Satake compactification

A type of compactification arising from work of I. Satake on the compactification of quotients of symmetric spaces by arithmetically-defined groups ([a9], [a10], [a11]). Below, the simplest case of this is presented first, to help suggest its generalization.

Let $H$ be the upper half-plane, the symmetric space of non-compact type for $G = \operatorname{SL} ( 2 , \bf R )$. For any subgroup $\Gamma$ of finite index in $\operatorname{SL} ( 2 , {\bf Z} )$ — these are arithmetic groups (cf. also Arithmetic group) — the quotient space $X = \Gamma {\color{blue} \backslash} H$ is a Riemann surface, a modular curve. A compactification $X ^ { * }$ of $X$ is obtained by first taking the $\operatorname{SL} ( 2 , \mathbf{Q} )$-invariant set

\begin{equation*} H ^ { * } = H {\color{blue} \bigcup }{\bf P} ^ { 1 } ({\bf Q} ) \subset {\bf P} ^ { 1 } ({\bf C} ), \end{equation*}

with an $\operatorname{SL} ( 2 , \mathbf{Q} )$-equivariant topology that is the given one on $H$ and makes $\mathbf{P} ^ { 1 } ( \mathbf{Q} )$ discrete; a deleted neighbourhood base for $\infty \in H ^ { * }$ is given by

\begin{equation*} H ^ { L } = \{ z \in H : \operatorname { Im } z > L \} \text { for } L > 0. \end{equation*}

Then $X ^ { * }$ is taken to be $\Gamma \backslash H ^ { * }$. It is a compact Riemann surface (thus automatically an algebraic curve). An important ingredient, both here and in Satake's generalization, is reduction theory. Relative to the point $\infty \in H ^ { * }$, it asserts that if $Z \in H$ and $\gamma \in \Gamma$ satisfy $\operatorname { Im } z > 1$ and $\operatorname { Im } ( \gamma z ) > 1$, then $\gamma$ lies in the group of real translations (equivalently, in the parabolic group $P$ of upper-triangular matrices, which is the stabilizer of $\infty$). This gives an embedding of the punctured disc $( \Gamma \cap P ) \backslash H ^ { 1 }$ in $X$, and one is inserting the missing origin by adjoining $\infty$ to $H$.

There was great interest in doing something similar for $A _ { g }$, the moduli space of Abelian varieties (cf. also Moduli theory; Abelian variety), which is the quotient $X _ { g } = \operatorname { Sp } ( 2 g , \mathbf{Z} ) \backslash H _ { g }$; here, $H _ { g }$ is the Siegel upper half-space of genus $g$, which is the symmetric space for $G = \operatorname { Sp } ( 2 g , \mathbf{R} )$, the rank-$g$ group of $( 2 g ) \times ( 2 g )$ symplectic matrices. For $g = 1$ one has $H _ { 1 } = H$ (from the preceding paragraph). Satake first observed, in [a9], that $X _ { g } ^ { * } = {\color{blue} \cup} _ { r \leq g } X _ { r }$ could be topologized in a way that makes it a compact space with "hereditary" structure: the closure of $X_r$ in $X _ { g } ^ { * }$ is homeomorphic to $X_r ^ { * }$. Some refer to this space as "the" Satake compactification of $X_{g}$.

Satake compactifications, in the sense of [a13] (after [a11] and [a3]), are defined from the following setting [a10]. Let $D$ be the symmetric space of non-compact type for the real semi-simple Lie group $G$ (cf. also Lie group, semi-simple). Each faithful finite-dimensional representation of $G$ (cf. also Representation of a Lie algebra) determines an embedding of $D$ in some real projective space, so let $\overline{ D }$ be the closure of $D$. The boundary of $\overline{ D }$ consists of pieces, called boundary components, that are homogeneous under a class of parabolic subgroups of $G$. If $r$ is the (real) rank of $G$, there are only $2 ^ { r } - 1$ distinct such spaces $\overline{ D }$ up to homeomorphism, corresponding to the non-empty subsets $S$ of a set of simple roots, and as such they form a semi-lattice; if one then writes $\overline { D } = \overline { D } _ { S }$, the identity mapping of $D$ extends to a continuous mapping $\overline { D } _ { S } \rightarrow \overline { D } _ { T }$ whenever $S \supset T$.

When $G$ is the real Lie group associated to an algebraic group defined over $\mathbf{Q}$, a boundary component is said to be rational when its normalizing parabolic subgroup is defined over $\mathbf{Q}$. Likewise, the structure over $\mathbf{Q}$ determines the class of arithmetic subgroups $\Gamma$ of $G$. Take $D ^ { * }$ to be the union of $D$ and its rational boundary components. Then, with a suitable topology on $D ^ { * }$, $X ^ { * } = \Gamma \backslash D ^ { * }$ is, under mild hypotheses, a compactification of $X = \Gamma \backslash D$. The collection of these inherit the semi-lattice structure from the above. There is a precise sense in which the topology of $X ^ { * }$ is induced from that of the closure of a Siegel set in $\overline{ D }$.

The Baily–Borel compactification is one of the minimal Satake compactifications in the case where $D$ is Hermitian, i.e., has a $G$-invariant complex structure (cf. also Hermitian symmetric space). This includes the case of $X _ { g } ^ { * }$ above. Here, $D$ gets embedded as a bounded symmetric domain. By means of automorphic forms of sufficiently high weight, $X ^ { * }$ gets embedded as a normal algebraic subvariety of complex projective space [a2]. This fact has rather strong consequences, for the singular locus of $D ^ { * }$ has "high" codimension in general, and one can invoke general results from algebraic geometry. It implies the existence of big families of Abelian varieties that do not degenerate. This compactification also enters into the topological interpretation of the $L^{2}$-cohomology of $X$, as conjectured in [a12] and proved by E. Looijenga, and L. Saper and M. Stern.

The applications of Satake compactifications cover a range of other areas. The stable cohomology of "the" Satake compactification was determined by R. Charney and R. Lee [a5] for application in $K$-theory. Non-Hermitian Satake compactifications occur in the combinatorial data of the toroidal compactifications of [a1] (see [a8], § 2), which provide resolution of the singularities of $X ^ { * }$ (cf. also Resolution of singularities).

Another compactification, the reductive Borel–Serre, a simple quotient of the manifold-with-corners constructed in [a4] (see, e.g., [a7]), dominates all Satake compactifications, and often coincides with the unique maximal one. It has played an increasing role in the theory of automorphic forms. It is the natural place to study the $L ^ { p }$-cohomology [a12] and to define weighted cohomology [a6].

The spaces $\overline{ D }$ (also defined in a different manner by H. Furstenberg) themselves have played an important role in harmonic analysis, rigidity theory and potential theory.

References

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[a10]	I. Satake, "On representations and compactifications of symmetric Riemannian spaces." Ann. of Math. , 71 (1960) pp. 77–110 MR0118775 Zbl 0094.34603
[a11]	I. Satake, "On compactifications of the quotient spaces for arithmetically defined discontinuous groups" Ann. of Math. , 72 (1960) pp. 555–580 MR0170356 Zbl 0146.04701
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