Symmetry-Based Subvarieties

• Symmetry-based subvarieties are geometric objects defined by intrinsic symmetries that compactify parameter spaces and stratify orbits in algebraic geometry.
• They play a key role in representation theory by encoding rigidity phenomena and facilitating the classification of orbit types in symmetric pairs.
• Explicit examples, such as varieties of reductions and Shimura subvarieties, illustrate their Fano properties and applications in moduli problems and deformation analysis.

A symmetry-based subvariety is a geometric object arising as a subvariety or compactification in an ambient parameter space, defined by intrinsic symmetries of the underlying structure. In the context of algebraic geometry and representation theory, notable families include varieties of reductions for symmetric pairs, orbit-type subvarieties associated with group actions, and loci defined by symmetry constraints in moduli or parameter spaces. These subvarieties play a critical role as geometric compactifications, orbit stratifications, and as testing grounds for rigidity and representation-theoretic phenomena, often exhibiting rich Fano-theoretic, combinatorial, or cohomological properties.

1. Varieties of Reductions for Reductive Symmetric Pairs

Let $G$ be a connected complex reductive group and $\theta: G \to G$ an involutive automorphism. The subgroup $K=(G^\theta)^\circ$ and the Lie algebra decomposition $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ (Cartan decomposition) define the reductive symmetric pair $(G, K)$.

The fundamental symmetry-based subvariety here is the variety of reductions $\Red$, which compactifies the space of Cartan subspaces in $\mathfrak{p}$: $\Red = \overline{\Red_o} \subset \mathrm{Grass}(r,\mathfrak{p})$ where $r = \operatorname{rank}(G,\theta)$ and $\Red_o$ is the $K$-orbit of Cartan subspaces of $\mathfrak{p}$. Key properties of $\Red$:

• $\Red$ is a projective variety with $\dim \Red = \dim \mathfrak{p} - r$ if $\dim \mathfrak{p} = \dim G - \dim K$.
• The open stratum $\Red_o$ is isomorphic to the affine symmetric space $K/N_K(\mathfrak{a})$, where $\mathfrak{a}$ is a Cartan subspace.
• The singular locus is a quadric section defined by the Killing form, with $\Red_s = \Red \cap Q$ a divisor.
• Minimal rational curves give positivity of $-K_{\Red}$; in favorable cases, $\Pic(\Red)\cong\mathbb{Z}$ and $\Red$ is a Fano variety.
• All geometric invariants of $\Red$—dimension, singular locus, anticanonical class, and orbit stratification—are driven by the representation theory of $(G,\theta)$, making $\Red$ the prototypical symmetry-based subvariety in this context (Grünewald, 2010).

2. Fano Properties and Explicit Compactifications

For low rank symmetric pairs, the variety of reductions has been explicitly identified:

• $(\mathrm{SL}_3, \mathrm{SO}_3)$, $(\mathrm{SL}_3\times\mathrm{SL}_3, \mathrm{SL}_3)$, $(\mathrm{SL}_6, \mathrm{Sp}_6)$, $(E_6,F_4)$ each yield a smooth Fano $\Red$ that compactifies an affine space of dimension $r$ with $b_2=1$.
• Companion analyses give further examples: $(\mathrm{SL}_4,\mathrm{SO}_4)$, Cartesian squares of $\mathrm{Sp}_4$, $G_2$ produce new Fano varieties.
• Minimal rational curves (lines) through a general point in $\Red$ provide estimates for the Fano index, encoding positivity of the anticanonical bundle, and the geometry of these curves is determined by the group-theoretic structure (Grünewald, 2010).

3. Rigidity and Deformation Phenomena

A key rigidity theorem governs the behavior of abelian subalgebras (Cartan-type or “anisotropic tori”) in $\mathfrak{p}$ under degeneration:

• If $\Gamma_1\subset \mathfrak{p}$ is an abelian subalgebra and $\Gamma_0$ lies in the closure of its $K$-orbit, unless $\Gamma_0$ is purely nilpotent, every semisimple element of $\Gamma_0$ arises as a limit of semisimple elements in $\Gamma_1$.
• This “rigidification” ensures the structural integrity of the semisimple part under deformations and underpins the stratification of $\Red$ and its orbit geometry (Grünewald, 2010).
• The proof leverages geometric invariant theory, Smith normal form for modules over local rings, and analysis of Chevalley–Jordan decomposition stability.

4. Stratification and Orbit Correspondence

The geometry of symmetry-based subvarieties such as $\Red$ is tightly linked to representation-theoretic stratifications:

• The centralizer map sends a regular element $x\in\mathfrak{p}$ to its anisotropic centralizer, inducing a bijection between regular decomposition classes in $\mathfrak{p}$ and regular $K$-orbits in $\Red$.
• The closure of a decomposition class in $\mathfrak{p}$ is always a union of decomposition classes; this structure is mirrored by the stratification of $\Red$ by $K$-orbit types and relates directly to representation-theoretic invariants like branching rules (Grünewald, 2010).
• Degeneration of orbits is reflected on both algebraic and geometric sides, enabling comparison of representation-theoretic and geometric compactifications.

5. Generalizations: Symmetric $k$-Varieties and Beyond

The essential concepts extend naturally to symmetric $k$-varieties for arbitrary fields $k$:

• For $(G,\theta, H)$ over $k$, the symmetric variety $X_k = G_k/H_k$ inherits a partition structure for minimal parabolic $k$-orbits expressed using Weyl group quotients indexed by $\theta$-stable $k$-split tori.
• The rational double coset orbits $P_k\backslash G_k/H_k$ always embed into their geometric (algebraically closed) counterparts, and surjectivity is characterized by the splitting properties of the fixed point group $H$—specifically, $\mathrm{krank}(H) = \mathrm{rank}(H)$ (Hunnell, 2019).
• This framework illustrates how symmetry-based subvarieties organize orbit combinatorics and reveal obstructions to “lifting” orbit types across field extensions.

6. Connections to Representation Theory and Moduli

Symmetry-based subvarieties serve as universal models for interpreting structure in representation theory and related parameter spaces:

• In quiver varieties, fixed loci under diagram automorphisms and involutive automorphisms model Springer-type resolutions, tying the geometry of symmetric spaces to universal enveloping algebras of subalgebras in symmetric pairs (Li, 2018).
• In moduli problems, such as Shimura subvarieties in the Prym locus, symmetry by a finite group action induces special subvarieties whose geometry and period maps reflect the representation content of associated covers (Grosselli et al., 2021).
• Symmetry-based structures are also central in the classification of invariant subvarieties under large group actions (e.g., the infinite symmetric group in infinite-dimensional affine or matrix spaces), dictating both the equations and combinatorial structure of irreducible components (Nagpal et al., 2020, Draisma et al., 2021).

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In summary, symmetry-based subvarieties—exemplified by varieties of reductions, orbit-type loci, or compactifications defined via group actions—form a unifying class of geometric objects whose structure, singularities, and positivity properties are fully controlled by underlying symmetries. Their paper provides deep connections between algebraic geometry, representation theory, and the theory of algebraic groups, continuously motivating new compactifications and revealing rigidity and orbit structures that drive classification results and applications across multiple fields (Grünewald, 2010, Hunnell, 2019, Grosselli et al., 2021, Nagpal et al., 2020, Draisma et al., 2021, Li, 2018).