Matsuki Duality in Algebraic Geometry

• Matsuki duality is a framework establishing a bijection between Borel orbits and symmetric subgroup orbits on flag varieties, pivotal for understanding representation theory.
• It is integrated with Luna–Vust theory by encoding combinatorial invariants such as weight lattices and spherical roots that classify orbit decompositions.
• The duality underpins the classification of spherical subgroups and informs orbit dimension analysis and deformation stability across varying characteristics.

A Matsuki duality in representation theory and algebraic geometry refers to intricate correspondences between orbits of subgroups on certain algebraic varieties, most notably flag varieties, with applications in the structure and classification of spherical subgroups. In the modern context, Matsuki duality has deep connections to the geometry of spherical varieties, Luna–Vust theory, and the classification of subgroups with open orbits on flag varieties. The following exposition uses the rigorous framework and terminology of the Luna–Vust theory and subsequent developments, as detailed in the classification of spherical subgroups and their orbits on flag varieties, particularly in simple algebraic groups over algebraically closed fields (Knop et al., 2013).

1. Spherical Subgroups and Flag Varieties

A subgroup $H$ of a simple algebraic group $G$ over an algebraically closed field $k$ is called spherical if it admits a dense orbit on the flag variety $G/B$. The flag variety $G/B$ is a smooth projective variety upon which $G$ acts transitively; the multiplicity-free property for the associated coordinate rings or representations is equivalent to the existence of an open $B$-orbit on $G/H$ (Knop et al., 2013).

Several characterizations are equivalent:

• $H$ has a dense orbit on $G/B$;
• The Borel subgroup $B$ acts on $G/H$ with an open dense orbit;
• The homogeneous space $G/H$ has only finitely many $B$-orbits;
• (When $\operatorname{char}k=0$) $k[G/H]$ is a multiplicity-free $G$-module.

These properties underpin the duality phenomena between various types of orbits and subgroups.

2. Matsuki Duality: Geometric and Combinatorial Aspects

In the context of real and $p$-adic groups, Matsuki duality describes the correspondence between orbits of a real form $K$ of $G$ and orbits of the Borel subgroup $B$ on the flag manifold $G/B$. The precise setting in the algebraic category generalizes this to a bijective correspondence between $K$-orbits and $B$-orbits, typically parametrized by the Weyl group or other combinatorial data.

The duality can be viewed in terms of orbits:

• For each $B$-orbit $O_B$ in $G/H$, there exists a unique $K$-orbit $O_K$ such that the intersection $O_B \cap O_K$ is non-empty and typically transverse.
• Such a duality reflects the decomposition of the flag variety into stratifications by both types of orbits.
• The intersection poset structure, closure relations, and corresponding dimension data exhibit a form of "duality" in the parametrization.

3. Luna–Vust Theory and Spherical Systems

Matsuki duality is naturally integrated into the combinatorial classification framework of Luna–Vust theory:

• The Luna datum $(\Lambda, \Sigma, S^p, D^a)$ encodes the weight lattice, the set of spherical roots, the parabolic subset, and the set of type-a colors for any spherical homogeneous space $G/H$ (Knop et al., 2013).
• Given dual pairs $(G,H)$ and their corresponding homogeneous data, the Luna–Vust machinery formalizes the duality as a combinatorial relationship between parameterizations of $B$-orbits and $K$-orbits.
• In the case of flag varieties, $B$-orbits correspond to Schubert cells, while symmetric subgroups $K$ give rise to Matsuki duals with their own orbits—both described categorically in terms of the Luna data.

4. Classification and Dimension Analysis

The classification of spherical subgroups, as given by Krämer (char $0$) and extended to positive characteristic by Brundan and Knop–Röhrle, directly identifies the cases where Matsuki duality exhibits its most robust form (Knop et al., 2013). The key tools include:

• Orbit dimension estimates: If $H\subset G$ is spherical, then

$$\dim H \ge \dim G/B = \dim G - \operatorname{rk} G,$$

and for a point $x_0$ in the open $B$-orbit,

$\dim H = \dim G/B + \dim C_B(x_0).$

• The combinatorics of the $B$-orbits, $H$-orbits, and their closures are encoded in the data of the Luna–Vust invariants and reflect the duality between the parametrizations of orbits.

This classification shows that, up to isogeny and conjugacy, all pairs $(G,H)$ admitting such duality (orbits bijection) are captured in explicit tables, and the only genuinely new cases in positive characteristic (not occurring in char $0$) are described, e.g., $$G_2\times \operatorname{Sp}(2)\subset\operatorname{Sp}(8)$$ for $p=2$.

5. Deformation Theorem and Stability of Duality

A structural result central to the extension and stability of Matsuki duality is the deformation theorem for subgroup schemes [(Knop et al., 2013), Theorem 5.1]:

• The property of being a spherical pair (and thus subject to Matsuki duality) is open in algebraic families: if a fibre $H_s \subset G_s$ is spherical for some geometric point $s$, then all fibres are spherical.
• This establishes that the duality phenomena persist over families of simple algebraic groups, and the portfolio of dual pairs remains stable under specialization.

The deformation theorem further explains why combinatorially-defined dualities (parametrized by Luna–Vust data) are robust across characteristics and base change, preserving the structure of orbit correspondences and their dimension-theoretic properties.

6. Significance, Applications, and Open Directions

Matsuki duality, especially as understood through the Luna–Vust framework and explicit orbit classification, has several impactful consequences:

• It informs the structure and geometry of all spherical varieties and has critical applications in representation theory (e.g., the description of multiplicity-free modules, harmonic analysis, and invariant theory).
• The duality is instrumental in double-coset decomposition finiteness problems and geometric constructions in positive characteristic.
• Open directions include extending the duality and classification to disconnected or non-reductive subgroup-schemes, analyzing geometry in bad characteristics, and generalizing deformation results to more general bases.

7. Broader Perspectives and Connections

While the original formulation of Matsuki duality was in the setting of real groups, its modern incarnations interact with the elaborate theory of spherical subgroups, colored fans, valuation cones, and the theory of wonderful compactifications:

• The duality connects to the geometry of orbits on flag varieties, closure relations, and the combinatorial structure of colored fans.
• The resulting invariant theory has consequences for the topology of embeddings, orbit closures, and their cohomological invariants.

The entire duality framework thus occupies a central position in the classification and geometry of homogeneous spaces, symmetric spaces, and their orbit stratifications in various group-theoretic and geometric settings (Knop et al., 2013).