fix a dominant weight \(\lambda\) and the corresponding line bundle \(\mathcal L_\lambda\) on \(X\). Then Chirivì and Maffei provide a basis of \(H^0(X,\mathcal L_\lambda)\) consisting of ‘standard monomials’ with certain properties.
The author shows that in the present situation one can do more. First of all, the standard monomials are shown to behave well with respect to restriction to \(B\times B\)-orbit closures, not just \(G\times G\)-orbit closures. Recall that there are finitely many \(B\times B\)-orbits and that they have been classified by T. A. Springer [J. Algebra 258, No. 1, 71-111 (2002; Zbl 1110.14047)]. There are degrees of freedom in the construction of Chirivì, Maffei and these the author exploits to arrange more properties familiar from the classical standard monomial theory for flag varieties.
The basis of \(H^0(X,\mathcal L_\lambda)\) is indexed by LS-paths again. And if \(Z\) is a \(G\times G\)-orbit closure, or more generally a \(B\times B\)-orbit closure, then the standard monomials that do not vanish on \(Z\) form a basis of \(H^0(Z,\mathcal L_\lambda)\). They can be characterized combinatorially. The many ingredients that are needed in the proof are explained clearly.
The author shows that in the present situation one can do more. First of all, the standard monomials are shown to behave well with respect to restriction to \(B\times B\)-orbit closures, not just \(G\times G\)-orbit closures. Recall that there are finitely many \(B\times B\)-orbits and that they have been classified by T. A. Springer [J. Algebra 258, No. 1, 71-111 (2002; Zbl 1110.14047)]. There are degrees of freedom in the construction of Chirivì, Maffei and these the author exploits to arrange more properties familiar from the classical standard monomial theory for flag varieties.
The basis of \(H^0(X,\mathcal L_\lambda)\) is indexed by LS-paths again. And if \(Z\) is a \(G\times G\)-orbit closure, or more generally a \(B\times B\)-orbit closure, then the standard monomials that do not vanish on \(Z\) form a basis of \(H^0(Z,\mathcal L_\lambda)\). They can be characterized combinatorially. The many ingredients that are needed in the proof are explained clearly.
Reviewer: Wilberd van der Kallen (Utrecht)
MSC:
| 20G10 | Cohomology theory for linear algebraic groups |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 14M17 | Homogeneous spaces and generalizations |
Keywords:
standard monomials; group compactifications; wonderful compactifications; semisimple groups of adjoint type; orbit closuresReferences:
| [1] | K. Appel, Standardmonome für wundervolle Kompaktifizierungen von Gruppen, PhD thesis, Wuppertal, 2006; K. Appel, Standardmonome für wundervolle Kompaktifizierungen von Gruppen, PhD thesis, Wuppertal, 2006 · Zbl 1196.14001 |
| [2] | Brion, M., The behaviour at infinity of the Bruhat decomposition, Comment. Math. Helv., 73, 1, 137-174 (1998) · Zbl 0935.14029 |
| [3] | Brion, M.; Kumar, S., Frobenius Splitting Methods in Geometry and Representation Theory, Progr. Math., vol. 231 (2005), Birkhäuser: Birkhäuser Boston · Zbl 1072.14066 |
| [4] | Brion, M.; Polo, P., Large Schubert varieties, Represent. Theory, 4, 97-126 (2000) · Zbl 0947.14026 |
| [5] | Chirivi‘, R.; Maffei, A., The ring of sections of a complete symmetric variety, J. Algebra, 261, 2, 310-326 (2003) · Zbl 1055.14052 |
| [6] | De Concini, C.; Procesi, C., Complete symmetric varieties, (Invariant Theory. Invariant Theory, Lecture Notes in Math., vol. 996 (1983), Springer), 1-44 · Zbl 0581.14041 |
| [7] | De Concini, C.; Springer, T. A., Compactification of symmetric varieties, Transform. Groups, 4, 2-3, 273-300 (1999) · Zbl 0966.14035 |
| [8] | Dabrowski, R., A simple proof of a necessary and sufficient condition for the existence of nontrivial global sections of a line bundle on a Schubert variety, (Kazhdan-Lusztig Theory and Related Topics. Kazhdan-Lusztig Theory and Related Topics, Contemp. Math., vol. 139 (1992), Amer. Math. Soc.), 113-120 · Zbl 0797.20036 |
| [9] | He, X.; Thomsen, J. F., Geometry of \(B \times B\)-orbit closures in equivariant embeddings, preprint, available at arXiv: · Zbl 1133.14053 |
| [10] | Littelmann, P., Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras, J. Amer. Math. Soc., 11, 3, 551-567 (1998) · Zbl 0915.20022 |
| [11] | Ramanathan, A., Equations defining Schubert varieties and Frobenius splitting of diagonals, Publ. Math. Inst. Hautes Études Sci., 65, 61-90 (1987) · Zbl 0634.14035 |
| [12] | Springer, T. A., Intersection cohomology of \(B \times B\)-orbit closures in group compactifications, J. Algebra, 258, 71-111 (2002) · Zbl 1110.14047 |
| [13] | Strickland, E., A vanishing theorem for group compactifications, Math. Ann., 277, 165-171 (1987) · Zbl 0595.14037 |
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