[Paper Review] On the wonderful compactification

Sam Evens, Benjamin F. Jones|ArXiv.org|Jan 3, 2008

Advanced Algebra and Geometry14 references22 citations

TL;DR

This paper provides a self-contained exposition of the wonderful compactification of a complex semisimple group of adjoint type, constructing it via a projective embedding using a regular highest weight representation. The key result establishes that the compactification is smooth, with $G \times G$-orbit closures corresponding to intersections of smooth divisors, and each orbit closure fibers over a partial flag variety with a symmetric space compactification as fiber.

ABSTRACT

These lecture notes explain the construction and basic properties of the wonderful compactification of a complex semisimple group of adjoint type. An appendix discusses the more general case of a semisimple symmetric space.

Motivation & Objective

• To provide an accessible, self-contained introduction to the wonderful compactification of complex semisimple groups of adjoint type for researchers without extensive background in algebraic groups.
• To clarify the geometric and group-theoretic structure of the compactification, particularly its $G \times G$-orbit decomposition and smoothness.
• To establish the relationship between the compactification and symmetric space compactifications via an appendix on general symmetric spaces.
• To compute the cohomology of the compactification using $T \times T$-fixed points and toric methods.
• To demonstrate that the compactification is smooth and that all $G$-orbit closures are smooth and correspond to intersections of divisors $D_I$.

Proposed method

• Construct the compactification via a projective embedding of a regular highest weight representation $V(\lambda)$ of $G$.
• Define the open affine piece $\mathbb{P}_0(V)$ as the set of points with nonzero coefficient of the highest weight vector $v_0$.
• Use the action of $U^{-}$ to identify $\mathbb{P}_0(V)$ with $\mathbb{C}^n$, showing it is isomorphic to the unipotent radical.
• Construct an isomorphism $\chi: N^{-} \times \mathbb{C}^r \to X_0$ using a $G$-equivariant morphism $\nu$ and a $N^{-}$-equivariant retraction $\beta$.
• Apply the valuative criterion and Hilbert-Mumford criterion via Proposition 5.2 to show the $G$-orbit closure $V$ is projective, hence equal to $X$.
• Use toric geometry to show the closure of $T$-orbits is complete, leveraging the Weyl chamber decomposition and the little Weyl group $W_A$.

Experimental results

Research questions

• RQ1 How can the wonderful compactification of a complex semisimple group of adjoint type be constructed in a way accessible to non-experts?
• RQ2 What is the structure of the $G \times G$-orbits and their closures in the compactification?
• RQ3 Why is the wonderful compactification smooth, and how does this follow from the orbit structure and toric geometry?
• RQ4 How does the cohomology of the compactification relate to the $T \times T$-fixed points and the Weyl group?
• RQ5 What is the relationship between the group compactification and the more general symmetric space compactification?

Key findings

• The wonderful compactification $X$ of a complex semisimple group $G$ of adjoint type is smooth, as shown by constructing an isomorphism $\chi: N^{-} \times \mathbb{C}^r \to X_0$ and using completeness of toric varieties.
• The open $G \times G$-orbit $X_0$ is isomorphic to $N^{-} \times \mathbb{C}^r$, and $X_0$ is dense in $X$, with $X$ being projective and equal to the closure of the $G$-orbit of $X_0$.
• All $G$-orbit closures in $X$ are smooth and correspond to intersections $D_I = \bigcap_{i \in I} D_i$ of smooth divisors $D_i$, where $D_i$ is the closure of a $G$-orbit in $X_0$.
• The closure of the $T$-orbit in $X$ is complete, as it contains a dense $A/D$-stable toric variety with a complete fan, implying projectivity via standard toric results.
• Each $G$-orbit closure $D_I$ fibers over a partial flag variety $G/Q_I$ with fiber isomorphic to the wonderful compactification of a symmetric space associated to the Levi factor of $Q_I$.
• The cohomology of $X$ is computed via $T \times T$-fixed points, and the structure of the $T \times T$-fixed locus is tied to the Weyl group and the Weyl chamber decomposition.