G-Stability Theory & Its Applications

• G-Stability theory is a mathematical framework that formalizes stability notions for group actions using orbit geometry, numerical invariants, and group-theoretic criteria.
• It employs methods like one-parameter subgroup analysis, Kempf–Ness functions, and Harder–Narasimhan filtrations to classify objects in geometric invariant theory, harmonic analysis, and model theory.
• The theory spans diverse domains—including tensor invariants, equivariant homotopy theory, and moduli spaces—offering universal stratification and stabilization insights.

G-stability theory encompasses a set of mathematical frameworks for the analysis of stability phenomena where a group or group-like structure (denoted conventionally as $G$) acts on algebraic, geometric, analytic, or categorical data. The term refers to interrelated but distinct paradigms, each motivated by the equivariant stabilization of geometric invariant theory, moduli problems, model theory, topological dynamics, and higher category theory. Each context deploys "stability" as a precise notion—typically governed by the geometry of orbits, the asymptotics of group actions, or the combinatorics of invariants—ultimately underpinning classification, rigidity, and moduli results with intrinsic group-theoretic content.

1. G-Stability in Geometric Invariant Theory and Affine G-Varieties

In the classical setting of algebraic group actions, G-stability theory formalizes the geometric and numerical criteria for stability, semistability, and polystability of points in affine $G$-varieties. For a reductive affine algebraic group $G$ acting on an affine algebraic variety $X$, the points in $X$ are stratified into unstable, semistable, and stable loci according to their orbit closures and stabilizer properties. A key tool is the correspondence between algebraic subgroups determined intrinsically by the point $x \in X$, the action of one-parameter subgroups ($1$-PS), and the parabolic subgroups of $G$:

• The set $\Lambda_x$ of $1$-PS $\lambda$ such that $\lim_{t\to0}\lambda(t)\cdot x$ exists, and the subgroup $H_x = \bigcap_{\lambda\in \Lambda_x} P(\lambda)$.
• Polystability coincides with $H_x$ being completely reducible, i.e., if $H_x\subset P$ for a parabolic $P$, then $H_x$ is conjugate into a Levi factor of $P$.
• Stability corresponds to $H_x$ being irreducible.

This framework recovers the Hilbert–Mumford numerical criterion: stability is equivalent to negativity of the minimum weight $\mu(v,\lambda)$ for all nontrivial $\lambda$, while semistability corresponds to nonpositivity (Casimiro et al., 2011). For representation varieties and character varieties, this perspective produces GIT interpretations for irreducibility and complete reducibility of representation images.

2. G-Stability in Real Reductive Lie Groups and Harmonic Analysis

For actions of real reductive Lie groups on analytic or topological spaces, G-stability extends the classical framework to settings where a Kempf–Ness-type function can be constructed. Given a real reductive group $G$, maximal compact subgroup $K$, a topological space or real submanifold $M$, and a $G$-equivariant gradient map $\Phi$, the key object is the "maximal weight" function:

• For each compactification direction $p \in \partial_\infty(G/K)$, the maximal weight $\mu(x,p)$ captures the asymptotic convexity of the corresponding exhaustion function $v_x$ associated with $x \in M$.
• Stability: $x$ is stable if and only if $\mu(x,p) > 0$ for every $p \in \partial_\infty(G/K)$.
• Polystability requires $\mu(x,p) \ge 0$ for all $p$, with the zero-locus of $\mu$ corresponding to a totally geodesic submanifold.

This analytic G-stability theory leads to a Hilbert–Mumford-type criterion for real group actions, Morse-theoretic stratification by critical values of the moment map, and extends to actions on probability measures and real submanifolds of Kähler varieties, preserving the equivariant geometric structure in infinite-dimensional contexts (Biliotti et al., 2016, Biliotti et al., 2022).

3. G-Stability in Model Theory and Topological Dynamics

Model-theoretic stability in a group context formalizes when definable sets or relations avoid encoding infinite linear orderings (half-graphs). For a subset $A\subseteq G$, stability is equivalent to the failure to realize infinite half-graphs in the bipartite relation $R(x, y) :\!\Longleftrightarrow xy\in A$. The central characterization connects:

• Grothendieck's double-limit theorem: a family of functions is relatively weakly compact if and only if iterated limits commute.
• Ben Yaacov's theorem: a formula is stable if and only if the corresponding family of functions on S-type spaces is relatively weakly compact (Conant, 2019).
• The Stone space $S(B)$ of a Boolean algebra of stable subsets admits a weakly almost periodic (\text{WAP}) $G$-flow structure, and the Ellis semigroup gives rise to a profinite group of generic types.
• Unique, invariant, finitely additive probability measures exist on left-invariant Boolean algebras of stable subsets, and genericity coincides with positive measure.

Extensions clarify that these structural results hold both globally and in the relative context of stability "in a model", facilitating measure-theoretic and combinatorial decompositions of definable sets.

4. G-Stable Rank and Hilbert–Mumford Theory for Tensors

In the theory of tensors and invariant theory, G-stability manifests as the G-stable rank, generalizing the Hilbert–Mumford criterion from projective GIT. Given a group $$G = \mathrm{GL}(V_1) \times \ldots \times \mathrm{GL}(V_d)$$ acting on the tensor product $V_1 \otimes \cdots \otimes V_d$:

• For a tensor $v$, the G-stable rank $\operatorname{rk}_G(v)$ is the minimal normalized slope achieved by any one-parameter degeneration, encoded in a linear optimization problem over the set of weights (Derksen, 2020).
• The Hilbert–Mumford criterion states that $v$ is semistable if and only if it is not annihilated under degenerations by any $1$-PS.
• The G-stable rank sharpens or bounds other combinatorial invariants (tensor rank, slice rank, border rank), and is Zariski-closed under certain conditions.

This framework is directly linked to recent advances in algebraic combinatorics, including applications to cap set-type problems.

5. G-Stability in Parametrized and Equivariant Homotopy Theory

Higher-categorical G-stability arises in the stabilization of G-categories and in the foundations of equivariant stable homotopy theory. The relevant setting is an orbital $\infty$-category, such as the orbit category $\mathcal{O}_G$ of a finite group $G$:

• $G$-categories are cocartesian fibrations over $\mathcal{O}_G^{op}$, encoding equivariance fiberwise.
• The parametrized suspension $\Sigma_G$ and loops $\Omega_G$ define stability: a pointed $G$-category is $G$-stable if these are inverses and finite $G$-colimits (and limits) exist (Nardin, 2016).
• The stabilization functor $\Sigma_G^\infty$ embeds $G$-spaces into $G$-spectra, with universal property among $G$-linear functors.
• Spectral Mackey functors and equivariant infinite loop space machinery are naturally encoded by this framework, and the universal characterization of genuine $G$-spectra as the $G$-stabilization of $G$-spaces forms the axis of the modern abstract approach to equivariant homotopy theory.

6. G-Stability in Moduli Problems and Constellations

In the context of moduli spaces of sheaves or $G$-constellations, G-stability appears as slope-stability concepts parametrized by invariants of the group. For an infinite reductive group $G$, these concepts bifurcate into:

• $\theta$-stability: defined via infinite collections of rational weights, with associated Harder–Narasimhan filtrations and polygons.
• GIT-stability: via finite-parameter GIT quotients (depending on finite subsets $D$), leading to corresponding finite-parameter Harder–Narasimhan data.
• The two stability notions need not coincide; the infinite-parameter picture can only be recovered as a limit of the finite-parameter polygons (Terpereau et al., 2015).

This dichotomy directly impacts moduli-theoretic constructions, the behavior of walls and filtrations, and the structure of moduli of $G$-equivariant sheaves.

7. Synthesis and Thematic Connections

G-stability theory, regardless of the specific mathematical domain, is unified by the central theme of capturing robust, well-behaved objects under group actions using a mixture of algebraic, analytic, dynamic, and categorical machinery. The various stability notions often admit equivalent geometric, combinatorial, or analytic characterizations (via orbits, numerical invariants, compactness, or convergence), leading to decomposition results, unique invariant structures (measures, stratifications, module decompositions), and universal properties.

Key methodological tools include:

• One-parameter subgroup analysis and parabolic reduction (GIT).
• Weakly almost periodicity and Ellis semigroup theory (topological dynamics).
• Maximal weight/Kempf–Ness-like functions (real and complex analytic actions).
• Slope-stability and Harder–Narasimhan filtrations (moduli and representation spaces).
• Stabilization functors and universal properties in higher category theory.

These frameworks continue to inform developments at the interface of algebra, geometry, dynamics, and topology, and they serve as structural pillars for modern approaches to equivariance, moduli, and spectral classification (Casimiro et al., 2011, Biliotti et al., 2022, Biliotti et al., 2016, Terpereau et al., 2015, Conant, 2019, Derksen, 2020, Nardin, 2016).