Definable Connected Cartan Subgroups

• Definable connected Cartan subgroups are maximal definable nilpotent subgroups whose identity component (a Carter subgroup) is uniquely linked to a Cartan subalgebra.
• They mirror the properties of Cartan subgroups in real Lie groups, with invariant dimensions equal to the Lie algebra rank and finitely many conjugacy classes.
• Their classification relies on o-minimal cell decomposition, root-space decomposition, and methods that bridge definable groups with classical Lie theory.

A definable connected Cartan subgroup is a maximal definable nilpotent subgroup of a group definable in an o-minimal structure, equipped with additional properties and structural correspondences with classical Lie theory. In the setting where the o-minimal structure is an expansion of a real closed field, the theory of Cartan subgroups mirrors and extends pivotal aspects of the theory of Cartan subgroups in real Lie groups: existence, conjugacy, dimension invariants, dense coverage, and root-theoretic characterization through the associated definable Lie algebras.

1. O-minimal Structures and Definable Groups

An o-minimal structure $\mathcal{M}$ is a structure $\langle M, <, \ldots\rangle$ such that every definable $X \subseteq M$ is a finite Boolean combination of intervals. If $G$ is a group definable in $\mathcal{M}$, it carries a canonical topology (the t-topology), making it into a definable $M$-manifold. Definable connectedness means $G$ admits no proper definable subgroup of finite index; thus, the definably connected component $G^\circ$ is the smallest definable subgroup of finite index in $G$.

A key structural result is the existence of a largest definable normal solvable subgroup, the radical $R(G)$, with $G/R(G)$ a direct product of finitely many definably simple groups of “Lie-type.”

2. Definitions: Cartan and Carter Subgroups

Let $G$ be a (definably connected) group definable in an o-minimal expansion $M$ of a real closed field $R$, and let $\mathfrak{g} = \operatorname{Lie}(G)$.

• Cartan subalgebra of $\mathfrak{g}$: A subalgebra $\mathfrak{h} \subset \mathfrak{g}$ is a Cartan subalgebra if it is nilpotent and self-normalizing, i.e., $$\mathfrak{n}_{\mathfrak{g}}(\mathfrak{h}) = \mathfrak{h}$$, or equivalently, if the zero root space $\mathfrak{g}^0(\mathfrak{h})$ equals $\mathfrak{h}$ after root-space decomposition over $K=R(i)$.
• Cartan subgroup ("Chevalley style"): A definable subgroup $H \leq G$ is a Cartan subgroup if:
  1. $H$ is maximal among definable nilpotent subgroups of $G$,
  2. For every $X \trianglelefteq H$ of finite index, $[N_G(X):X] < \infty$. If $H^\circ$ denotes the definably connected component, $H^\circ$ is a Carter subgroup—a definably connected nilpotent subgroup normal in its normalizer and of finite index in $N_G(H^\circ)$. One has $H = C_G(H^\circ) \cdot H^\circ$ (Baro et al., 2017, Baro et al., 2011).

3. Existence, Uniqueness, and the Lie Correspondence

Every Cartan subalgebra $\mathfrak{h} \subseteq \mathfrak{g}$ arises as the Lie algebra of a unique definable Cartan subgroup $H \leq G$: $$H = C(\mathfrak{h}) := \{g \in N_G(\mathfrak{h}): \forall \lambda \in \Lambda(\mathfrak{g},\mathfrak{h}),\ \lambda \circ \operatorname{Ad}(g)|_{\mathfrak{h}_K} = \lambda\}.$$ Here, $C(\mathfrak{h})$ is nilpotent, definable, and its identity component $C(\mathfrak{h})^\circ$ is a Carter subgroup. The nilpotency is established by reducing to the linear case via the adjoint representation and invoking the classical fact that the centralizer of a semisimple torus in an algebraic group is a Cartan subgroup (Baro et al., 2017).

Two Cartan subgroups with the same Lie algebra must coincide: their identity components agree, and a Cartan subgroup is determined by its identity component.

4. Dimension, Conjugacy, and Covering Properties

All Cartan subgroups in a definably connected $G$ have the same dimension, namely the rank $\mathrm{rk}\ \mathfrak{g}$ (dimension of any Cartan subalgebra). Each Cartan subgroup $H$ has a definably connected component $H^\circ$ that is a Carter subgroup.

There exist only finitely many $G$-conjugacy classes of Cartan subgroups. In the solvable case there is a single class; in the semisimple (finite radical) case, the classification reduces to that of Cartan subgroups in connected real Lie groups, which are all of the same dimension and satisfy $Q_1 = Q_2$ if and only if $Q_1^\circ = Q_2^\circ$ (Baro et al., 2011).

The union of Cartan subgroups,

$\bigcup_{H \in \mathrm{Cartan}(G)} H,$

is a dense, definably large subset of $G$ in the t-topology (and dimension-theoretically syndetic: finitely many translates cover $G$). This largeness is made precise in the o-minimal dimension: $\dim(G \setminus \bigcup H^G) < \dim G$, and more detailed stratification yields finitely many large disjoint definable "slices", each contained in some Cartan coset (Baro et al., 2011, Baro et al., 2017).

5. Root-Space Decomposition and Characterizations

Let $\mathfrak{h} \leq \mathfrak{g}$ be nilpotent. The root-space decomposition for the adjoint representation $$\mathrm{ad}:\mathfrak{g}_K \rightarrow \operatorname{End}(\mathfrak{g}_K)$$ restricted to $\mathfrak{h}_K$ yields generalized eigenspaces $\mathfrak{g}_K^\lambda(\mathfrak{h}_K)$ indexed by roots $\lambda: \mathfrak{h}_K \rightarrow K$. The decomposition

$$\mathfrak{g}_K = \bigoplus_{\lambda \in \Lambda} \mathfrak{g}_K^\lambda(\mathfrak{h}_K)$$

satisfies that $\mathfrak{h}$ is a Cartan subalgebra if and only if $\mathfrak{h} = \mathfrak{g}^0(\mathfrak{h})$, the zero root space.

In the linear case, Cartan subgroups of $G \leq GL(n,R)$ are the intersections $H' \cap G$ where $H'$ is a Cartan subgroup of the Zariski closure $\mathrm{Cl}^{\mathrm{Zar}}(G)$. Thus, the o-minimal theory aligns tightly with the algebraic group perspective (Baro et al., 2017).

6. Regular Points and Uniqueness

For a definable representation $\rho:G\rightarrow GL(V)$, regular points are defined via the characteristic polynomial. The rank function $r(g)$, giving the smallest $j$ with nonzero coefficient $a_j(g)$ in the characteristic polynomial of $\rho(g)-(1+T)\operatorname{Id}$, is upper semi-continuous. The set of regular points $\mathrm{Reg}_\rho(G)$ is open and dense, and for $\rho = \operatorname{Ad}$, this yields the set of regular elements $\mathrm{Reg}(G)$. The "Cartan test" states that for $g \in G$:

• $g$ is regular,

• $$\dim_R V^1(\operatorname{Ad}(g)) = \mathrm{rk}\ \mathfrak{g}$$,
• the generalized eigenspace $\mathfrak{g}^1(\operatorname{Ad}(g))$ is a Cartan subalgebra, are equivalent.

Each regular point $g \in G$ belongs to exactly one Cartan subgroup, which is explicitly $H = C(\mathfrak{g}^1(\operatorname{Ad}(g)))$ (Baro et al., 2017).

7. Examples and Interplay with the o-minimal Levi Decomposition

For $G = SL_2(\mathbb{R})$, up to conjugacy there are precisely two Cartan subgroups: the split torus $$Q_\mathrm{split} = \{\operatorname{diag}(\lambda, \lambda^{-1}) : \lambda \neq 0 \} \cong \mathbb{R}^\times$$ (non-connected; its definably connected component $\mathbb{R}^{>0}$ is a Carter subgroup) and the compact torus $Q_\mathrm{compact} = SO_2(\mathbb{R})$ (connected). Both have dimension $1$, and the union of their conjugates is "large": matrices with $|\operatorname{tr}| > 2$ (split) or $|\operatorname{tr}| < 2$ (compact) lie in a Cartan coset, and each coset covers an open dense subset of $G$ (Baro et al., 2011).

For general $GL_n(\mathbb{R})$, Cartan subgroups are centralizers of maximal split or compact tori, have finitely many conjugacy classes, and their union is dense.

In groups admitting an o-minimal Levi decomposition $G^\circ = R(G) \rtimes S$, Cartan subgroups of $G$ correspond to lifts of Cartan subgroups in each simple factor, and in the radical $R(G)$, Cartan subgroups are self-normalizing and conjugate. Cartan subgroups of $G$ project to Cartan subgroups of $S=G/R(G)$ (Baro et al., 2011).

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The o-minimal theory of definable connected Cartan subgroups thus yields a complete structural analogue to the classical theory for real Lie groups, relying on o-minimal cell decomposition, definable choice, definable Lie theory, dimension theory, and algebraic group methods, without the use of exponential maps.