2. General Hermitian symmetric domain [04YB]

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2. General Hermitian symmetric domain

Let $\mathbb{G}$ be a reductive algebraic group over $\mathbb{Q}$ , $G=\mathbb{G}(\mathbb{R})$ , $K$ (one of) its maximal compact subgroup, and $D:=G/K$ , which we suppose to have a Hermitian symmetric domain structure. We moreover assume $D$ is irreducible so that $G$ is simple as a Lie group. Suppose that $\Gamma$ is an arithmetic subgroup of $\mathbb{G}(\mathbb{Q})$ , which acts on $D$ . Hence we can discuss Hermitian locally symmetric space $\Gamma\backslash D$ .

Satake [Sat60a], [Sat60b] constructed compactifications of Riemannian locally symmetric spaces $G/K$ associated to irreducible projective representations $\tau\colon G\to PGL(\mathbb{C})$ satisfying certain conditions. They are stratified as:

\overline{\Gamma\backslash D}^{\rm Sat,\tau}=\Gamma\backslash D\sqcup\bigsqcup_{P}(\Gamma\cap Q(P))\backslash M_{P}/(K\cap M_{P}).

Here, $P$ runs over all the $\mu(\tau)$ -connected rational parabolic subgroups, $P=N_{P}A_{P}M_{P}$ denotes the Langlands decomposition, and $Q(P)$ is the $\mu(\tau)$ -saturation of $P$ . We are particularly interested in the case when $\tau$ is the adjoint representation $\tau_{\rm ad}$ .

On the other hand, given any toroidal compactification [AMRT75] for $\Gamma\backslash D$ , we can apply the Morgan-Shalen type compactification to it as [Od16, Appendix] (following [MS84, BJo17]). The Morgan-Shalen type compactification $\overline{\Gamma\backslash D}^{\rm MSBJ}$ obtained in this way is independent of the cone decomposition for the toroidal compactification [Od16, A.13, A.14].

We now compare these two compactifications.

Theorem 2.1.

Let $\Gamma\backslash D$ be a locally Hermitian symmetric space. Consider its toroidal compactification and the associated (generalised) Morgan-Shalen compactification $\overline{\Gamma\backslash D}^{\rm MSBJ}$ . Then this is homeomorphic to the Satake compactification $\overline{(\Gamma\backslash D)}^{\rm Sat,\tau_{\rm ad}}$ for the adjoint representation $\tau_{\rm ad}$ of $G$ .

In the following we make an “elementary” but important observation on a rationality phenomenon of the limits along one parameter holomorphic family, which we expect to fit well with the recent approach to extend the theta functions in [GS12] etc.

Proposition 2.2.

Suppose $U\subset\overline{U}^{\rm hyb}(\mathcal{X})$ is a Morgan-Shalen-Boucksom-Jonsson compactification associated to an arbitrary dlt stacky pair $(\mathcal{X},\mathcal{D})$ of boundary coefficients $1$ ([Od16]) with $\mathcal{U}:=\mathcal{X}\setminus\mathcal{D}$ , its coarse moduli space $\mathcal{U}\to U$ . Then for any holomorphic morphism $\Delta^{*}:=\{z\in\mathbb{C}\mid 0<|z|<1\}\to\mathcal{U}$ which extend to $\Delta:=\{z\in\mathbb{C}\mid|z|<1\}\to\mathcal{X}$ , it induces a continuous map $\Delta\to\overline{U}^{\rm hyb}(\mathcal{X})$ , i.e., the limit exists. Furthermore, such possible limits in $\Delta(\mathcal{D})$ are characterized as points with rational coordinates.

Corollary 2.3 (corollary to Theorem 2.1 and Proposition 2.2).

Take an arbitrary holomorphic map $f\colon\Delta^{*}\to\Gamma\backslash D$ , which extends to a map to a toroidal compactification of $\Gamma\backslash D$ . Then $f$ also extends to a map $\Delta\to\overline{\Gamma\backslash D}^{\rm Sat,\tau_{ad}}$ where $0$ is sent to a point with rational coordinates, i.e., a point in the dense subset $(C(F)\cap U(F)\otimes\mathbb{Q})/\mathbb{Q}_{>0}\subset C(F)/\mathbb{R}_{>0}$ .

This is partially proved in the case of $A_{g}$ in [Od16] by using degeneration data in [FC90].

Remark 2.4.

Although we assume that $G$ is simple in this section, our Morgan-Shalen type compactification construction [Od16, Appendix] still works for non-simple $G$ . Thus, our construction also gives a new Satake-type compactification for non-simple $G$ , e.g., of the Hilbert modular varieties.