Compactifications of reductive groups as moduli stacks of bundles (Q2787620)

For a split connected reductive group \(G\) there is a well-studied notion of toroidal compactifications of \(G\), see e.g.\ [\textit{D. A. Timashev}, Sb. Math. 194, No. 4, 589--616 (2003; Zbl 1074.14043); translation from Mat. Sb. 194, No. 4, 119--146 (2003)]. A new viewpoint on these compactifications via moduli theory is presented in the paper under review.NEWLINENEWLINEMore precisely, a \textit{chain of framed bundles} is defined as a \(\mathbb{G}_m\)-equivariant principal \(G\)-bundle over a chain of \(n+1\) projective lines equipped with the standard \(\mathbb{G}_m\)-action such that the intersection points of the lines are fixed and the fibers at the two remaining fixed points \(p_{\pm}\) are \(\mathbb{G}_m\)-invariantly trivialized (i.e., identified with \(G\)). Such a framed bundle chain is determined up to an isomorphism by a collection \((\beta_1,\dots,\beta_n)\in\Lambda^n\), where \(\Lambda\) is the cocharacter lattice of a split maximal torus \(T\subset G\) and \(\beta_i\) define the \(\mathbb{G}_m\)-action in the fibers over the intersection points. The sequence \((\beta_1,\dots,\beta_n)\in\Lambda^n\) is determined up to the action of the Weyl group \(W=N_G(T)/Z_G(T)\). A notion of stability for chains of framed bundles is defined by a choice of a simplicial fan \(\Sigma\) supported in the positive Weyl chamber \(\Lambda^+_{\mathbb Q}\) and of nonzero lattice vectors \(\lambda_1,\dots,\lambda_N\) on the rays of \(\Sigma\): a framed bundle chain is \textit{\(\Sigma\)-stable} if, up to the \(W\)-action, \(\beta_1,\dots,\beta_n\) is a subsequence of \(\lambda_1,\dots,\lambda_N\) (in the same order) corresponding to the rays of some \(n\)-dimensional cone \(\sigma\in\Sigma\).NEWLINENEWLINEIt is proved that \(\Sigma\)-stable framed bundle chains are represented by a smooth separated tame Artin stack of finite presentation \({\mathcal M}_G(\Sigma)\), which is proper if and only if \(\Sigma\) covers the whole \(\Lambda^+_{\mathbb Q}\). The coarse moduli space \(M_G(\Sigma)\) of \({\mathcal M}_G(\Sigma)\) is the toroidal spherical \((G\times G)\)-equivariant embedding of \(G\) corresponding to the uncolored fan \(w_0\Sigma\), where \(w_0\) is the longest element in \(W\). Each toroidal embedding of \(G\) with quotient singularities, in particular, each toric orbifold (for \(G=T\)) arises in this way. An alternative construction of \({\mathcal M}_G(\Sigma)\) as a global quotient by a torus action of an open subset in a reductive algebraic monoid generalizing \textit{E. B. Vinberg}'s enveloping semigroup [in: Lie groups and Lie algebras: E. B. Dynkin's seminar. Providence, RI: American Mathematical Society. 145--182 (1995; Zbl 0840.20041)] is considered. For \(G=\mathrm{PGL}_n\) and \(\Sigma\) the fan with the single maximal cone \(\Lambda^+_{\mathbb Q}\) and with the primitive lattice vectors (fundamental coweigths) on the rays, the closure of \(T\) in \({\mathcal M}_G(\Sigma)\) is the Losev-Manin moduli space of genus zero weighted pointed stable curves [\textit{A. Losev} and \textit{Y. Manin}, Mich. Math. J. 48, 443--472 (2000; Zbl 1078.14536)]. The respective functor from weighted pointed stable curves to framed bundle chains is described. A relation with \textit{I. Kausz}'s compactification of \(\mathrm{GL}_n\) [Doc. Math. 5, 553--594 (2000; Zbl 0971.14029)] is discussed.