1 Introduction↩︎

The closure \(\bar H\) of a maximal torus \(H\) of a semisimple complex group \(G\) of adjoint type in its wonderful compactification \(X\) was studied by De Concini and Procesi [1]. It is a smooth toric variety with fan given by the Weyl chamber decomposition, and this property is closely related to the smoothness of the wonderful compactification. In a series of papers [2], [3], the first author and Lu embedded \(X\) inside the variety of Lagrangian subalgebras of \(\mathop{\mathrm{\mathfrak g}}\oplus \mathop{\mathrm{\mathfrak g}}\), where \(\mathop{\mathrm{\mathfrak g}}\) is the Lie algebra of \(G\) and used the smoothness of \(X\) to prove that all irreducible components of the variety of Lagrangian subalgebras are smooth. Motivations from Poisson geometry suggest that one should also study the variety \(\mathop{\mathrm{\mathcal{L}}}\) of Lagrangian subalgebras in a degeneration of \(\mathop{\mathrm{\mathfrak g}}\oplus \mathop{\mathrm{\mathfrak g}}\) given by the semidirect sum of \(\mathop{\mathrm{\mathfrak g}}\) with its coadjoint representation \(\mathop{\mathrm{\mathfrak g^{\ast}}}.\) It is elementary to observe that the variety \(\mathop{\mathrm{\mathcal{L}}}\) contains a subvariety \(\bar {\mathop{\mathrm{\mathfrak g}}}^{\ast}\) given by the closure of \(\mathop{\mathrm{\mathfrak g^{\ast}}}.\) While we are currently unable to understand the geometry of \(\bar {\mathop{\mathrm{\mathfrak g}}}^{\ast}\), in this paper we obtain a rather complete understanding of the closure \(\bar \mathop{\mathrm{\mathfrak h}}\) of a Cartan subalgebra \(\mathop{\mathrm{\mathfrak h}}\) of \(\mathop{\mathrm{\mathfrak g}}\) embedded into \(\mathop{\mathrm{\mathcal{L}}}.\) We view \(\bar \mathop{\mathrm{\mathfrak h}}\) as the natural additive analogue of \(\bar H\), and its study leads in a natural way to the theory of additive toric varieties [4]. In support of this perspective, in this paper we prove that \(\bar \mathop{\mathrm{\mathfrak h}}\) is a normal variety with finitely many \(\mathop{\mathrm{\mathfrak h}}\)-orbits, which gives an affine paving of \(\bar \mathop{\mathrm{\mathfrak h}}\), and determine a number of invariants of the topology of \(\bar \mathop{\mathrm{\mathfrak h}}.\)

In more detail, following ideas from [2], [3], consider the Lie algebra \(\mathop{\mathrm{\mathfrak d}}:= \mathop{\mathrm{\mathfrak g}}\ltimes \mathop{\mathrm{\mathfrak g^{\ast}}}\), where \(\mathop{\mathrm{\mathfrak g}}\) acts on its dual space \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) by the coadjoint action, and the factor \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) is regarded as an abelian Lie algebra. The Lie algebra \(\mathop{\mathrm{\mathfrak d}}\) comes equipped with a symmetric nondegenerate invariant bilinear form \((~,~)\) defined by \[((x, \phi), (y, \psi)) := \phi(y) + \psi(x),\] for all \(x, y \in \mathop{\mathrm{\mathfrak g}}\) and \(\phi, \psi \in \mathop{\mathrm{\mathfrak g^{\ast}}}.\) We call a Lie subalgebra \(\mathop{\mathrm{\mathfrak l}}\) of \(\mathop{\mathrm{\mathfrak d}}\) Lagrangian if \(\mathop{\mathrm{\mathfrak l}}\) is a maximal isotropic subspace of \(\mathop{\mathrm{\mathfrak d}}\) relative to the above symmetric form, and a triple \((\mathop{\mathrm{\mathfrak d}}, \mathop{\mathrm{\mathfrak l}}_1, \mathop{\mathrm{\mathfrak l}}_2)\) is called a Manin triple if \(\mathop{\mathrm{\mathfrak l}}_1\) and \(\mathop{\mathrm{\mathfrak l}}_2\) are transversal Lagrangian subalgebras of \(\mathop{\mathrm{\mathfrak d}}.\) Recall that Manin triples as above are closely related to Poisson Lie group structures on a connected group \(L_1\) with Lie algebra \(\mathop{\mathrm{\mathfrak l}}_1.\) The set \(\mathop{\mathrm{\mathcal{L}}}\) of Lagrangian subalgebras of \(\mathop{\mathrm{\mathfrak d}}\) has the structure of a (reduced) closed subvariety of the Grassmannian \(\mathop{\mathrm{Gr}}(\dim \mathop{\mathrm{\mathfrak g}}, \mathop{\mathrm{\mathfrak d}})\) of vector subspaces of \(\mathop{\mathrm{\mathfrak d}}\) with the same dimension as \(\mathop{\mathrm{\mathfrak g}}.\) The variety \(\mathop{\mathrm{\mathcal{L}}}\) is referred to as the variety of Lagrangian subalgebras of \(\mathop{\mathrm{\mathfrak d}}.\) The paper [3] gives a detailed understanding of the variety \(\mathop{\mathrm{\mathcal{L}}}(\mathop{\mathrm{\mathfrak g}}\oplus \mathop{\mathrm{\mathfrak g}})\) of Lagrangian subalgebras of \(\mathop{\mathrm{\mathfrak g}}\oplus \mathop{\mathrm{\mathfrak g}}\), which is a natural setting to study families of Poisson homogeneous spaces for the standard Poisson Lie group structure on \(G.\) The variety \(\mathop{\mathrm{\mathcal{L}}}\) is in a natural sense a degeneration of \(\mathop{\mathrm{\mathcal{L}}}(\mathop{\mathrm{\mathfrak g}}\oplus\mathop{\mathrm{\mathfrak g}})\), but the geometry of \(\mathop{\mathrm{\mathcal{L}}}\) seems to be much more difficult to understand. In a recent paper [5], the authors found an unexpected connection between \(\mathop{\mathrm{\mathcal{L}}}\) and the abelian ideals of a Borel subalgebra of \(\mathop{\mathrm{\mathfrak g}}\), and it would be of interest to relate the results of [5] to the present paper.

We may embed \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) into \(\mathop{\mathrm{\mathcal{L}}}\) via the embedding \[\phi \longmapsto \mathop{\mathrm{\mathfrak l}}_{\phi}:=\{(x, - \mathop{\mathrm{ad}}_x^*(\phi)): x \in \mathop{\mathrm{\mathfrak g}}\},\] and we refer to the closure \(\bar {\mathop{\mathrm{\mathfrak g}}}^{\ast}\) of \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) in \(\mathop{\mathrm{\mathcal{L}}}\) as the wonderful compactification of \(\mathop{\mathrm{\mathfrak g^{\ast}}}.\) Embed \(\mathop{\mathrm{\mathfrak h}}\) into \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) via the Killing form. Then the composition \[\mathop{\mathrm{\mathfrak h}}\lhook\joinrel\longrightarrow \mathop{\mathrm{\mathfrak g^{\ast}}}\lhook\joinrel\longrightarrow \mathop{\mathrm{\mathcal{L}}}\] embeds \(\mathop{\mathrm{\mathfrak h}}\) into \(\mathop{\mathrm{\mathcal{L}}}.\) The closure \(\bar \mathop{\mathrm{\mathfrak h}}\) of the image of this embedding in \(\mathop{\mathrm{\mathcal{L}}}\) is called the wonderful compactification of \(\mathop{\mathrm{\mathfrak h}}\) and is the main object of study of this paper. As a consequence of our construction, we will see (Proposition 6) that \(\bar \mathop{\mathrm{\mathfrak h}}\) is a special case of the matroid Schubert varieties studied in [6] and [7].

Write \(\Phi\) for the root system for \((\mathop{\mathrm{\mathfrak g}}, \mathop{\mathrm{\mathfrak h}}).\) Let \(\mathcal{S}\) be the set of closed root subsystems of \(\Phi\) of rank \(\mathop{\mathrm{rk}}\Phi - 1.\) We view \(\mathcal{S}\) as a poset, where the partial order is inclusion. A maximal element of the poset \((\mathcal{S}, \subseteq)\) is called a good root subsystem of \(\Phi.\) For a good root subsystem \(\Phi'\) of \(\Phi\), we let \(\mathop{\mathrm{\mathfrak g}}'\) be a semisimple Lie subalgebra of \(\mathop{\mathrm{\mathfrak g}}\) and \(\mathop{\mathrm{\mathfrak h}}'\) a Cartan subalgebra of \(\mathop{\mathrm{\mathfrak g}}'\) such that the root system for \((\mathop{\mathrm{\mathfrak g}}', \mathop{\mathrm{\mathfrak h}}')\) is isomorphic to \(\Phi'.\) We prove the following result about the structure of \(\bar \mathop{\mathrm{\mathfrak h}}.\)

To analyze the singularities of \(\bar \mathop{\mathrm{\mathfrak h}}\), we prove the following result.

Using results in [6], one deduces that the variety \(\bar \mathop{\mathrm{\mathfrak h}}\) is Cohen-Macaulay. Hence, it satisfies Serre’s condition (S2). In view of Theorem 2 and Serre’s criterion for normality, we have

We further consider \(k\)-step good root subsystems of \(\Phi\), which are defined inductively as good root subsystems of \((k-1)\)-step good root subsystems of \(\Phi\), where \(\Phi\) is regarded as a \(0\)-step good root subsystem of \(\Phi.\) We then apply the above results to find an affine paving \[\bar \mathop{\mathrm{\mathfrak h}}= \bigsqcup \mathring C(\Psi),\] where the union is over all \(k\)-step good root subsystems \(\Psi\) of \(\Phi\) for \(k=0, \dots, r := \dim \mathop{\mathrm{\mathfrak h}}\) in Corollary 1. View \(\mathop{\mathrm{\mathfrak h}}\) as an additive algebraic group. We prove that the action of \(\mathop{\mathrm{\mathfrak h}}\) on itself by addition extends to an action of \(\mathop{\mathrm{\mathfrak h}}\) on \(\bar \mathop{\mathrm{\mathfrak h}}\). Further, the subvarieties \(\mathring C(\Psi)\) are the \(\mathop{\mathrm{\mathfrak h}}\)-orbits of \(\bar \mathop{\mathrm{\mathfrak h}}.\) In particular, there are finitely many \(\mathop{\mathrm{\mathfrak h}}\)-orbits in \(\bar \mathop{\mathrm{\mathfrak h}}\), and \(\mathring C(\Phi)\) is the open \(\mathop{\mathrm{\mathfrak h}}\)-orbit which is \(\mathop{\mathrm{\mathfrak h}}\)-equivariantly isomorphic to \(\mathop{\mathrm{\mathfrak h}}\). Due to the above results, we may regard \(\bar \mathop{\mathrm{\mathfrak h}}\) as an additive analogue of a toric variety.

As a consequence of the above affine paving, much of the structure of \(H^{\bullet}(\bar \mathop{\mathrm{\mathfrak h}}, \mathop{\mathrm{\mathbb{Z}}})\) is determined by the combinatorics of \(k\)-step good root subsystems. To study this combinatorics, we relate the order relation given by closures of strata in \(\bar \mathop{\mathrm{\mathfrak h}}\) to the Coxeter arrangements studied by Orlik and Solomon. More precisely, we give an isomorphism of posets between the poset consisting of strata of \(\bar \mathop{\mathrm{\mathfrak h}}\) with order relation given by closure, and the poset of subspaces of a Coxeter arrangement. As a consequence, the Betti numbers of \(\bar \mathop{\mathrm{\mathfrak h}}\) can be deduced from the number of subspaces of given rank in the corresponding Coxeter arrangement, and these are computed in [8] and [9]. In more detail, for a root system \(\Phi\), we let \(f(\Phi,k) := \mathop{\mathrm{rk}}H^{2(r-k)}(\bar \mathop{\mathrm{\mathfrak h}}, \mathop{\mathrm{\mathbb{Z}}})\), which is the number of \(k\)-step good root subsystems. We let \(S(n,k)\) denote the Stirling number of the second kind, which counts the number of partitions of the set \(\{ 1, \dots, n \}\) into \(k\) nonempty parts.

In Theorem 22, we give analogous formulas for types \(B\), \(C\), and \(D\) (note that the bijection with Coxeter arrangements implies that the answer is the same for types \(B\) and \(C\)). We also give formulas for the Betti numbers of \(\bar \mathop{\mathrm{\mathfrak h}}\) in the exceptional cases, which can be deduced from tables in [9], but we also computed them using the software SageMath.

We further show that the above isomorphism of posets is compatible with natural Weyl group actions. We use these results to determine the Weyl group representation on \(H^{\bullet}(\bar \mathop{\mathrm{\mathfrak h}}, \mathop{\mathrm{\mathbb{C}}})\) in Corollary 8. Finally, we determine the cup product structure of \(H^{\bullet}(\bar \mathop{\mathrm{\mathfrak h}}, \mathop{\mathrm{\mathbb{Z}}})\) in terms of a natural basis given by the stratification, which we denote by \(\xi_X\), where \(X\) is a subspace in the intersection lattice \(L(\mathcal{A})\) of the Coxeter arrangement corresponding to a stratum. Our result is the same as in [10], but our method is much more elementary.

We discuss the genesis of this paper, and its relation to other papers in the literature. We established most of the results of this paper without being aware of the notion of matroid Schubert variety from [6] and [7], but have incorporated results from this literature where it is useful to streamline the exposition. In particular, we use results from [6] in an essential way to prove normality of \(\bar \mathop{\mathrm{\mathfrak h}}.\) Since our interest is focused on Lie theoretic aspects of \(\bar \mathop{\mathrm{\mathfrak h}},\) the relation of \(\bar \mathop{\mathrm{\mathfrak h}}\) to \(\bar H,\) and potential applications to the geometry of the resulting compactification of \(\mathop{\mathrm{\mathfrak g^{\ast}}},\) we generally have not emphasized perspectives from matroid Schubert varieties. However, the stratification we describe for \(\bar \mathop{\mathrm{\mathfrak h}}\), which we define using root subsystems, may also be described as a standard stratification for matroid Schubert varieties, and, as stated above, Theorem 5 is a special case of the main result of [10], although our proof is much simpler. We also note that there is an open affine subvariety of \(\bar \mathop{\mathrm{\mathfrak h}}\) which meets all singular strata, and which is an example of the reciprocal varieties introduced by Proudfoot and Speyer [11]. In an earlier draft, we were unaware of the computations in [9] and [8], and we are grateful to the referee and Matt Douglass for informing us of these computations and of the connection between \(k\)-step good root subsystems and parabolic root systems. As a result, we previously developed a variant of the work of Borel and de Siebenthal from [12] in order to count good root subsystems, and this earlier draft included a lengthy study of the combinatorics of \(k\)-step good root subsystems in order to compute the Betti numbers for \(\bar \mathop{\mathrm{\mathfrak h}}.\) In view of [8], we have omitted this combinatorics, along with our variant of the work of Borel and de Siebenthal.

Due to the analogy with \(\bar H,\) we view our results showing that \(\bar \mathop{\mathrm{\mathfrak h}}\) is normal and has finitely many \(\mathop{\mathrm{\mathfrak h}}\)-orbits as indicating that \(\bar \mathop{\mathrm{\mathfrak h}}\) should be regarded as an additive toric variety. To develop this theory, it would be desirable if each additive toric variety is uniquely determined by a set of combinatorial data such that the combinatorial data determining \(\bar \mathop{\mathrm{\mathfrak h}}\) is the Coxeter arrangement. See Colin Crowley’s paper [4] for general results about additive toric varieties. Forthcoming related work by the second author, Ana Balibanu, and Crowley constructs a flat degeneration such that the generic fiber is the toric variety associated to a rational central essential hyperplane arrangement and the special fiber is a non-reduced union of matroid Schubert varieties. In this context, \(\bar \mathop{\mathrm{\mathfrak h}}\) is a component of the special fiber for the family with generic fiber \(\bar H\), and this family is expected to be a subfamily of the degeneration of \(\mathop{\mathrm{\mathcal{L}}}(\mathop{\mathrm{\mathfrak g}}\oplus\mathop{\mathrm{\mathfrak g}})\) to \(\mathop{\mathrm{\mathcal{L}}}.\) In further related work, we note that the paper [13] gives a combinatorial construction of the real locus of \(\bar \mathop{\mathrm{\mathfrak h}}\) as a permutohedron modulo parallel faces for the Cartan subalgebra of a general complex semisimple \(\mathop{\mathrm{\mathfrak g}}.\) In forthcoming work of the second author with Leo Jiang, the authors give an analogous construction for any matroid Schubert variety of a central essential real hyperplane arrangement.

This paper is organized as follows. In Section 2, we set up basic notation and define objects of interest in the study of \(\bar \mathop{\mathrm{\mathfrak h}}.\) In Section 3 we prove that \(\bar \mathop{\mathrm{\mathfrak h}}\) is normal and has an affine paving given by \(\mathop{\mathrm{\mathfrak h}}\)-orbits. In particular, Theorems 1, 2 and 3 are proved in this section. In Section 4, we relate our stratification of \(\bar \mathop{\mathrm{\mathfrak h}}\) to Coxeter arrangements, and use this to compute the Betti numbers of \(\bar \mathop{\mathrm{\mathfrak h}}\) for an arbitrary semisimple Lie algebra. Theorem 4 is proved in this section. In Section 5, we compute the Weyl group representation on \(H^{\bullet}(\bar \mathop{\mathrm{\mathfrak h}}, \mathop{\mathrm{\mathbb{C}}})\) and determine the cohomology ring of \(\bar \mathop{\mathrm{\mathfrak h}}\) in terms of the basis given by the strata. Theorem 5 is proved in this section.

Acknowledgments. We are grateful to Ana Balibanu, Colin Crowley, Victor Ginzburg, Boming Jia, Leo Jiang, Joel Kamnitzer, Jacob Matherne, Eckhard Meinrenken, Nick Proudfoot, Nick Salter, and Shuddhodan Vasudevan for many stimulating discussions. We would also like to thank Jacob Matherne and Nick Proudfoot for encouraging us to write up this paper, and we are especially grateful to Matt Douglass for his careful reading of an earlier draft of this paper and for his incisive and helpful comments, and the referee for several comments which have significantly improved our paper.

2 Construction of \(\bar \mathop{\mathrm{\mathfrak h}}\)↩︎

In this section, we define \(\bar \mathop{\mathrm{\mathfrak h}}\) and give an embedding of \(\bar \mathop{\mathrm{\mathfrak h}}\) into a product of projective lines. As a consequence, we show that \(\bar \mathop{\mathrm{\mathfrak h}}\) is a matroid Schubert variety.

Throughout this paper, we work over the complex numbers. Let \(G\) be a connected semisimple algebraic group of adjoint type and \(\mathop{\mathrm{\mathfrak g}}:= \text{Lie} (G)\) the Lie algebra of \(G.\) Write \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) for the dual space of \(\mathop{\mathrm{\mathfrak g}}.\) The adjoint (resp. coadjoint) action of \(G\) on \(\mathop{\mathrm{\mathfrak g}}\) (resp. \(\mathop{\mathrm{\mathfrak g^{\ast}}}\)) will be denoted by \(\mathop{\mathrm{Ad}}\) (resp. \(\mathop{\mathrm{Ad}}^{\ast}\)). The adjoint (resp. coadjoint) action of \(\mathop{\mathrm{\mathfrak g}}\) on \(\mathop{\mathrm{\mathfrak g}}\) (resp. \(\mathop{\mathrm{\mathfrak g^{\ast}}}\)) will be denoted by \(\mathop{\mathrm{ad}}\) (resp. \(\mathop{\mathrm{ad}}^{\ast}\)). Define the algebraic group \(D\) to be the semidirect product \(G \ltimes \mathop{\mathrm{\mathfrak g^{\ast}}}\), where \(G\) acts on \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) by the coadjoint action. Specifically, as a set \(D\) is the Cartesian product \(G \times \mathop{\mathrm{\mathfrak g^{\ast}}}\), and group multiplication in \(D\) is given by \[(g, \phi) (h, \psi) := (gh, \mathop{\mathrm{Ad}}^{\ast}_{h^{-1}} \phi + \psi) \quad \forall g, h \in G \text{ and } \phi, \psi \in \mathop{\mathrm{\mathfrak g^{\ast}}}.\] Inversion in \(D\) is given by \[(g, \phi)^{-1} := (g^{-1}, - \mathop{\mathrm{Ad}}^{\ast}_g \phi) \quad \forall g \in G \text{ and } \phi \in \mathop{\mathrm{\mathfrak g^{\ast}}}.\]

Consider the Lie algebra \(\mathop{\mathrm{\mathfrak d}}\) of \(D\), which is the semidirect sum \(\mathop{\mathrm{\mathfrak g}}\ltimes \mathop{\mathrm{\mathfrak g^{\ast}}}.\) As a vector space, \(\mathop{\mathrm{\mathfrak d}}\) is the Cartesian product \(\mathop{\mathrm{\mathfrak g}}\times \mathop{\mathrm{\mathfrak g^{\ast}}}.\) The Lie bracket in \(\mathop{\mathrm{\mathfrak d}}\) is given by \[[(x, \phi), (y, \psi)] := ([x, y], \mathop{\mathrm{ad}}^{\ast}_x \psi - \mathop{\mathrm{ad}}^{\ast}_y \phi) \quad \forall x, y \in \mathop{\mathrm{\mathfrak g}}\text{ and } \phi, \psi \in \mathop{\mathrm{\mathfrak g^{\ast}}}.\]

The exponential map \(\text{Exp}: \mathop{\mathrm{\mathfrak d}}\rightarrow D\) is given by \[\text{Exp} (x, \phi) = \bigl( \text{exp}(x), \sum_{i=0}^{\infty} \frac{(-1)^{i}}{(i+1)!} (\mathop{\mathrm{ad}}^*_x)^i \phi \bigr) \quad \forall x \in \mathop{\mathrm{\mathfrak g}}\text{ and } \phi \in \mathop{\mathrm{\mathfrak g^{\ast}}},\] where \(\text{exp}: \mathop{\mathrm{\mathfrak g}}\rightarrow G\) is the exponential map for \(G.\) This formula corrects an error in the formula for the exponential map that we presented in [5], but this mistake does not affect the rest of that paper. By abuse of notation, the adjoint action of \(D\) on \(\mathop{\mathrm{\mathfrak d}}\) is also denoted by \(\mathop{\mathrm{Ad}}.\) Using the formulas above, one verifies that the adjoint action of \(D\) on \(\mathop{\mathrm{\mathfrak d}}\) is given by \[\begin{align} \label{adj} \mathop{\mathrm{Ad}}_{(g, \phi)} (x, \psi) = (\mathop{\mathrm{Ad}}_g x, - \mathop{\mathrm{Ad}}^{\ast}_g \mathop{\mathrm{ad}}^{\ast}_x \phi + \mathop{\mathrm{Ad}}^{\ast}_g \psi) \quad \forall g \in G, x \in \mathop{\mathrm{\mathfrak g}}\text{ and } \phi, \psi \in \mathop{\mathrm{\mathfrak g^{\ast}}}. \end{align}\tag{1}\]

The algebraic group \(D\) has the following important algebraic subgroups. It is easy to check that the homomorphism of algebraic groups \[G \longrightarrow D, ~ g \longmapsto (g, 0)\] is an embedding of \(G\) into \(D.\) We will regard \(G\) as an algebraic subgroup of \(D\) via this homomorphism. View \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) as an additive algebraic group. It is easy to check that the homomorphism of algebraic groups \[\mathop{\mathrm{\mathfrak g^{\ast}}}\longrightarrow D, ~ \phi \longmapsto (e, \phi),\] where \(e\) stands for the identity element of \(G\), is an embedding of \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) into \(D.\) We will regard \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) as an algebraic subgroup of \(D\) via this homomorphism. At the Lie algebra level, we view \(\mathop{\mathrm{\mathfrak g}}\) (resp. \(\mathop{\mathrm{\mathfrak g^{\ast}}}\)) as a Lie subalgebra of \(\mathop{\mathrm{\mathfrak d}}\) via the embedding \(\mathop{\mathrm{\mathfrak g}}\rightarrow \mathop{\mathrm{\mathfrak d}}, ~ x \mapsto (x, 0)\) (resp. \(\mathop{\mathrm{\mathfrak g^{\ast}}}\rightarrow \mathop{\mathrm{\mathfrak d}}, ~ \phi \mapsto (0, \phi)\)).

Introduce a bilinear form \((~, ~): \mathop{\mathrm{\mathfrak d}}\otimes \mathop{\mathrm{\mathfrak d}}\rightarrow \mathbb{C}\) by defining \[((x, \phi), (y, \psi)) := \phi (y) + \psi (x) \quad \forall x, y \in \mathop{\mathrm{\mathfrak g}}\text{ and } \phi, \psi \in \mathop{\mathrm{\mathfrak g^{\ast}}}.\] It is easy to verify that the bilinear form \((~, ~)\) is symmetric, nondegenerate and \(D\)-invariant. Here, \(D\)-invariance means that \[\begin{align} \label{inv} (\mathop{\mathrm{Ad}}_{(g, \phi)} (x, \psi), \mathop{\mathrm{Ad}}_{(g, \phi)} (y, \chi)) = ((x, \psi), (y, \chi)) \quad \forall g \in G, x,y \in \mathop{\mathrm{\mathfrak g}}\text{ and } \phi, \psi, \chi \in \mathop{\mathrm{\mathfrak g^{\ast}}}. \end{align}\tag{2}\]

Let \(V\) be a finite dimensional vector space and \((~, ~)\) a nondegenerate bilinear form on \(V.\) A vector subspace \(W\) of \(V\) is called isotropic if \((w, w') = 0\) for all \(w, w' \in W.\) An isotropic subspace \(W\) of \(V\) is Lagrangian if \(W\) is maximal, with respect to inclusion, among all isotropic subspaces of \(V.\) When \(V\) is even dimensional, this condition is equivalent to saying that \(\dim W = \frac{1}{2} \dim V.\)

A Lagrangian subalgebra of \(\mathop{\mathrm{\mathfrak d}}\) is a Lie subalgebra \(\mathop{\mathrm{\mathfrak l}}\) of \(\mathop{\mathrm{\mathfrak d}}\) which is also a Lagrangian vector subspace of \(\mathop{\mathrm{\mathfrak d}}\) with respect to the nondegenerate bilinear form \((~, ~)\) on \(\mathop{\mathrm{\mathfrak d}}\) introduced above. For example, it is easy to check that \(\mathop{\mathrm{\mathfrak g}}\) and \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) are Lagrangian subalgebras of \(\mathop{\mathrm{\mathfrak d}}.\) Moreover, if \(\mathop{\mathrm{\mathfrak l}}\) is a Lagrangian subalgebra of \(\mathop{\mathrm{\mathfrak d}}\) and \((g, \phi)\) is an element of \(D\), then \(\mathop{\mathrm{Ad}}_{(g, \phi)} \mathop{\mathrm{\mathfrak l}}\) is also a Lagrangian subalgebra of \(\mathop{\mathrm{\mathfrak d}}\) by Equation (2 ).

Let \(\mathop{\mathrm{Gr}}(n, \mathop{\mathrm{\mathfrak d}})\) be the Grassmannian of \(n\)-dimensional vector subspaces of \(\mathop{\mathrm{\mathfrak d}}\), where \(n := \dim \mathop{\mathrm{\mathfrak g}}= \frac{1}{2} \dim \mathop{\mathrm{\mathfrak d}}.\) For an \(n\)-dimensional vector subspace of \(\mathop{\mathrm{\mathfrak d}}\), the property of being a Lie subalgebra (resp. being an isotropic vector subspace) of \(\mathop{\mathrm{\mathfrak d}}\) is a closed condition. Therefore, the set of Lagrangian subalgebras of \(\mathop{\mathrm{\mathfrak d}}\) has a natural structure of a reduced closed subvariety of \(\mathop{\mathrm{Gr}}(n, \mathop{\mathrm{\mathfrak d}})\), to be denoted by \(\mathop{\mathrm{\mathcal{L}}}.\) Since \(\mathop{\mathrm{Gr}}(n, \mathop{\mathrm{\mathfrak d}})\) is a projective variety, the variety \(\mathcal{L}\) is projective as well. We will call \(\mathcal{L}\) the variety of Lagrangian subalgebras of \(\mathop{\mathrm{\mathfrak d}}.\) By Equation (2 ), the image of the map \[D \times \mathop{\mathrm{\mathcal{L}}}\longrightarrow \mathop{\mathrm{Gr}}(n, \mathop{\mathrm{\mathfrak d}}), ~ ((g, \phi), \mathop{\mathrm{\mathfrak l}}) \longmapsto \mathop{\mathrm{Ad}}_{(g, \phi)} \mathop{\mathrm{\mathfrak l}}\] is in \(\mathop{\mathrm{\mathcal{L}}}\) and, hence, gives an action of the algebraic group \(D\) on the variety \(\mathop{\mathrm{\mathcal{L}}}.\) By abuse of notation, this action will be denoted by \(\mathop{\mathrm{Ad}}\). The following lemma is easy to prove.

From Lemma 1, we see that we can identify \(D/G\) with the \(D\)-orbit in \(\mathop{\mathrm{\mathcal{L}}}\) through the point \(\mathop{\mathrm{\mathfrak g}}.\) Since the subgroups \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) and \(G\) of \(D\) intersect only at the identity element \((e, 0)\) of \(D\), the morphism \[\mathop{\mathrm{\mathfrak g^{\ast}}}\longrightarrow D/G\] sending an element \(\phi\) of \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) to the coset of \((e,\phi)\) is an isomorphism of varieties. Therefore, we get an embedding \[\mathop{\mathrm{\mathfrak g^{\ast}}}\lhook\joinrel\longrightarrow{} \mathop{\mathrm{\mathcal{L}}}, ~ \phi \longmapsto \mathop{\mathrm{Ad}}_{(e,\phi)} \mathop{\mathrm{\mathfrak g}}\] of \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) into \(\mathop{\mathrm{\mathcal{L}}}.\) More concretely, by Equation (1 ), this embedding sends an element \(\phi\) of \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) to the Lagrangian subalgebra \[\{(x, - \mathop{\mathrm{ad}}^{\ast}_x \phi): x \in \mathop{\mathrm{\mathfrak g}}\}\] of \(\mathop{\mathrm{\mathfrak d}}.\)

In the following example, we compute \(\bar \mathop{\mathrm{\mathfrak g}}^{\ast}\) in the case where \(\mathop{\mathrm{\mathfrak g}}\) is \(\mathfrak {sl}_2.\)

Example 1 is somewhat misleading in the sense that, in general, \(\bar \mathop{\mathrm{\mathfrak g}}^{\ast}\) is expected to be a singular variety. It was pointed out in [5] that the geometry of \(\mathop{\mathrm{\mathcal{L}}}\) is far more intricate than that of the variety of Lagrangian subalgebras of \(\mathop{\mathrm{\mathfrak g}}\oplus \mathop{\mathrm{\mathfrak g}}\), equipped with the bilinear form \[(\mathop{\mathrm{\mathfrak g}}\oplus \mathop{\mathrm{\mathfrak g}}) \otimes (\mathop{\mathrm{\mathfrak g}}\oplus \mathop{\mathrm{\mathfrak g}}) \longrightarrow \mathbb{C}, ~ ((x_1,y_1), (x_2,y_2)) \longmapsto \kappa(x_1,x_2) - \kappa(y_1,y_2),\] where \(\kappa\) stands for the Killing form. In particular, the geometry of \(\bar \mathop{\mathrm{\mathfrak g}}^{\ast}\) is expected to be more complicated than that of the wonderful compactification \(X\) of \(G\) studied by De Concini and Procesi [1]. It was observed in the literature, cf. [14], that to study the geometry of \(X\), it is usually helpful if one first studies the geometry of the wonderful compactification \(\bar H\) of a Cartan subgroup \(H\) of \(G.\) As a first step towards understanding the geometry of \(\bar \mathop{\mathrm{\mathfrak g}}^{\ast}\), we introduce an analogue of \(\bar H\) as follows.

Fix a Cartan subalgebra \(\mathop{\mathrm{\mathfrak h}}\) of \(\mathop{\mathrm{\mathfrak g}}.\) We have an embedding \[\mathop{\mathrm{\mathfrak h}}\lhook\joinrel\longrightarrow{} \mathop{\mathrm{\mathfrak g^{\ast}}}, ~ h \longmapsto \kappa(h, -).\] Composing this with the embedding \(\mathop{\mathrm{\mathfrak g^{\ast}}}\hookrightarrow \mathop{\mathrm{\mathcal{L}}}\), we get an embedding \[{\rm emb}: \mathop{\mathrm{\mathfrak h}}\lhook\joinrel\longrightarrow{} \mathop{\mathrm{\mathfrak g^{\ast}}}\lhook\joinrel\longrightarrow{} \mathop{\mathrm{\mathcal{L}}}\subseteq \mathop{\mathrm{Gr}}(n, \mathop{\mathrm{\mathfrak d}}).\]

Identify \(\mathop{\mathrm{\mathfrak g}}\) with \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) via \[\mathop{\mathrm{\mathfrak g}}\cong \mathop{\mathrm{\mathfrak g^{\ast}}}, ~ x \longmapsto \kappa(x, -).\] This induces an identification of \(\mathop{\mathrm{Gr}}(n, \mathop{\mathrm{\mathfrak d}})\) with \(\mathop{\mathrm{Gr}}(n, \mathop{\mathrm{\mathfrak g}}\times \mathop{\mathrm{\mathfrak g}}).\) Under this identification, if an element \(\phi\) of \(\mathop{\mathrm{\mathfrak g^{\ast}}}\) is identified with \(y \in \mathop{\mathrm{\mathfrak g}}\), then the Lagrangian subalgebra \[\{(x, - \mathop{\mathrm{ad}}^{\ast}_x \phi): x \in \mathop{\mathrm{\mathfrak g}}\}\] of \(\mathop{\mathrm{\mathfrak d}}\) is identified with the vector subspace \[\{(x, [y,x]): x \in \mathop{\mathrm{\mathfrak g}}\}\] of \(\mathop{\mathrm{\mathfrak g}}\times \mathop{\mathrm{\mathfrak g}}.\) The \(\mathop{\mathrm{\mathfrak g^{\ast}}}\)-action on \(\mathop{\mathrm{\mathfrak d}}\) from Equation (1 ) induces a \(\mathop{\mathrm{\mathfrak g}}\)-action on \(\mathop{\mathrm{\mathfrak g}}\times \mathop{\mathrm{\mathfrak g}}\) given by \((z,(x,u))\mapsto (x,[z,x]+u).\) Let \(\Phi\) be the root system for \((\mathop{\mathrm{\mathfrak g}}, \mathop{\mathrm{\mathfrak h}}).\) For each root \(\lambda \in \Phi\), choose a root vector \(e_{\lambda}.\) When \(y\) is in \(\mathop{\mathrm{\mathfrak h}}\), the vector space \(\{(x, [y,x]): x \in \mathop{\mathrm{\mathfrak g}}\}\) is clearly spanned by \(\mathop{\mathrm{\mathfrak h}}\oplus 0\) and \((e_{\lambda}, \lambda(y) e_{\lambda})\) for all \(\lambda \in \Phi.\) This proves the following

By abuse of notation, the composition in Lemma 2 will also be denoted by \({\rm emb}.\) Note that \({\rm emb}\) is \(\mathop{\mathrm{\mathfrak h}}\)-equivariant for the translation action of \(\mathop{\mathrm{\mathfrak h}}\) on itself and the induced action of \(\mathop{\mathrm{\mathfrak h}}\) on \(\mathop{\mathrm{Gr}}(n, \mathop{\mathrm{\mathfrak g}}\times \mathop{\mathrm{\mathfrak g}}).\)

Let \(V\) be a vector space and \(\{v_1, \ldots, v_m\}\) a basis of \(V.\) Define a morphism \[f: (\mathbb{P}^1)^m \longrightarrow \mathop{\mathrm{Gr}}(m, V \times V), ~ f \bigl( ([x_{i,0}: x_{i,1}] )_{1 \le i \le m} \bigr) := {\rm Span} ((x_{i,0}v_i, x_{i,1}v_i): 1 \le i \le m).\] It is easy to see that this morphism is well-defined. Arguing in local coordinates, it is also easy to see that this morphism is a closed embedding.

Consider the case where \(V = \mathop{\mathrm{\mathfrak g}}.\) For this, consider a basis \[\{h_1, \ldots, h_r, e_{\lambda}: \lambda \in \Phi\}\] of \(\mathop{\mathrm{\mathfrak g}},\) where \(\{h_1, \ldots, h_r\}\) is a basis of \(\mathop{\mathrm{\mathfrak h}}.\) The morphism \({\rm emb}: \mathop{\mathrm{\mathfrak h}}\hookrightarrow \mathop{\mathrm{Gr}}(n, \mathop{\mathrm{\mathfrak g}}\times \mathop{\mathrm{\mathfrak g}})\) factors through the morphism \(f: (\mathbb{P}^1)^n \hookrightarrow \mathop{\mathrm{Gr}}(n, \mathop{\mathrm{\mathfrak g}}\times \mathop{\mathrm{\mathfrak g}})\) explained above. In fact, there is a commutative diagram \[\begin{tikzcd} \mathop{\mathrm{\mathfrak h}}\arrow[d, "{\rm emb}'"] \arrow[dr, "{\rm emb}"] \\ (\mathbb{P}^1)^n \arrow[r, "f"] & \mathop{\mathrm{Gr}}(n, \mathop{\mathrm{\mathfrak g}}\times \mathop{\mathrm{\mathfrak g}}), \end{tikzcd}\] where \({\rm emb}'\) sends an element \(h\) of \(\mathop{\mathrm{\mathfrak h}}\) to the point of \((\mathbb{P}^1)^n\) whose \(h_i\)-component is \([1:0]\) for all \(1 \le i \le r\) and whose \(\lambda\)-component is \([1:\lambda (h)]\) for all \(\lambda \in \Phi.\) Further, if \(\mathop{\mathrm{\mathfrak h}}\) acts on \((\mathbb{P}^1)^n\) by the formula \[y\cdot ([x_{i,0}:x_{i,1}])_{i=1, \ldots, r} \times ([x_{\lambda,0},x_{\lambda,1}])_{\lambda \in \Phi} = ([x_{i,0}:x_{i,1}])_{i=1, \ldots, r} \times ([x_{\lambda,0},\lambda(y)x_{\lambda,0}+x_{\lambda,1}])_{\lambda \in \Phi},\] then the morphisms \({\rm emb}'\) and \(f\) are \(\mathop{\mathrm{\mathfrak h}}\)-equivariant.

We identify the open set consisting of \(x=[x_0:x_1]\) of \({\mathbb{P}^1}\) with \(x_0\not= 0\) with \(\mathop{\mathrm{\mathbb{C}}}\) using the map \(x\mapsto \frac{x_1}{x_0} \in \mathop{\mathrm{\mathbb{C}}}\), and write \(\infty\) for the point \([0:1].\)

Choose a decomposition \(\Phi = \Phi^+ \sqcup \Phi^-\) of \(\Phi\) into the sets of positive and negative roots. Let \(d\) be the cardinality of \(\Phi^+.\) Introduce a closed embedding \(l: (\mathbb{P}^1)^d \rightarrow (\mathbb{P}^1)^n\) by sending the point \((x_{\lambda})_{\lambda \in \Phi^+}\) of \((\mathbb{P}^1)^d\) to the point of \((\mathbb{P}^1)^n\) whose \(h_i\)-component is \(0\) for all \(1 \le i \le r\), whose \(\lambda\)-component is \(x_{\lambda}\) and whose \((-\lambda)\)-component is \(-x_{\lambda}\) for all \(\lambda \in \Phi^+.\) Then it is clear that \({\rm emb}': \mathop{\mathrm{\mathfrak h}}\hookrightarrow (\mathbb{P}^1)^n\) factors through \(l: (\mathbb{P}^1)^d \rightarrow (\mathbb{P}^1)^n\), namely, there is a commutative diagram \[\begin{tikzcd} \mathop{\mathrm{\mathfrak h}}\arrow[d, "{\rm emb}''"] \arrow[dr, "{\rm emb}'"] \\ (\mathbb{P}^1)^d \arrow[r, "l"] & (\mathbb{P}^1)^n, \end{tikzcd}\] where \({\rm emb}''\) sends an element \(h\) of \(\mathop{\mathrm{\mathfrak h}}\) to the point of \((\mathbb{P}^1)^d\) whose \(\lambda\)-component is \([1:\lambda (h)]\) for all \(\lambda \in \Phi^+.\) These arguments prove the following

Another way of stating Proposition 6 is as follows. View \(\mathbb{C}^d\) as the vector space of \((\Phi^+)\)-tuples of complex numbers. View \(\mathbb{C}^d\) also as the affine chart of \((\mathbb{P}^1)^d\) consisting of points all of whose components are not equal to \(\infty.\) The linear map \[\mathop{\mathrm{\mathfrak h}}\longrightarrow \mathbb{C}^d, ~ h \longmapsto (\lambda(h))_{\lambda \in \Phi^+}\] is clearly injective. Via this injective linear map, we view \(\mathop{\mathrm{\mathfrak h}}\) as a vector subspace of \(\mathbb{C}^d.\) Then Proposition 6 says that \(\bar \mathop{\mathrm{\mathfrak h}}\) is isomorphic to the closure of the vector subspace \(\mathop{\mathrm{\mathfrak h}}\) of \(\mathbb{C}^d\) in \((\mathbb{P}^1)^d.\)

Thus, \(\bar \mathop{\mathrm{\mathfrak h}}\) is a matroid Schubert variety. Matroid Schubert varieties were first considered in [6] and play a crucial role in the proof of the top-heavy conjecture by Braden, Huh, Matherne, Proudfoot, and Wang [15].

3 Algebro-geometric Properties of \(\bar \mathop{\mathrm{\mathfrak h}}\)↩︎

In this section, we prove that \(\bar \mathop{\mathrm{\mathfrak h}}\) is a normal variety by showing it is regular in codimension one and using results from [6] to show that \(\bar \mathop{\mathrm{\mathfrak h}}\) is Cohen-Macaulay. In the process, we show that there are finitely many \(\mathop{\mathrm{\mathfrak h}}\)-orbits on \(\bar \mathop{\mathrm{\mathfrak h}}\), and that the orbits give an affine paving of \(\bar \mathop{\mathrm{\mathfrak h}}.\) For this reason, it is reasonable to think of \(\bar \mathop{\mathrm{\mathfrak h}}\) as an additive analogue of a toric variety. We also extend the action of the Weyl group \(W\) on \(\mathop{\mathrm{\mathfrak h}}\) to a \(W\)-action on \(\bar \mathop{\mathrm{\mathfrak h}}\), and we explicitly compute \(\bar \mathop{\mathrm{\mathfrak h}}\) in the case of \(\mathfrak{sl}_3\), in which case \(\bar \mathop{\mathrm{\mathfrak h}}\) is a singular hypersurface in \((\mathbb{P}^1)^3.\)

3.1 Defining Ideal and Boundary Components of \(\bar \mathop{\mathrm{\mathfrak h}}\)↩︎

Let \(S := \mathbb{C}[x_{\lambda,0}, x_{\lambda,1}: \lambda \in \Phi^+]\) be the multi-homogeneous coordinate ring of \((\mathbb{P}^1)^d.\) The ring \(S\) is \((\mathbb{Z}_{\ge 0})^d\)-graded so that the variables \(x_{\lambda,0}, x_{\lambda,1}\), for \(\lambda \in \Phi^+\), have multi-degree \[(0, \ldots, 0, 1, 0, \ldots, 0),\] where the only \(1\) appears in the \(\lambda\)-component. An element of \(S\) is said to be multi-homogeneous if all its terms have the same multi-degree which we refer to as “degree” in the sequel. It is easy to see that every element \(p\) of \(S\) can be uniquely written in the form \[p = \sum \limits_{u \in (\mathbb{Z}_{\ge 0})^d} p_u,\] where \(p_u\) is multi-homogeneous of degree \(u\) for each \(u \in (\mathbb{Z}_{\ge 0})^d.\) The \(p_u\)’s will be referred to as the multi-homogeneous components of \(p.\) Let \(J\) be an ideal of \(S.\) We say that \(J\) is a multi-homogeneous ideal if it can be generated by multi-homogeneous polynomials. Equivalently, \(J\) is multi-homogeneous if whenever an element of \(S\) is in \(J\), so are all its multi-homogeneous components.

As in projective geometry, for each multi-homogeneous ideal \(J\) of \(S\), the vanishing locus \(V(J)\) of \(J\) in \((\mathbb{P}^1)^d\) is well-defined. Choose multi-homogeneous generators \(p_1, \ldots, p_k\) of \(J.\) Then the set \[\{x = ([x_{\lambda,0}: x_{\lambda,1}])_{\lambda \in \Phi^+} \in (\mathbb{P}^1)^d: p_1 \bigl( (x_{\lambda,0},x_{\lambda,1})_{\lambda \in \Phi^+} \bigr) = 0, \ldots, p_k \bigl( (x_{\lambda,0},x_{\lambda,1})_{\lambda \in \Phi^+} \bigr) = 0\}\] is well-defined and independent of the choice of multi-homogeneous generators of \(J.\) This set is the vanishing locus \(V(J).\) Our first goal is to find a multi-homogeneous ideal \(J\) of \(S\), as simple as possible, whose vanishing locus is \(\bar \mathop{\mathrm{\mathfrak h}}\), and use it to analyze the boundary \(\bar \mathop{\mathrm{\mathfrak h}}- \mathop{\mathrm{\mathfrak h}}\) of \(\mathop{\mathrm{\mathfrak h}}.\)

Let \(R := \mathbb{C}[x_{\lambda}: \lambda \in \Phi^+]\) be the coordinate ring of \(\mathbb{C}^d,\) which we regard as an affine open set of \((\mathbb{P}^1)^d.\) For \(P \in R\), the multi-homogenization of \(P\) is the multi-homogeneous polynomial \(P^h \in S\) such that \[P \bigl( (\frac{x_{\lambda,1}}{x_{\lambda,0}})_{\lambda \in \Phi^+} \bigr) = \frac{P^h \bigl( (x_{\lambda,0}, x_{\lambda,1})_{\lambda \in \Phi^+} \bigr)}{\prod \limits_{\lambda \in \Phi^+} x_{\lambda,0}^{a_{\lambda}}},\] where \(a_{\lambda} \in \mathbb{Z}_{\ge 0}\) for all \(\lambda \in \Phi^+\) and the numerator and denominator of the right-hand side have no common prime divisors. For an ideal \(I\) of \(R\), its multi-homogenization is the ideal \[I^h := \langle P^h: P \in I \rangle\] of \(S\) generated by the multi-homogenization of elements of \(I.\) We remark that, in general, if \(P_1, \ldots, P_k\) are generators of \(I\), the multi-homogeneous ideal \(\langle P_1^h, \ldots, P_k^h \rangle\) could be properly contained in \(I^h.\)

Let \(I(\mathop{\mathrm{\mathfrak h}}) \lhd R\) be the ideal consisting of all polynomials vanishing on \(\mathop{\mathrm{\mathfrak h}}\), where we have regarded \(\mathop{\mathrm{\mathfrak h}}\) as a closed subvariety of \(\mathbb{C}^d\) via \({\rm emb}''.\) By Proposition 6, it is clear that \(\bar \mathop{\mathrm{\mathfrak h}}\) is isomorphic to the vanishing locus \(V(I(\mathop{\mathrm{\mathfrak h}})^h)\) of the multi-homogenization of \(I(\mathop{\mathrm{\mathfrak h}}).\) We aim to find a multi-homogeneous ideal \(J\) (possibly properly) contained in \(I(\mathop{\mathrm{\mathfrak h}})^h\) such that the vanishing locus of \(J\) is \(\bar \mathop{\mathrm{\mathfrak h}}\), at least set theoretically, and computation with \(J\) is easier than with \(I(\mathop{\mathrm{\mathfrak h}})^h.\)

Write \(Z \subseteq (\mathbb{P}^1)^d\) for the vanishing locus of \(J(\Phi).\) It is clear from the definition of \(J(\Phi)\) that \[Z \supseteq \bar \mathop{\mathrm{\mathfrak h}}\supseteq \mathop{\mathrm{\mathfrak h}}.\] To analyze the difference \(Z - \mathop{\mathrm{\mathfrak h}}\), we need to introduce some notation. A root subsystem \(\Phi'\) of \(\Phi\) is called closed if for all \(\lambda, \mu \in \Phi'\) with \(\lambda + \mu \in \Phi\), one has \(\lambda + \mu \in \Phi'.\) Let \((x_{\lambda})_{\lambda \in \Phi^+}\) be a point of \(Z - \mathop{\mathrm{\mathfrak h}}\subseteq (\mathbb{P}^1)^d.\) Define \[\begin{align} \label{findef} {\rm Fin} := \{\lambda \in \Phi^+: x_{\lambda} \neq \infty\}. \end{align}\tag{3}\] The following lemma can be easily deduced from the fact that if \(\lambda, \mu, \lambda+\mu \in \Phi^+\), then \(x_{\lambda+\mu}-x_{\lambda}-x_{\mu}\) vanishes on \(\mathop{\mathrm{\mathfrak h}}.\)

Denote by \(\mathcal{S}\) the set of all closed root subsystems of \(\Phi\) of rank \(r - 1,\) where \(r = \mathop{\mathrm{rk}}\Phi.\) The set \(\mathcal{S}\), together with inclusion, is a poset.

Let \[P = \sum\limits_{\lambda \in \Phi^+} c_{\lambda} x_{\lambda}\] be a linear polynomial in \(R.\) The support \(\text{supp}(P)\) of \(P\) is defined to be the set \[\{\lambda \in \Phi^+: c_{\lambda} \neq 0\}.\]

For each good root subsystem \(\Phi'\) of \(\Phi\), choose, once and for all, a semisimple Lie subalgebra \(\mathop{\mathrm{\mathfrak g}}'\) of \(\mathop{\mathrm{\mathfrak g}}\) and a Cartan subalgebra \(\mathop{\mathrm{\mathfrak h}}'\) of \(\mathop{\mathrm{\mathfrak g}}'\) such that the root system for \((\mathop{\mathrm{\mathfrak g}}', \mathop{\mathrm{\mathfrak h}}')\) is isomorphic to \(\Phi'.\) We may choose \((\mathop{\mathrm{\mathfrak g}}', \mathop{\mathrm{\mathfrak h}}')\) such that \(\mathop{\mathrm{\mathfrak h}}'\) is a Lie subalgebra of \(\mathop{\mathrm{\mathfrak h}}.\) Fix a good root subsystem \(\Phi'\) of \(\Phi.\) A linear polynomial \(P \in R\) that vanishes on \(\mathop{\mathrm{\mathfrak h}}\) is said to be of type I if \(\text{supp}(P) \subseteq \Phi'\), and is said to be of type II if there are at least two elements of \(\text{supp}(P)\) which are not contained in \(\Phi'.\) Consider a point \(x = (x_{\lambda})_{\lambda \in \Phi^+}\) of \((\mathbb{P}^1)^d\) such that \(x_{\lambda} = \infty\) for all \(\lambda \notin \Phi'.\) Then, for every \(P\) of type II, the multi-homogenization \(P^h\) automatically vanishes at \(x.\) Hence, the point \(x\) is in \(Z\) if and only if \(P^h\) vanishes at \(x\) for all \(P\) of type I. Put \((\Phi')^+ := \Phi' \cap \Phi^+.\) Then \(x\) is in \(Z\) if and only if \((x_{\lambda})_{\lambda \in (\Phi')^+} \in (\mathbb{P}^1)^{(\Phi')^+}\) is in the vanishing locus of the multi-homogeneous ideal \(J(\Phi')\), defined similarly as in Definition 4.

By definition, Lemma 4, and our analysis above, it is clear that \[C({\Phi'}) \subseteq Z - \mathop{\mathrm{\mathfrak h}}\] for all good root subsystems \(\Phi'.\)

In the course of proving Theorem 8, we have proved

From now on, we view each \(\mathop{\mathrm{\mathfrak h}}'\) as a subset of \(\bar \mathop{\mathrm{\mathfrak h}}\) via the composition \[\begin{align} \mathop{\mathrm{\mathfrak h}}' \lhook\joinrel\longrightarrow \bar \mathop{\mathrm{\mathfrak h}}' \cong C(\Phi') \lhook\joinrel\longrightarrow \bar \mathop{\mathrm{\mathfrak h}}. \end{align}\] We remark that the ideal \(J(\Phi)\) that we used to compute the boundary of \(\mathop{\mathrm{\mathfrak h}}\), a priori, could be properly contained in \(I(\mathop{\mathrm{\mathfrak h}})^h.\) However, it is proved in [6] that these two ideals are actually equal.

For each \(k\)-step good root subsystem \(\Psi\), we define \(\mathring C(\Psi)\) as in Equation (4 ) and define \(C(\Psi)\) as in Definition 6.

We denote the resulting stratification of \(\bar \mathop{\mathrm{\mathfrak h}}\) by \({\mathcal{C}}.\) We are grateful to Matt Douglass for pointing out the relation between good and parabolic root subsystems.

The proof is immediate from Proposition 10, since maximal proper parabolic root subsystems are precisely those given by omitting one simple root, up to \(W\)-conjugacy.

In a previous version of this paper, we were unaware of the connection with parabolic root subsystems, and gave a construction of good root subsystems using a variant of the Borel-de Siebenthal algorithm [12]. Proposition 10 makes this construction unnecessary, so we have omitted it. We thank Matt Douglass and the referee for their help with this issue.

3.2 Normality of \(\bar \mathop{\mathrm{\mathfrak h}}\)↩︎

We now proceed to show \(\bar \mathop{\mathrm{\mathfrak h}}\) is regular in codimension one. It is evident that every point of \(\mathop{\mathrm{\mathfrak h}}\) is regular. Let \(x = (x_{\lambda})_{\lambda \in \Phi^+} \in (\mathbb{P}^1)^d.\) We consider affine charts of \((\mathbb{P}^1)^d\) containing \(x\) of the form \(\prod \limits_{\lambda \in \Phi^+} U_{\lambda}\), where for each \(\lambda \in \Phi^+\), \(U_{\lambda}\) is either \(\mathbb{C}\) or \(\mathbb{P}^1 - \{0\}\) and \(x_{\lambda} \in U_{\lambda}.\) Given an affine chart \(U\) containing \(x = (x_{\lambda})_{\lambda \in \Phi^+}\), for \(\lambda \in \Phi^+\), we let \(z_{\lambda}\) be the inhomogeneous coordinate on \(U_{\lambda}.\) Namely, we define \[\begin{align} z_{\lambda} := \begin{cases} x_{\lambda,1}/x_{\lambda,0} & ~ \text{if} ~ U_{\lambda} = \mathbb{C} \\ x_{\lambda,0}/x_{\lambda,1} & ~ \text{if} ~ U_{\lambda} = \mathbb{P}^1 - \{0\}. \end{cases} \end{align}\] Let \(R_U := \mathbb{C}[z_{\lambda}: \lambda \in \Phi^+]\) be the coordinate ring of \(U\) and \(\mathfrak p \lhd R_U\) the ideal of polynomials in \(R_U\) that vanish on \(\bar \mathop{\mathrm{\mathfrak h}}\cap U.\) Since \(\mathop{\mathrm{\mathfrak h}}\) is irreducible, so is \(\bar \mathop{\mathrm{\mathfrak h}}.\) It follows that \(\bar \mathop{\mathrm{\mathfrak h}}\cap U\) is irreducible and, hence, equidimensional. Therefore, we can apply the Jacobian criterion to prove regularity of \(x.\) Concretely, to prove that \(x\) is regular, we need to find \(d - r\) polynomials in \(\mathfrak p\) such that their differentials evaluated at \(x\) are linearly independent.

To prove that \(\bar \mathop{\mathrm{\mathfrak h}}\) is normal, we use the following results. Recall that \(S\) is the multi-homogeneous coordinate ring of \((\mathbb{P}^1)^d.\)

3.3 Weyl Group Action on \(\bar \mathop{\mathrm{\mathfrak h}}\)↩︎

Let \(W\) be the Weyl group for \((\mathop{\mathrm{\mathfrak g}}, \mathop{\mathrm{\mathfrak h}}).\) The group \(W\) acts on \((\mathbb{P}^1)^{\Phi}\) by \[\begin{align} \label{Waction} w \cdot (x_{\lambda})_{\lambda \in \Phi} := (x_{w^{-1}(\lambda)})_{\lambda \in \Phi}. \end{align}\tag{6}\] With this \(W\)-action, the morphism \(\mathop{\mathrm{\mathfrak h}}\rightarrow (\mathbb{P}^1)^{\Phi}\) sending \(h \in \mathop{\mathrm{\mathfrak h}}\) to the point whose \(\lambda\)-component is \(\lambda(h)\) is \(W\)-equivariant. From this it follows that \(\bar \mathop{\mathrm{\mathfrak h}}\) is a \(W\)-stable subset of \((\mathbb{P}^1)^{\Phi}.\) Hence we have proved

Let \((\mathbb{P}^1 - \{0\})^d\) be the affine chart of \((\mathbb{P}^1)^d\) consisting of points all of whose components are nonzero. The intersection \(\bar \mathop{\mathrm{\mathfrak h}}\cap (\mathbb{P}^1 - \{0\})^d\) is exactly the reciprocal variety of \(\mathop{\mathrm{\mathfrak h}}\) studied in [11], and is an affine open neighborhood of the most singular point \((\infty, \ldots, \infty)\) of \(\bar \mathop{\mathrm{\mathfrak h}}\), as is evident in the following example.

4 Cohomology of \(\bar{\mathop{\mathrm{\mathfrak h}}}\) and the Coxeter Arrangement↩︎

In this section, we establish a bijection between the poset given by the strata of \(\bar \mathop{\mathrm{\mathfrak h}}\) with the intersection lattice of the Coxeter arrangement defined by the root system \(\Phi.\) As a consequence, we deduce the Betti numbers of \(\bar \mathop{\mathrm{\mathfrak h}}\) from known results in [8] in the classical cases, and from results in [9] or alternatively SageMath computations in the exceptional cases.

Recall the stratification \(\mathcal{C}\) of \(\bar \mathop{\mathrm{\mathfrak h}}= \sqcup \;\mathring C(\Psi)\) from Corollary 1, and note the following easy consequence of this stratification. For \(\Psi\) a \(k\)-step good root subsystem of \(\Phi\), we consider the corresponding homology fundamental class \(\mathop{\mathrm{Fund}}_{\Psi} \in H_{2(r-k)}(\bar \mathop{\mathrm{\mathfrak h}}, \mathbb{Z})\) determined by the \((r-k)\)-dimensional subvariety \(C(\Psi)\), and the corresponding cohomology class \(\xi_{\Psi} \in H^{2(r-k)}(\bar \mathop{\mathrm{\mathfrak h}}, \mathbb{Z})\) defined by the property that \(\langle \xi_{\Psi}, \mathop{\mathrm{Fund}}_{\Theta} \rangle = \delta_{\Psi, \Theta}\) for each \(k\)-step good root subsystem \(\Theta\) of \(\Phi.\)

Thus, in order to determine the Betti numbers of \(\bar \mathop{\mathrm{\mathfrak h}}\), we need to count the number of strata in \(\mathcal{C}\) in each dimension. After a previous version of this paper was posted on the arXiv, we learned from Matt Douglass that, based on results proved in that version of this paper, the counting can be deduced from work of Orlik and Terao [8].

4.1 \(\bar{\mathop{\mathrm{\mathfrak h}}}\) and the Coxeter Arrangement↩︎

We start by giving another characterization of the \(k\)-step good root subsystems.

An easy consequence of Corollary 1 and Lemma 5 is

Let \(\mathcal{A}\) be the Coxeter arrangement. In other words, \(\mathcal{A}\) consists of the root hyperplanes in \(\mathop{\mathrm{\mathfrak h}}\), where by a root hyperplane, we mean a hyperplane in \(\mathop{\mathrm{\mathfrak h}}\) of the form \[\lambda^{\perp} := \{x \in \mathop{\mathrm{\mathfrak h}}: \lambda (x) = 0\} \text{ for some } \lambda \in \Phi^+.\]

It is evident that the \(W\)-action on \(\mathop{\mathrm{\mathfrak h}}\) induces a \(W\)-action on \(L(\mathcal{A}).\)

Define maps \[\begin{align} F: L(\mathcal{A}) \longrightarrow \mathcal{C}, ~ X \longmapsto \mathring C(X^{\perp} \cap \Phi) \quad \text{and} \quad G: \mathcal{C} \longrightarrow L(\mathcal{A}), ~ \mathring C(\Psi) \longmapsto \Psi^{\perp}. \end{align}\]

Using Theorem 19, we can interpret the Betti numbers of \(\bar \mathop{\mathrm{\mathfrak h}}\) in terms of \(L(\mathcal{A}).\)

As a consequence of Theorem 19, we have

It is worth noticing that \(w_k(L(\mathcal{A}))\) also has topological meaning. To explain this, consider the complex manifold \[M := \mathop{\mathrm{\mathfrak h}}- \bigcup \limits_{\lambda \in \Phi^+} \lambda^{\perp}.\] The space \(M\) was studied extensively by Artin [18], Brieskorn [19], and Orlik and Solomon [20]. For \(0 \le k \le \dim \mathop{\mathrm{\mathfrak h}}\), write \(f'(\Phi, k)\) for the \((\dim \mathop{\mathrm{\mathfrak h}}- k)\)th Betti number of \(M.\) Orlik and Solomon proved that

The two kinds of Whitney numbers are related by a Möbius inversion formula, which is a form of duality in combinatorics. It is interesting to look for a topological/geometric duality between the spaces \(\bar \mathop{\mathrm{\mathfrak h}}\) and \(M\) that explains the combinatorial duality of their Betti numbers.

4.2 Betti Numbers of \(\bar \mathop{\mathrm{\mathfrak h}}\)↩︎

In this section, we use the results of the previous section to determine the Betti numbers of \(\bar{\mathop{\mathrm{\mathfrak h}}}\) for each simple Lie algebra. When the root system \(\Phi\) is of classical type, we give the answer in terms of familiar combinatorial invariants using formulas from [8], and when \(\Phi\) is of exceptional type, we deduce the answers from [9].

By Corollary 4, the Betti numbers of \(\bar \mathop{\mathrm{\mathfrak h}}\) when \(\Phi\) is of classical type can be computed by counting the subspaces in the corresponding hyperplane arrangement, which are determined in [8]. We are grateful to Matt Douglass for pointing out these formulas to us. In a previous version of this paper, we developed an intricate combinatorics in order to count the \(k\)-step good root subsystems, which is now unnecessary in view of [8].

We computed the Betti numbers for the exceptional Lie algebras using SageMath, and are grateful to the referee for pointing out that they can be deduced from the tables in [9]. In fact, in [9], the row labeled \(A_0\) counts parabolic root subsystems in the given root subsystem in a particular \(W\)-orbit of parabolic root subsystem. Thus, the \(2(r-k)\) Betti number of \(\bar \mathop{\mathrm{\mathfrak h}}\) can be computed from the corresponding table in [9], by summing over \(W\)-orbits with a parabolic root subsystem of rank \(k\) in the table. For example, the entry \(24\) for \(f(F_4,3)\) may be obtained from table V in [9] by summing the entries in the columns for \(A_1\) and \(\tilde{A}_1\), and the entry \(122\) for \(f(F_4,2)\) by summing the entries from the 4 columns for rank 2 root systems in table V.

The generating function of the numbers \(f(\Phi,k)\) can be computed using Theorem 22.

Using the well-known formula \[\sum_{n=0}^{\infty} \sum_{k=0}^n S(n,k)\frac{x^n}{n!}y^k = e^{y(e^x - 1)}\] for the Stirling numbers of the second kind, we obtain the following corollary of Theorem 22.

For types \(B\), \(C\) and \(D\), the following slight modification of \(F_A(q,t)\) is more convenient.

The following corollary follows easily from Theorem 22 and [21].

5 Topology of \(\bar{\mathop{\mathrm{\mathfrak h}}}\)↩︎

In this section, we determine the Weyl group representation on \(H^{\bullet}(\bar \mathop{\mathrm{\mathfrak h}}, \mathop{\mathrm{\mathbb{C}}})\) and determine the cup product structure of \(H^{\bullet}(\bar \mathop{\mathrm{\mathfrak h}}, \mathop{\mathrm{\mathbb{Z}}}).\)

5.1 Weyl Group Representation↩︎

Recall the definition of \(L(\mathcal{A})\) in Definition 10. Let \(\mathbb{C} \cdot L(\mathcal{A})\) be the \(\mathbb{C}\)-vector space with \(L(\mathcal{A})\) as a basis. It is graded by declaring that \(X \in L(\mathcal{A})\) is in degree \(2\mathop{\mathrm{rk}}X.\) It follows from Theorem 19 that

For \(X \in L(\mathcal{A})\), define \[W^X := \{w \in W: w ~ \text{fixes every point of} ~ X\}.\] By definition, \(W^X\) is a parabolic subgroup of \(W.\) By Steinberg [22], \(W^X\) is the reflection group generated by the reflections associated to \(\lambda \in X^{\perp}.\) The following theorem is due to Orlik and Solomon [9].

Moreover, let \(N(W^X)\) be the normalizer of \(W^X\) in \(W.\) Orlik and Solomon [9] also proved

Detailed analysis of the structure of the group \(N(W^X)\) can be found in the work of Howlett [23]. In particular, Howlett proved that \(N(W^X)\) is the semidirect product of \(W^X\) with the subgroup of \(W\) consisting of those elements which map a given set of Coxeter generators of \(W^X\) to itself.

Following Orlik and Solomon [9], we refer to the \(W\)-conjugacy class of a Coxeter element of a parabolic subgroup of \(W\) as a parabolic class. By Theorem 24, the map \[\begin{align} \{W \text{-orbits in} ~ L(\mathcal{A})\} & \longrightarrow \{\text{parabolic classes in} ~ W\} \\ \quad W \cdot X & \longmapsto ~ \text{conjugacy class of a Coxeter element of} ~ W^X \end{align}\] is a well-defined bijection.

The set \(\mathcal{P}\) of parabolic classes in \(W\) has been classified by many different authors [9], [24], [25]. For a parabolic class \(c\), choose an element \(X\) of \(L(\mathcal{A})\) such that \(c\) is a Coxeter element of \(W^X.\) This is always possible by the bijection in the previous paragraph. For the sake of simplicity, we write \(N(c)\) for the normalizer \(N(W^X).\) We define the rank \(\mathop{\mathrm{rk}}N(c)\) of \(N(c)\) to be the rank of the parabolic subgroup \(W^X.\) Although \(N(c)\) is only well-defined up to conjugation in \(W\), its rank is well-defined.

As a corollary of Theorem 19, Theorem 24 and Proposition 25, we have

Let \(\bar H\) be the toric variety associated with the fan whose maximal cones are the Weyl chambers. It is known that there is a \(W\)-action on \(\bar H\) extending the \(W\)-action on \(H\), so there is an induced \(W\)-action on \(H^{\bullet}(\bar H, \mathop{\mathrm{\mathbb{C}}}).\) It is interesting to compare Corollary 8 with the following result of Stembridge [26]. In particular, it is interesting to relate the set \(\mathcal{P}\) of parabolic classes to the \(W\)-orbits in the set \(\mathcal{Q}\) that appears below, noting that \(\mathcal{P}\) has cardinality at most \(2^{\mathop{\mathrm{rk}}\Phi}\).

5.2 Cup Product↩︎

For \(X \in L(\mathcal{A})\), recall that \(F(X)= \mathring C(\Psi)\) for some \((r-k)\)-step good root subsystem \(\Psi=\Psi_X\) of \(\Phi\) and let \(k = \dim(\mathring C(\Psi)).\) Let \(\mathop{\mathrm{Fund}}_X =[C(\Psi_X)] \in H_{2k}(\bar \mathop{\mathrm{\mathfrak h}}, \mathop{\mathrm{\mathbb{Z}}})\) denote the cellular homology class represented by the stratum \(F(X) \in \mathcal{C},\) and let \(\{ \xi_X : X \in L(\mathcal{A})\}\) denote the dual basis of cohomology. Recall that \[H_{\bullet}(\bar \mathop{\mathrm{\mathfrak h}}, \mathop{\mathrm{\mathbb{Z}}}) = \bigoplus \limits_{X \in L(\mathcal{A})} \mathbb{Z} \cdot \mathop{\mathrm{Fund}}_X \quad \text{and} \quad H^{\bullet}(\bar \mathop{\mathrm{\mathfrak h}}, \mathop{\mathrm{\mathbb{Z}}}) = \bigoplus \limits_{X \in L(\mathcal{A})} \mathbb{Z} \cdot \xi_X.\] The cup product of the \(\xi_X\)’s is computed in the following

To prove this result, we recall some standard facts about intersection theory on the smooth variety \((\mathbb{P}^1)^d,\) where \(d=|\Phi^+|.\) If \(\eta \subseteq \Phi^+\), we consider the subvariety \[Q_{\eta} =\{(x_{\lambda} ) \in (\mathbb{P}^1)^d : x_{\lambda}=0 \;\forall \lambda \not\in \eta \}.\] We note that \(Q_{\eta} \cong (\mathbb{P}^1)^{|\eta|}\) and let \([Q_{\eta}] \in H_{2|\eta|} ((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}})\) be the homology class defined by the submanifold \(Q_{\eta}\), and let \(\{ u_{\eta} \}\) denote the corresponding dual basis of \(H^{\bullet}((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}}).\) Then \(H^2((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}})\) is generated by the elements \(u_{\lambda} \in H^2((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}}),\) and \[H^{\bullet} ((\mathbb{P}^1)^{d}, \mathop{\mathrm{\mathbb{Z}}}) \cong \prod \limits_{\lambda \in \Phi^+} \mathbb{Z}[u_{\lambda}]/(u_{\lambda}^2) \cong \bigoplus \limits_{\substack{\{ \lambda_1, \ldots, \lambda_k\} \subseteq \Phi^+ }} \mathbb{Z} \cdot u_{\lambda_1} \cdots u_{\lambda_k}.\] There is a Poincaré duality isomorphism \(\mathcal{D}:H_{2\bullet}((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}}) \to H^{2(d-\bullet)}((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}})\), and \(\mathcal{D}([Q_{\eta}])=u_{\Phi^+ - \eta}.\) We consider also the intersection pairing \[H_{2k}((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}}) \otimes H_{2\ell}((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}}) \longrightarrow H_{2(k + \ell - d)}((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}}), \;[X]\cdot [Y]:=(\mathcal{D})^{-1}(\mathcal{D}(X)\smile \mathcal{D}(Y)),\] for \([X] \in H_{2k}((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}})\) and \([Y] \in H_{2\ell}((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}}).\) Further, for \(\eta_1, \eta_2 \subseteq \Phi^+,\) \[\begin{align} u_{\eta_1} \smile u_{\eta_2}= \begin{cases} u_{\eta_1 \cup \eta_2} & ~ \text{if} ~ \eta_1 \cap \eta_2 = \emptyset \\ 0 & ~ \text{else}, \end{cases} \end{align}\] and if \(|\eta_2|=d-|\eta_1|,\) \[\label{pairing} \begin{align} [Q_{\eta_1}]\cdot [Q_{\eta_2}]= \begin{cases} [pt] & ~ \text{if} ~ \eta_2 = \Phi^+ - \eta_1 \\ 0 & ~ \text{else}. \end{cases} \end{align}\tag{7}\]

As a consequence, if \(\eta = \{ \lambda_1, \dots, \lambda_k \}\), then \[u_{\eta}= \prod \limits_{i=1, \dots, k} u_{\lambda_i}.\]

Note also that there is a cycle map \(A^{\bullet}((\mathbb{P}^1)^d) \to H^{2\bullet}((\mathbb{P}^1)^d, \mathop{\mathrm{\mathbb{Z}}})\) which is a ring isomorphism ([27]). In particular, we can use assertions about the product in the Chow ring to compute the intersection pairing in homology.

Consider the natural inclusion \(\iota: \bar \mathop{\mathrm{\mathfrak h}}\hookrightarrow (\mathbb{P}^1)^d.\) For a \((r-k)\)-step good root subsystem \(\Psi\) of \(\Phi^+\), let \(LI(\Psi)\) be the set of \(k\)-element subsets \(\eta = \{ \lambda_1, \dots, \lambda_k \}\) such that \(\lambda_1, \dots, \lambda_k\) is a basis of a Cartan subalgebra \(\mathop{\mathrm{\mathfrak h}}_{\Psi}\) contained in \(\mathop{\mathrm{\mathfrak h}}.\) Recall that \(\mathop{\mathrm{\mathfrak h}}_{\Psi}\) is a Cartan subalgebra of a semisimple Lie algebra \(\mathop{\mathrm{\mathfrak g}}_{\Psi}\) such that \(\Psi\) is the roots of \(\mathop{\mathrm{\mathfrak g}}_{\Psi}\) with respect to \(\mathop{\mathrm{\mathfrak h}}_{\Psi}.\) We prove

Sam Evens: Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556. Email: sevens@nd.edu

Yu Li: Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556. Email: yli234@nd.edu

References↩︎