Semistable locus of a group compactification
Let \(G\) be a connected, semi-simple algebraic group of adjoint type over an algebraically closed field. One can consider \(G\) as a \(G\times G\)-variety by left and right translation. Then there exists a canonical \(G\times G\)-equivariant embedding \(X\) of \(G\) which is called the wonderful compactification [\textit{C. De Concini} and \textit{C. Procesi}, Lect. Notes Math. 996, 1--44 (1983; Zbl 0581.14041)]. The variety \(X\) is an irreducible, smooth projective \((G\times G)\)-variety with finitely many \(G \times G\)-orbits indexed by the subsets of the set of simple roots of \(G\). The boundary \(X\setminus G\) is a union of smooth divisors with normal crossing.NEWLINENEWLINEThe classification of the orbits by the action on \(X\) of the diagonal copy of \(G\) in \(G\times G\) was obtained by \textit{G. Lusztig} [Mosc. Math. J. 4, No. 4, 869--896 (2004; Zbl 1103.20041)] in terms of \(G\)-stable pieces.NEWLINENEWLINEThe authors give an explicit description of the semi-stable locus of \(X\) (for the diagonal \(G\)-action) using Lusztig's \(G\)-stable pieces and calculate the geometric quotient \(X/\!/G\). More generally, they consider the case where the action is twisted by a diagram automorphism. Let \(\sigma\) be a diagram automorphism of \(G\), in particular an automorphism on \(G\) such that \(\sigma(B) = B\) and \(\sigma(T) = T\). The author consider the variety \(X_\sigma\) isomorphic \(X\), but with the action twisted by \((g_1,g_2)\rightarrow(g_1,\sigma(g_2))\).NEWLINENEWLINEIn [\textit{C. De Concini, S. Kannan} and \textit{A. Maffei}, Mosc. Math. J. 8, No. 4, 667--696 (2008; Zbl 1161.14036)] is described the semi-stable locus and geometric quotient for complete symmetric varieties (which includes as a special case the nontwisted conjugation action of \(G\) on its wonderful compactification), in terms of the \(G\)-invariants of the Cox ring.NEWLINENEWLINEIn this work it is used a different approach based on the geometry of the nilpotent cone of \(X_\sigma\). This allows the authors to give an explicit description of the semi-stable locus as a union of Lusztig's \(G\)-stable pieces. Such an explicit description plays an important role in the study of character sheaves on the wonderful compactification [\textit{X. He}, Adv. Math. 225, No. 6, 3258--3290 (2010; Zbl 1252.20046)]. The case of the twisted action on the wonderful compactification is not included in the work of De Concini, Kannan and Maffei.NEWLINENEWLINEReviewer's remark: The first author used in [Zbl 1252.20046] only the description of the semi-stable locus in the first part of this work. In the second part of this work the authors study the geometric quotient \(X_\sigma/\!/G\), using some results of the first author [loc. cit.], but no logical loops appear.
scientific article; zbMATH DE number 6043189
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