[Paper Review] Invariant Orders on Hermitian Lie Groups

Gabi Ben Simon, Tobias Hartnick|arXiv (Cornell University)|Nov 15, 2010

Geometric and Algebraic Topology21 references10 citations

TL;DR

This paper establishes the equivalence of three distinct bi-invariant partial orders—geometric, Lie semigroup, and maximal order—on the universal covering group of $\mathrm{SL}_2(\mathbb{R})$ and more generally on Hermitian Lie groups of tube type. It demonstrates that these orders coincide via inclusion relations, with the maximal order defined by the translation number quasimorphism, and proves that the Lie semigroup is maximal and definable by a single continuous function.

ABSTRACT

We study three natural bi-invariant partial orders on a certain covering group of the automorphism group of a bounded symmetric domain of tube type; these orderings are defined using the geometry of the Shilov boundary, Lie semigroup theory and quasimorphisms respectively. Our main result shows that these orders are related by two inclusion relations. In the case of $SL_2(\R)$ we can show that they coincide.

Motivation & Objective

• To investigate the relationship between three natural bi-invariant partial orders on the universal covering group of $\mathrm{SL}_2(\mathbb{R})$ and more general Hermitian Lie groups.
• To determine whether the geometric, Lie semigroup, and maximal orders are distinct or coincident in the context of Hermitian Lie groups.
• To establish that the Lie semigroup order is maximal among conjugation-invariant subsemigroups and can be defined by a single continuous function.
• To generalize the concept of translation number to higher-rank Hermitian Lie groups using the Maslov quasimorphism.
• To unify geometric, Lie-theoretic, and quasimorphism-based orderings through a common framework in symmetric space theory.

Proposed method

• Define the geometric order via the action of the group on the real line, lifting the boundary action of $\mathrm{PU}(1,1)$ on $S^1$.
• Define the Lie semigroup order using the unique pair of $\mathrm{Ad}$-invariant pointed convex cones in the Lie algebra, exponentiating the positive cone to form the order semigroup.
• Define the maximal order using the translation number quasimorphism $T(g) = \lim_{n\to\infty} \frac{g^n.x - x}{n}$, which generates a maximal conjugation-invariant subsemigroup $G^{+}_{\max}$.
• Establish inclusion relations: $G^{+}_{\mathrm{geom}} \subset G^{+}_{\max}$ and $G^{+}_{\mathrm{cont}} \subset G^{+}_{\mathrm{geom}}$, using the Kaneyuki causal structure on the Shilov boundary.
• Use the triple decomposition $\mathfrak{g}(T_\Omega) = \mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$ and the Cayley transform to identify the homogeneous space $\check{R} \cong H/P$ and relate the causal structure to the cone of non-negative definite quadratic forms.
• Prove continuity of the Kaneyuki order by showing the preimage of the positive cone under the quotient map is a locally topologically generated Lie semigroup.

Experimental results

Research questions

• RQ1 Are the geometric, Lie semigroup, and maximal orders on the universal cover of $\mathrm{SL}_2(\mathbb{R})$ distinct or equivalent?
• RQ2 Can the Lie semigroup order be characterized as maximal among conjugation-invariant subsemigroups?
• RQ3 Does the translation number quasimorphism generate a maximal conjugation-invariant subsemigroup in the universal cover of $\mathrm{SL}_2(\mathbb{R})$?
• RQ4 How do the geometric, continuous, and maximal orders relate in higher-rank Hermitian Lie groups?
• RQ5 Can the geometric order on the universal cover of a Hermitian Lie group be described via a single continuous function?

Key findings

• The geometric, Lie semigroup, and maximal orders on the universal covering group of $\mathrm{SL}_2(\mathbb{R})$ coincide, as proven in Proposition 3.13.
• The Lie semigroup $G^+$ is a maximal conjugation-invariant subsemigroup of $G$, and can be described by a single continuous function—the translation number $T$.
• The maximal order $G^{+}_{\max}$, defined via the quasimorphism $T$, refines both the geometric and Lie semigroup orders.
• The geometric order on the universal cover of a Hermitian Lie group is induced by the action on the universal cover of the Lagrangian Grassmannian, with positivity modeled on the cone of non-negative definite quadratic forms.
• The Kaneyuki causal structure on the Shilov boundary induces a continuous order on $\check{R} \cong H/P$, and this order is continuous in the sense of Definition 4.2.
• The continuous order on $\check{R}$ is uniquely determined up to inversion and satisfies $x \preceq y \Leftrightarrow \exists g \in G^{+}_{\mathrm{cont}}: gx = y$, which implies $G^{+}_{\mathrm{cont}} \subset G^{+}_{\mathrm{geom}}$.