View source for Baily-Borel compactification

''Satake–Baily–Borel compactification''

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300101.png" /> be a semi-simple [[Linear algebraic group|linear algebraic group]] (cf. also [[Semi-simple algebraic group|Semi-simple algebraic group]]) defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300102.png" />, meaning that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300103.png" /> can be embedded as a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300104.png" /> such that each element is diagonalizable (cf. also [[Diagonalizable algebraic group|Diagonalizable algebraic group]]), and that the equations defining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300105.png" /> as an [[Algebraic variety|algebraic variety]] have coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300106.png" /> (and that the group operation is an algebraic morphism). Further, suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300107.png" /> contains a torus (cf. [[Algebraic torus|Algebraic torus]]) that splits over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300108.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b1300109.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001010.png" />-rank at least one), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001011.png" /> is of Hermitian type, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001012.png" /> can be given a complex structure with which it becomes a symmetric domain, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001013.png" /> denotes the real points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001015.png" /> is a maximal compact subgroup. Finally, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001016.png" /> be an arithmetic subgroup (cf. [[Arithmetic group|Arithmetic group]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001017.png" />, commensurable with the integer points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001018.png" />. Then the arithmetic quotient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001019.png" /> is a [[Normal analytic space|normal analytic space]] whose Baily–Borel compactification, also sometimes called the Satake–Baily–Borel compactification, is a canonically determined projective normal [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001020.png" />, defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001021.png" />, in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001022.png" /> is Zariski-open (cf. also [[Zariski topology|Zariski topology]]) [[#References|[a1]]] [[#References|[a2]]] [[#References|[a15]]] [[#References|[a16]]].

To describe <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001023.png" /> in the complex topology, first note that the Harish–Chandra realization [[#References|[a6]]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001024.png" /> as a bounded symmetric domain may be compactified by taking its topological closure. Then a rational boundary component of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001025.png" /> is a boundary component whose [[Stabilizer|stabilizer]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001026.png" /> is defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001027.png" />; based on a detailed analysis of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001029.png" />-roots and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001031.png" />-roots of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001032.png" />, there is a natural bijection between the rational boundary components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001033.png" /> and the proper maximal parabolic subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001034.png" /> defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001035.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001036.png" /> denote the union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001037.png" /> with all its rational boundary components. Then (cf. [[#References|[a18]]]) there is a unique topology, the Satake topology, on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001038.png" /> such that the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001039.png" /> extends continuously and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001040.png" /> with its quotient topology compact and Hausdorff. It also follows from the construction that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001041.png" /> is a finite disjoint union of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001042.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001043.png" /> for some rational boundary component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001044.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001045.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001046.png" /> is the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001047.png" /> with the stabilizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001048.png" />. In addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001049.png" /> and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001050.png" /> has a natural structure as a [[Normal analytic space|normal analytic space]]; the closure of any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001051.png" /> is the union of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001052.png" /> with some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001053.png" />s of strictly smaller dimension; and it can be proved that every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001054.png" /> has a fundamental system of neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001055.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001056.png" /> is connected for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001057.png" />.

In order to describe the structure sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001058.png" /> (cf. also [[Scheme|Scheme]]) with which it becomes a normal analytic space and a projective variety, define an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001060.png" />-function on an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001061.png" /> to be a continuous complex-valued function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001062.png" /> whose restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001063.png" /> is analytic, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001064.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001065.png" />. Then, associating to each open <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001066.png" /> the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001067.png" />-module of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001068.png" />-functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001069.png" /> determines the [[Sheaf|sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001070.png" /> of germs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001071.png" />-functions. Further, for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001072.png" /> the sheaf of germs of restrictions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001073.png" />-functions to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001074.png" /> is the structure sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001075.png" />. Ultimately it is proved [[#References|[a2]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001076.png" /> is a normal analytic space which can be embedded in some complex projective space as a projective, normal algebraic variety. The proof of this last statement depends on exhibiting that in the collection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001077.png" />-functions there are enough automorphic forms for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001078.png" />, more specifically, Poincaré-Eisenstein series, which generalize both Poincaré series and Eisenstein series (cf. also [[Theta-series|Theta-series]]), to separate points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001079.png" /> as well as to provide a projective embedding.

===History and examples.===
The simplest example of a Baily–Borel compactification is when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001080.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001081.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001082.png" /> is the complex upper half-plane, on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001083.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001084.png" /> acts by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001085.png" />. (The bounded realization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001086.png" /> is a unit disc, to which the upper half-plane maps by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001087.png" />.) The properly discontinuous action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001088.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001089.png" /> extends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001090.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001091.png" /> is a smooth projective curve. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001092.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001093.png" />-rank one, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001094.png" /> is a finite set of points, referred to as cusps.

Historically the next significant example was for the Siegel modular group, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001095.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001096.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001097.png" /> consisting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001098.png" /> symmetric complex matrices with positive-definite imaginary part; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b13001099.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010100.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010101.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010102.png" />. I. Satake [[#References|[a17]]] was the first to describe a compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010103.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010104.png" /> endowed with its Satake topology (cf. also [[Satake compactification|Satake compactification]]). Then Satake, H. Cartan and others (in [[#References|[a19]]]) and W.L. Baily [[#References|[a13]]] further investigated and exhibited the analytic and algebraic structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010105.png" />, using automorphic forms as mentioned above. Baily [[#References|[a14]]] also treated the Hilbert–Siegel modular group, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010106.png" /> for a totally real [[Number field|number field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010107.png" />.

In the meanwhile, under only some mild assumption about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010108.png" />, Satake [[#References|[a18]]] constructed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010109.png" /> with its Satake topology, while I.I. Piateckii-Shapiro [[#References|[a10]]] described a normal analytic compactification whose topology was apparently weaker than that of the Baily–Borel compactification. Later, P. Kiernan [[#References|[a7]]] showed that the topology defined by Piateckii-Shapiro is homeomorphic to the Satake topology used by Baily and Borel.

===Other compactifications.===
Other approaches to the compactification of arithmetic quotients of symmetric domains to which the Satake and Baily–Borel approach may be compared are the Borel–Serre compactification [[#References|[a3]]], see the discussion in [[#References|[a20]]], and the method of toroidal embeddings [[#References|[a12]]].

===Cohomology.===
Zucker's conjecture [[#References|[a21]]] that the (middle perversity) intersection cohomology [[#References|[a4]]] (cf. also [[Intersection homology|Intersection homology]]) of the Baily–Borel compactification coincides with its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010111.png" />-cohomology, has been given two independent proofs (see [[#References|[a8]]] and [[#References|[a11]]]); see also the discussion and bibliography in [[#References|[a5]]].

===Arithmetic and moduli.===
In many cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010112.png" /> has an interpretation as the moduli space for some family of Abelian varieties (cf. also [[Moduli theory|Moduli theory]]), usually with some additional structure; this leads to the subject of Shimura varieties (cf. also [[Shimura variety|Shimura variety]]), which also addresses arithmetic questions such as the field of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010114.png" />. Geometrically, the strata of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010115.png" /> parameterize different semi-Abelian varieties, i.e., semi-direct products of algebraic tori with Abelian varieties, into which the Abelian varieties represented by points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010116.png" /> degenerate. For an example see [[#References|[a9]]], where this is thoroughly worked out for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010117.png" />-forms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010118.png" />, especially for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010119.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W.L. Baily, Jr.,   A. Borel,   "On the compactification of arithmetically defined quotients of bounded symmetric domains"  ''Bull. Amer. Math. Soc.'' , '''70'''  (1964)  pp. 588–593</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.L. Baily, Jr.,   A. Borel,   "Compactification of arithmetic quotients of bounded symmetric domains"  ''Ann. of Math. (2)'' , '''84'''  (1966)  pp. 442–528</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Borel,   J.P. Serre,   "Corners and arithmetic groups"  ''Comment. Math. Helv.'' , '''48'''  (1973)  pp. 436–491</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Goresky,   R. MacPherson,   "Intersection homology, II"  ''Invent. Math.'' , '''72'''  (1983)  pp. 135–162</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Goresky,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010120.png" />-cohomology is intersection cohomology"  R.P. Langlands (ed.)  D. Ramakrishnan (ed.) , ''The Zeta Functions of Picard Modular Surfaces'' , Publ. CRM  (1992)  pp. 47–63</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  Harish-Chandra,   "Representations of semi-simple Lie groups. VI"  ''Amer. J. Math.'' , '''78'''  (1956)  pp. 564–628</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P. Kiernan,   "On the compactifications of arithmetic quotients of symmetric spaces"  ''Bull. Amer. Math. Soc.'' , '''80'''  (1974)  pp. 109–110</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  E. Looijenga,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010121.png" />-cohomology of locally symmetric varieties"  ''Computers Math.'' , '''67'''  (1988)  pp. 3–20</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  "The zeta functions of Picard modular surfaces"  R.P. Langlands (ed.)  D. Ramakrishnan (ed.) , Publ. CRM  (1992)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I.I. Piateckii-Shapiro,   "Arithmetic groups in complex domains"  ''Russian Math. Surveys'' , '''19'''  (1964)  pp. 83–109  ''Uspekhi Mat. Nauk.'' , '''19'''  (1964)  pp. 93–121</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  L. Saper,   M. Stern,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010122.png" />-cohomology of arithmetic varieties"  ''Ann. of Math.'' , '''132''' :  2  (1990)  pp. 1–69</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  A. Ash,   D. Mumford,   M. Rapoport,   Y. Tai,   "Smooth compactifications of locally symmetric varieties" , Math. Sci. Press  (1975)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  W.L. Baily, Jr.,   "On Satake's compactification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010123.png" />"  ''Amer. J. Math.'' , '''80'''  (1958)  pp. 348–364</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  W.L. Baily, Jr.,   "On the Hilbert–Siegel modular space"  ''Amer. J. Math.'' , '''81'''  (1959)  pp. 846–874</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  W.L. Baily, Jr.,   "On the orbit spaces of arithmetic groups" , ''Arithmetical Algebraic Geometry (Proc. Conf. Purdue Univ., 1963)'' , Harper and Row  (1965)  pp. 4–10</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  W.L. Baily, Jr.,   "On compactifications of orbit spaces of arithmetic discontinuous groups acting on bounded symmetric domains" , ''Algebraic Groups and Discontinuous Subgroups'' , ''Proc. Symp. Pure Math.'' , '''9''' , Amer. Math. Soc.  (1966)  pp. 281–295</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  I. Satake,   "On the compactification of the Siegel space"  ''J. Indian Math. Soc. (N.S.)'' , '''20'''  (1956)  pp. 259–281</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  I. Satake,   "On compactifications of the quotient spaces for arithmetically defined discontinuous groups"  ''Ann. of Math.'' , '''72''' :  2  (1960)  pp. 555–580</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  "Fonctions automorphes" , ''Sém. H. Cartan 10ième ann. (1957/8)'' , '''1–2''' , Secr. Math. Paris  (1958)  (Cartan)</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  S. Zucker,   "Satake compactifications"  ''Comment. Math. Helv.'' , '''58'''  (1983)  pp. 312–343</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  S. Zucker,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130010/b130010124.png" />-cohomology of warped products and arithmetic groups"  ''Ann. of Math.'' , '''70'''  (1982)  pp. 169–218</TD></TR></table>