Symmetric spaces for groups over involutive algebras and applications to Higgs bundles

Technical Summary: Symmetric Spaces for Groups over Involutive Algebras and Applications to Higgs Bundles

Problem Statement
The paper addresses the challenge of generalizing the theory of Lie groups and their associated symmetric spaces to the setting of noncommutative rings, specifically involutive algebras $(A, \sigma)$. While the theory of Lie groups over commutative fields is well-established, the noncommutative case presents significant difficulties, such as the lack of a canonical determinant, which complicates the definition of subgroups like $SL_n$. The authors focus on symplectic groups $Sp_2(A, \sigma)$ and indefinite orthogonal groups $O(1,1)(A, \sigma)$ over Hermitian involutive algebras and their complexifications. The primary goal is to construct explicit geometric models for the Riemannian symmetric spaces $G/K$ associated with these groups, describe their tangent spaces and diffeomorphisms, and apply this framework to the theory of Higgs bundles to study representations of fundamental groups of Riemann surfaces.

Methodology
The paper is divided into two distinct parts:

• Part 1: Geometric Models of Symmetric Spaces:
  The authors systematically construct and analyze four distinct models for the symmetric spaces associated with groups over Hermitian algebras:

  1. Space of Indefinite Involutive Operators: Defined via operators $J$ satisfying $J^2 = \text{Id}$ (or $J^2 = -\text{Id}$ for complex structures) and compatibility with a sesquilinear form.
  2. Projective Space Model: Realized as open domains within the projective space of lines in $A^2$ (or its complex/quaternionic extensions), specifically those lines satisfying positivity or isotropy conditions relative to the defining forms.
  3. Half-Space Models: Affine charts derived from the projective models, identified with subsets of the algebra $A$ (or its extensions) defined by conditions on the real and imaginary parts of elements (e.g., $y \in A^\sigma_+$).
  4. Precompact Models: Bounded domains obtained via Möbius transformations of the half-space models, analogous to the unit ball model in hyperbolic geometry.

  For each model, the authors explicitly compute the (complexified) tangent spaces, describe the action of the group $G$ and its maximal compact subgroup $K$, and derive the differentials of the equivariant diffeomorphisms connecting these models. They also define the invariant Riemannian metrics on these spaces. The analysis covers groups over real Hermitian algebras ($Sp_2(A, \sigma)$, $O(1,1)(A, \sigma)$, $A^\times$) and their complexifications ($Sp_2(A^C, \sigma^C)$, $O(A^C, \sigma^C)$).

• Part 2: Applications to Higgs Bundles:
  Leveraging the geometric models from Part 1, the authors reinterpret the data of $G$-Higgs bundles. Given a reductive representation $\rho: \pi_1(X) \to G$, Corlette's theorem guarantees a $\rho$-equivariant harmonic map to the symmetric space $G/K$. The pullback of the complexified tangent bundle yields the Higgs field. By choosing different geometric models for $G/K$, the authors provide alternative geometric realizations of the Higgs bundle data $(E, \phi)$.

  • Component Counting: Using the topology of the maximal compact subgroups $K$ rather than standard Morse–Bott theory, the authors derive exact counts for the connected components of the moduli spaces of polystable Higgs bundles for $Sp_2(A^C, \sigma^C)$ and $O(A^C, \sigma^C)$.
  • Hitchin Morphism Factorization: The authors construct a factorization of the Hitchin morphism for these groups. This factorization passes through intermediate affine Geometric Invariant Theory (GIT) quotients. The map is induced by quadratic norm maps from the tangent models to formally real Jordan algebras (specifically, algebras of Hermitian or symmetric elements).

Key Contributions and Results

1. Explicit Geometric Models: The paper provides a unified framework for realizing symmetric spaces of groups over involutive algebras. It explicitly details the tangent spaces and transition maps between the operator, projective, half-space, and precompact models for $Sp_2$, $O(1,1)$, and their complexifications.
2. Topological Invariants: The authors establish the number of connected components and the fundamental groups of the groups $Sp_2(A^C, \sigma^C)$ and $O(A^C, \sigma^C)$ in terms of the structure of the underlying algebra $A$ (specifically the number of quaternionic and complex factors in its Artin–Wedderburn decomposition).
3. Moduli Space Component Count: A new method is introduced to count the connected components of Higgs bundle moduli spaces by analyzing the topology of the maximal compact subgroups, bypassing the need for Morse–Bott theory.
4. Factorization of the Hitchin Morphism: The paper demonstrates that the Hitchin morphism for $Sp_2(A^C, \sigma^C)$-Higgs bundles (and real analogues) factors through intermediate GIT quotients. This factorization is governed by quadratic norm maps to Jordan algebras.
5. Dimensional Efficiency: The authors show that the target spaces of these norm maps (the complexified Jordan algebras) have strictly smaller dimensions than the original complexified tangent models. This reduction in algebraic complexity allows for a more explicit characterization of the Hitchin base and the image of the Hitchin morphism.

Significance and Claims
The paper claims that this noncommutative perspective offers a "geometric dictionary" for the Hitchin base, interpreting abstract holomorphic differentials as invariants derived from quadratic norm maps on Jordan-algebraic targets. The authors state that this approach is particularly fruitful for studying positive representations of fundamental groups of closed surfaces, motivated by generalizations of total positivity.

The significance of the work lies in its ability to:

• Unify the treatment of various classical Lie groups (such as $Sp_{2n}(\mathbb{R})$, $O(n,n)$, $U(n,n)$, $Sp(n,n)$, etc.) under the umbrella of groups over involutive algebras.
• Provide explicit, computable models for symmetric spaces in noncommutative settings, which were previously less developed.
• Simplify the algebraic complexity involved in describing the Hitchin base for these groups through the factorization of the Hitchin morphism.
• Offer new geometric interpretations of Higgs bundle data that are directly tied to the specific geometry of the symmetric space models.

The authors remain modest regarding the scope, noting that the factorization and component counting results are specific to the groups $Sp_2(A^C, \sigma^C)$ and $O(A^C, \sigma^C)$ (and their real forms) within the context of the developed theory, rather than a universal solution for all noncommutative Lie groups. The work is presented as a foundational step for further applications in nonabelian Hodge theory and the study of positive representations.