Wonderful Compactification of a Cartan Subalgebra of a Semisimple Lie Algebra
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910277714313216 |
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| author | Evens, Sam Li, Yu |
| author_facet | Evens, Sam Li, Yu |
| contents | Let $\mathfrak h$ be a Cartan subalgebra of a complex semisimple Lie algebra $\mathfrak g.$ We define a compactification $\bar {\mathfrak h}$ of $\mathfrak h$, which is analogous to the closure $\bar H$ of the corresponding maximal torus $H$ in the adjoint group of $\mathfrak g$ in its wonderful compactification, which was introduced and studied by De Concini and Procesi \cite{DCP}. We observe that $\bar {\mathfrak h}$ is a matroid Schubert variety and prove that the irreducible components of the boundary $\bar {\mathfrak h} - \mathfrak h$ of $\mathfrak h$ are divisors indexed by root system data. We prove that $\bar {\mathfrak h}$ is a normal variety and find an affine paving of $\bar {\mathfrak h},$ where the strata are given by the orbits of $\mathfrak h.$ We show that the strata of $\bar {\mathfrak h}$ correspond bijectively to subspaces of the corresponding Coxeter hyperplane arrangement studied by Orlik and Solomon, and prove that the associated posets are isomorphic. As a consequence, we express the Betti numbers of $\bar {\mathfrak h}$ in terms of well-known combinatorial invariants in the classical cases. We show that the Weyl group $W$ acts on $\bar {\mathfrak h}$, and describe $H^{\bullet}(\bar {\mathfrak h}, \mathbb C)$ as a representation of $W$, and compute the cup product for $H^{\bullet}(\bar {\mathfrak h}, \mathbb Z)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_19936 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wonderful Compactification of a Cartan Subalgebra of a Semisimple Lie Algebra Evens, Sam Li, Yu Representation Theory Algebraic Geometry Combinatorics Symplectic Geometry Let $\mathfrak h$ be a Cartan subalgebra of a complex semisimple Lie algebra $\mathfrak g.$ We define a compactification $\bar {\mathfrak h}$ of $\mathfrak h$, which is analogous to the closure $\bar H$ of the corresponding maximal torus $H$ in the adjoint group of $\mathfrak g$ in its wonderful compactification, which was introduced and studied by De Concini and Procesi \cite{DCP}. We observe that $\bar {\mathfrak h}$ is a matroid Schubert variety and prove that the irreducible components of the boundary $\bar {\mathfrak h} - \mathfrak h$ of $\mathfrak h$ are divisors indexed by root system data. We prove that $\bar {\mathfrak h}$ is a normal variety and find an affine paving of $\bar {\mathfrak h},$ where the strata are given by the orbits of $\mathfrak h.$ We show that the strata of $\bar {\mathfrak h}$ correspond bijectively to subspaces of the corresponding Coxeter hyperplane arrangement studied by Orlik and Solomon, and prove that the associated posets are isomorphic. As a consequence, we express the Betti numbers of $\bar {\mathfrak h}$ in terms of well-known combinatorial invariants in the classical cases. We show that the Weyl group $W$ acts on $\bar {\mathfrak h}$, and describe $H^{\bullet}(\bar {\mathfrak h}, \mathbb C)$ as a representation of $W$, and compute the cup product for $H^{\bullet}(\bar {\mathfrak h}, \mathbb Z)$. |
| title | Wonderful Compactification of a Cartan Subalgebra of a Semisimple Lie Algebra |
| topic | Representation Theory Algebraic Geometry Combinatorics Symplectic Geometry |
| url | https://arxiv.org/abs/2411.19936 |