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Siegel sets covering arithmetic quotients

Recall our goal is to prove the following theorem.

Theorem 1 (Bakker–Klinger–Tsimerman, 1.1). Let $\mathbb{G}$ be a connected semisimple group over $\mathbb{Q}$, and let $G = \mathbb{G}(\mathbb{R})^+$. Let $\Gamma \subseteq \mathbb{G}(\mathbb{Q})^+$ be a torsion free arithmetic subgroup and let $M \subseteq G$ be a connected compact subgroup, so that we have the arithmetic quotient $$ S_{\Gamma,G,M} = \Gamma \backslash G / M. $$

  1. For each choice of maximal compact subgroup $M \subseteq K \subseteq G$, the quotient $S_{\Gamma,G,M}$ admits a natural $\mathbb{R}_ \mathrm{alg}$-definable manifold structure, characterized by the fact that for all semi-algebraic Siegel set $\mathfrak{S} \subseteq G/M$ associated to a parabolic $\mathbb{P} \subseteq \mathbb{G}$ and $K$, the induced map $$ \mathfrak{S} \to S_{\Gamma,G,M} $$ is $\mathbb{R}_ \mathrm{alg}$-definable. In particular, there exists a semi-algebraic fundamental set $\mathcal{F} \subseteq G$ for which $\mathcal{F} \to S_{\Gamma,G,M}$ is $\mathbb{R}_ \mathrm{alg}$-definable.
  2. Any morphism $$ S_{\Gamma^\prime,G^\prime,M^\prime} \to S_{\Gamma,G,M}; \quad \Gamma^\prime h M^\prime \mapsto \Gamma \phi(h) g M $$ of arithmetic quotients is $\mathbb{R}_ \mathrm{alg}$-definable. Here, a morphism is a $\phi \colon \mathbb{G}^\prime \to \mathbb{G}$ and $g \in G$ satisfying that $\phi(\Gamma^\prime) \subseteq \Gamma$, $\phi(M^\prime) \subseteq g M g^{-1}$, $\phi(K^\prime) \subseteq g K g^{-1}$, and the Cartan involution $\theta_{gKg^{-1}}$ preserves $\phi(\mathfrak{g}^\prime)$.

Remark 2. The $\mathbb{R}_ \mathrm{an}$-version of (1) follows from the existence of the Borel–Serre compactification, but (2) does not because the Borel–Serre compactification is not functorial.

Siegel sets#

Let $\mathbb{P} \subseteq \mathbb{G}$ be a parabolic subgroup over $\mathbb{Q}$, and let $\mathbb{N}_ \mathbb{P} \hookrightarrow \mathbb{P} \twoheadrightarrow \mathbb{L}_ \mathbb{P}$ the unipotent radical and Levi. Let $A_P \subseteq Z(L_P)$ be the connected component of the $\mathbb{R}$-points of the maximal split torus in $Z(\mathbb{L}_ \mathbb{P})$, and let $$ M_P = \bigcap_{\chi \in X^\ast(\mathbb{L}_ \mathbb{P})} \ker \chi^2 \subseteq L_P = \mathbb{L}_\mathbb{P}(\mathbb{R}). $$ Then we have the decomposition $L_P = A_P M_P$.

Let $X$ be the symmetric space for $G$, which parametrizes maximal compact subgroups, so that when we choose a maximal compact $K$ then we have $X \cong G/K$. Given a point $x \in X$, corresponding to a maximal compact $K_x \subseteq G$, gives rises to an algebraic involution $$ \theta_x \colon \mathbb{G} \to \mathbb{G} $$ so that $G^{\theta_x = 1} = K_x$.

Example 3. For $\mathbb{G} = \mathrm{SL}_ n$, we have $\theta (A) = (A^t)^{-1}$ corresponding to $\mathrm{SO}_ n(\mathbb{R})$.

So the choice of $x$ gives a unique Levi subgroup $\mathbb{L}_ {\mathbb{P},x} \subseteq \mathbb{P}_ \mathbb{R}$. Then we have $$ P = N_P L_{P,x} = N_P A_{P,x} M_{P,x}. $$ By the Iwasawa decomposition, we moreover have $$ G = PK_x = N_P A_{P,x} M_{P,x} K_x. $$ We denote by $\Phi(A_{P,x}, N_P)$ the set of roots of $A_{P,x}$ acting on the Lie algebra of $N_P$, and let $\Delta(A_{P,x}, N_P)$ be the subset of simple roots. For a root $\alpha$, we denote $$ \psi_\alpha \colon A_{P,x} \to \mathbb{R}^\times; \quad a \mapsto a^\alpha. $$

Definition 4. For $t \gt 0$, we define $$ (A_{P,x})_ t = \lbrace a \in A_{P,x} : a^\alpha \gt t \text{ for all } \alpha \in \Phi(A_{P,x}, N_P) \rbrace. $$ A Siegel set for $G$ is a set of the form $$ \mathfrak{S} = U \times (A_{P,x})_ t \times W $$ where $U \subseteq N_P$ and $W \subseteq M_{P,x} K$ are open relatively compact semi-algebraic sets.

Definition 5. A Siegel set for $G/M$ is a set of the form $\pi_x(\mathfrak{S})$ where $\mathfrak{S} \subseteq G$ is a Siegel set and $$ \pi_x \colon G \to G / gMg^{-1} \xrightarrow{\mathrm{conj}_{g^{-1}}} G/M $$ where $g M g^{-1} \subseteq K_x$.

For $G = \mathrm{SL}_ 2(\mathbb{R})$, we can take $M = K = \mathrm{SO}_ 2(\mathbb{R})$ and $G/M \cong \mathbb{H}$. Here for the $\mathbb{P}$ consisting of upper triangular matrices, we have $$ \mathfrak{S} = N_P (A_{P,x})_ t K \cong \lbrace \mathrm{Im} \subseteq (a, b), \mathrm{Re} \gt 0 \rbrace. $$ If we took $\Gamma = \Gamma_0(2)$ instead, we would have two cusps $0$ and $2$. Then we will also have to look at the $(\begin{smallmatrix} & -1 \br 1 & \end{smallmatrix}) \mathbb{P} (\begin{smallmatrix} & 1 \br -1 & \end{smallmatrix})$ which is the lower triangular matrices.

Proposition 6. There exist finitely many conjugacy classes of parabolic $\mathbb{Q}$-subgroups, say represented by $\mathbb{P}_ 1, \dotsc, \mathbb{P}_ k$.

  1. There exist Siegel sets $\mathfrak{S}_ i$ associated to $\mathbb{P}_ {i,x}$ such that $\bigcup_i \mathrm{im}(\mathfrak{S}_ i) = S_{\Gamma,G,M}$.
  2. For any two $\mathbb{P}_ i$ and $\mathbb{P}_ j$ with associated Siegel sets $\mathfrak{S}_ i$ and $\mathfrak{S}_ j$, the set $$ \Gamma_{\mathfrak{S}_ i, \mathfrak{S}_ j} = \lbrace \gamma \in \Gamma : \gamma \mathfrak{S}_ i, \mathfrak{S}_ j \neq \emptyset \rbrace $$ is finite.
  3. (Orr, 2018) If $\mathbb{H} \hookrightarrow \mathbb{G}$ is a closed subgroup and $\mathfrak{S}_ H$ is a Siegel set of $H(\mathbb{R})$, then there exists a finite set $C \subseteq G(\mathbb{Q})$ and a Siegel set $\mathfrak{S}_ G$ of $G$ such that $\mathfrak{G}_ H \subseteq C \mathfrak{G}_ G$.

Proof of the theorem#

Now using this proposition we can prove the theorem. For the first part, we use (1) of the proposition to find a finite collection of semi-algebraic Siegel sets $\mathfrak{S}_ i = U_i \times A_{P_i,t_i} \times W_i$ such that the image cover $S_{\Gamma,G,M}$. Then we have $$ S_{\Gamma,G,M} = \coprod_{i=1}^k \mathfrak{G}_ i / \lbrace x_1 \sim x_2 \text{ when } \gamma x_1 = x_2 \text{ for some } \gamma \in \Gamma \rbrace. $$ This equivalence relation is $\mathbb{R}_ \mathrm{alg}$-definably proper, where the transition maps are indeed definable because they are left multiplication by some elements in $\Gamma$. Technically, to get a proper equivalence relation we need to take the closures of Siegel sets and glue them together.

We now prove functoriality. Suppose we have $$ (\phi, g) \colon S_{\Gamma^\prime,G^\prime,M^\prime} \to S_{\Gamma,G,M}. $$ This can be decomposed as $(\mathrm{conj}_ {g^{-1}} \circ \phi, 1) \circ (\mathrm{conj}_ g, g)$ where the second is induced by left multiplication by $g$ and hence semi-algebraic. So now we can take care of $(\phi, 1)$. Again, we cover the source by Siegel sets $\mathfrak{S}_ i^\prime$ and show that $(\phi, 1) \vert_{\mathfrak{S}_ i^\prime}$ are $\mathbb{R}_ \mathrm{alg}$-definable.

  1. When $\mathbb{G}^\prime = \mathbb{G}$, this is easy.
  2. When $\mathbb{G}^\prime \to \mathbb{G}$ is surjective, we may reduce to the case when both $\mathbb{G}, \mathbb{G}^\prime$ are adjoint. Then $\mathbb{G}^\prime = \mathbb{G} \times \mathbb{H}$ and then change $\Gamma, M$ by the first case so that everything splits. Then the projection to one component is semi-algebraic.
  3. When $\mathbb{G}^\prime \hookrightarrow \mathbb{G}$, we use this result of Orr. Then $\mathfrak{S}_ i^\prime \subseteq C_i \mathfrak{S}_ i \to S_{\Gamma,G,M}$. Both maps are $\mathbb{R}_ \mathrm{alg}$-definable, and therefore so is the composition.