%% Papers Published
@article{fds359205,
Author = {Cox, D and Esnault, H and Hain, R and Harris, M and Ji, L and Saito, M-H and Saper, L},
Title = {Remembering Steve Zucker},
Journal = {Notices of the American Mathematical Society},
Volume = {68},
Number = {7},
Pages = {1156-1172},
Publisher = {American Mathematical Society},
Editor = {Cox, D and Harris, M and Ji, L},
Year = {2021},
Month = {August},
Key = {fds359205}
}
@article{fds320773,
Author = {Saper, L},
Title = {ℒ-modules and micro-support},
Journal = {to appear in Annals of Mathematics},
Year = {2018},
Key = {fds320773}
}
@article{fds320662,
Author = {Saper, L},
Title = {Perverse sheaves and the reductive Borel-Serre
compactification},
Volume = {39},
Pages = {555-581},
Booktitle = {Hodge Theory and L²-analysis},
Publisher = {International Press},
Editor = {Ji, L},
Year = {2017},
Abstract = {We briefly introduce the theory of perverse sheaves with
special attention to the topological situation where strata
can have odd dimension. This is part of a project to use
perverse sheaves on the topological reductive Borel-Serre
compactification of a Hermitian locally symmetric space as a
tool to study perverse sheaves on the Baily-Borel
compactification, a projective algebraic variety. We sketch
why the decomposition theorem holds for the natural map
between the reductive Borel-Serre and the Baily-Borel
compactifications. We demonstrate how to calculate
extensions of simple perverse sheaves on the reductive
Borel-Serre compactification and illustrate with the example
of Sp(4,R).},
Key = {fds320662}
}
@article{fds320536,
Author = {Ji, L and Murty, VK and Saper, L and Scherk, J},
Title = {The fundamental group of reductive Borel–Serre and Satake
compactifications},
Journal = {Asian Journal of Mathematics},
Volume = {19},
Number = {3},
Pages = {465-486},
Publisher = {International Press of Boston},
Year = {2015},
url = {http://www.intlpress.com/site/pub/pages/journals/items/ajm/content/vols/0019/0003/a004/},
Abstract = {Let G be an almost simple, simply connected algebraic group
defined over a number field k, and let S be a finite set of
places of k including all infinite places. Let X be the
product over v ε S of the symmetric spaces associated to
G(kv), when v is an infinite place, and the Bruhat-Tits
buildings associated to G(kv), when v is a finite place. The
main result of this paper is to compute explicitly the
fundamental group of the reductive Borel-Serre
compactification of Γ\X, where Γ is an S-arithmetic
subgroup of G. In the case that G is neat, we show that this
fundamental group is isomorphic to Γ/EΓ, where EΓ is the
subgroup generated by the elements of Γ belonging to
unipotent radicals of k-parabolic subgroups. Analogous
computations of the fundamental group of the Satake
compactifications are made. It is noteworthy that
calculations of the congruence subgroup kernel C(S, G) yield
similar results.},
Doi = {10.4310/AJM.2015.v19.n3.a4},
Key = {fds320536}
}
@article{fds37687,
Author = {Leslie D. Saper},
Title = { -modules and the conjecture of Rapoport
and Goresky-MacPherson},
Volume = {298},
Series = {Astérisque},
Pages = {319--334},
Booktitle = {Formes Automorphes (I) -- Actes du Semestre du Centre
Émile Borel, printemps 2000},
Publisher = {Société Mathématique de
France},
Editor = {J. Tilouine and H. Carayol and M. Harris and M.-F.
Vignéras},
Year = {2005},
MRNUMBER = {2141706},
url = {http://arxiv.org/abs/math/0112250},
Keywords = {Intersection cohomology • Shimura varieties •
locally symmetric spaces • compactifications},
Abstract = {<a href="http://smf.emath.fr/Publications/Asterisque/2005/298/html/smf_ast_298_319-334.html">click
here for abstract of published paper</a>},
Key = {fds37687}
}
@article{fds244104,
Author = {Saper, L},
Title = {ℒ-modules and the conjecture of Rapoport and
Goresky-Macpherson},
Volume = {298},
Number = {298},
Pages = {319-334},
Booktitle = {Formes Automorphes (I) — Actes du Semestre du Centre
Émile Borel, printemps 2000},
Editor = {Tilouine, J and Carayol, H and Harris, M and Vignéras,
M-F},
Year = {2005},
ISBN = {2-85629-172-4},
ISSN = {0303-1179},
Abstract = {Consider the middle perversity intersection cohomology
groups of various compactifications of a Hermitian locally
symmetric space. Rapoport and independently Goresky and
MacPherson have conjectured that these groups coincide for
the reductive Borel-Serre compactification and the
Baily-Borel-Satake compactification. This paper describes
the theory of ℒ-modulcs and how it is used to solve the
conjecture. More generally we consider a Satake
compactification for which all real boundary components are
equal-rank. Details will be given elsewhere, As another
application of ℒ-modules, we prove a vanishing theorem for
the ordinary cohomology of a locally symmetric space. This
answers a question raised by Tilouine.},
Key = {fds244104}
}
@article{fds305514,
Author = {Saper, L},
Title = {L²-cohomology of locally symmetric spaces.
I},
Journal = {Pure and Applied Mathematics Quarterly},
Volume = {1},
Number = {4},
Pages = {889-937},
Publisher = {International Press of Boston},
Year = {2005},
MRNUMBER = {2201005},
url = {http://arxiv.org/abs/math/0412353v3},
Abstract = {Let X be a locally symmetric space associated to a reductive
algebraic group G defined over Q. L-modules are a
combinatorial analogue of constructible sheaves on the
reductive Borel-Serre compactification of X; they were
introduced in [math.RT/0112251]. That paper also introduced
the micro-support of an L-module, a combinatorial invariant
that to a great extent characterizes the cohomology of the
associated sheaf. The theory has been successfully applied
to solve a number of problems concerning the intersection
cohomology and weighted cohomology of the reductive
Borel-Serre compactification [math.RT/0112251], as well as
the ordinary cohomology of X [math.RT/0112250]. In this
paper we extend the theory so that it covers L²-cohomology.
In particular we construct an L-module whose cohomology is
the L²-cohomology of X and we calculate its micro-support.
As an application we obtain a new proof of the conjectures
of Borel and Zucker.},
Doi = {10.4310/PAMQ.2005.v1.n4.a9},
Key = {fds305514}
}
@article{fds244105,
Author = {Saper, L},
Title = {Geometric rationality of equal-rank Satake
compactifications},
Journal = {Mathematical Research Letters},
Volume = {11},
Number = {5},
Pages = {653-671},
Publisher = {International Press of Boston},
Year = {2004},
MRNUMBER = {2106233},
url = {http://arxiv.org/abs/math/0211138v4},
Abstract = {Satake has constructed compactifications of symmetric spaces
D=G/K which (under a condition called geometric rationality
by Casselman) yield compactifications of the corresponding
locally symmetric spaces. The different compactifications
depend on the choice of a representation of G. One example
is the Baily-Borel-Satake compactification of a Hermitian
locally symmetric space; Baily and Borel proved this is
always geometrically rational. Satake compactifications for
which all the real boundary components are equal-rank
symmetric spaces are a natural generalization of the
Baily-Borel-Satake compactification. Recent work (see
math.RT/0112250, math.RT/0112251) indicates that this is the
natural setting for many results about cohomology of
compactifications of locally symmetric spaces. In this paper
we prove any Satake compactification for which all the real
boundary components are equal-rank symmetric spaces is
geometrically rational aside from certain rational rank 1 or
2 exceptions; we completely analyze geometric rationality
for these exceptional cases. The proof uses Casselman's
criterion for geometric rationality. We also prove that a
Satake compactification is geometrically rational if the
representation is defined over the rational
numbers.},
Doi = {10.4310/MRL.2004.v11.n5.a9},
Key = {fds244105}
}
@article{fds320537,
Author = {Saper, L},
Title = {On the Cohomology of Locally Symmetric Spaces and of their
Compactifications},
Pages = {219-289},
Booktitle = {Current developments in mathematics, 2002},
Publisher = {International Press},
Editor = {Jerison, D and Lusztig, G and Mazur, B and Mrowka, T and Schmid, W and Stanley, R and Yau, S-T},
Year = {2003},
MRNUMBER = {2062320},
url = {http://arxiv.org/abs/math/0306403},
Abstract = {This expository article is an expanded version of talks
given at the "Current Developments in Mathematics, 2002"
conference. It gives an introduction to the (generalized)
conjecture of Rapoport and Goresky-MacPherson which
identifies the intersection cohomology of a real equal-rank
Satake compactification of a locally symmetric space with
that of the reductive Borel-Serre compactification. We
motivate the conjecture with examples and then give an
introduction to the various topics that are involved:
intersection cohomology, the derived category, and
compactifications of a locally symmetric space, particularly
those above. We then give an overview of the theory of
L-modules and micro-support (see math.RT/0112251) which was
developed to solve the conjecture but has other important
applications as well. We end with sketches of the proofs of
three main theorems on L-modules that lead to the resolution
of the conjecture. The text is enriched with many examples,
illustrations, and references to the literature.},
Key = {fds320537}
}
@article{fds244108,
Author = {Saper, L},
Title = {Tilings and finite energy retractions of locally symmetric
spaces},
Journal = {Commentarii Mathematici Helvetici},
Volume = {72},
Number = {2},
Pages = {167-201},
Publisher = {Research Institute for Mathematical Sciences, Kyoto
University},
Year = {1997},
Month = {January},
MRNUMBER = {99a:22019},
url = {http://dx.doi.org/10.1007/pl00000369},
Abstract = {Let Γ\X̄ be the Borel-Serre compactification of an
arithmetic quotient Γ\X of a symmetric space of noncompact
type. We construct natural tilings Γ\X̄ = ∐P Γ\X̄P
(depending on a parameter b) which generalize the
Arthur-Langlands partition of Γ\X. This is applied to yield
a natural piecewise analytic deformation retraction of
Γ\X̄ onto a compact submanifold with corners Γ\X0 ⊂
Γ\X. In fact, we prove that Γ\X0 is a realization (under a
natural piecewise analytic diffeomorphism) of Γ\X̄ inside
the interior Γ\X. For application to the theory of harmonic
maps and geometric rigidity, we prove this retraction and
diffeomorphism have finite energy except for a few low rank
examples. We also use tilings to give an explicit
description of a cofinal family of neighborhoods of a face
of Γ\X̄, and study the dependance of tilings on the
parameter b and the degeneration of tilings.},
Doi = {10.1007/pl00000369},
Key = {fds244108}
}
@article{fds320484,
Author = {Saper, L},
Title = {L²-cohomology of the Weil-Peterson metric},
Volume = {150},
Pages = {345-360},
Booktitle = {Mapping Class Groups and Moduli Spaces of Riemann Surfaces
Proceedings of Workshops Held June 24-28, 1991, in
Göttingen, Germany, and August 6-10, 1991, in Seattle,
Washington},
Publisher = {Amer. Math. Soc.},
Editor = {Bödigheimer, C-F and Hain, R},
Year = {1993},
ISBN = {9780821851678},
MRNUMBER = {94j:32014},
url = {http://www.ams.org/mathscinet-getitem?mr=94j:32014},
Key = {fds320484}
}
@article{fds9231,
Author = {Leslie D. Saper and Mark Stern},
Title = {Appendix to: On the shape of the contribution of a fixed
point on the boundary. The case of Q-rank one, by M.
Rapoport},
Booktitle = {The Zeta Functions of Picard Modular Surfaces},
Publisher = {Les Publications CRM, Montréal},
Editor = {R. Langlands and D. Ramakrishnan},
Year = {1992},
MRNUMBER = {93e:11070b},
url = {http://www.ams.org/mathscinet-getitem?mr=93e:11070b},
Key = {fds9231}
}
@article{fds244107,
Author = {Saper, L},
Title = {L²-cohomology of Kähler varieties with isolated
singularities},
Journal = {Journal of Differential Geometry},
Volume = {36},
Number = {1},
Pages = {89-161},
Publisher = {International Press of Boston},
Year = {1992},
ISSN = {1945-743X},
MRNUMBER = {93e:32038},
url = {http://dx.doi.org/10.4310/jdg/1214448444},
Doi = {10.4310/jdg/1214448444},
Key = {fds244107}
}
@article{fds320485,
Author = {Saper, L and Stern, M},
Title = {Appendix to: On the shape of the contribution of a fixed
point on the boundary. The case of Q-rank one, by M.
Rapoport},
Pages = {489-491},
Booktitle = {The zeta functions of Picard modular surfaces based on
lectures delivered at a CRM Workshop in the spring of
1988},
Publisher = {Centre De Recherches Mathématiques},
Editor = {Langlands, R and Ramakrishnan, D},
Year = {1992},
Key = {fds320485}
}
@article{fds244103,
Author = {Habegger, N and Saper, L},
Title = {Intersection cohomology of cs-spaces and Zeeman's
filtration},
Journal = {Inventiones Mathematicae},
Volume = {105},
Number = {1},
Pages = {247-272},
Publisher = {Springer Nature},
Year = {1991},
ISSN = {0020-9910},
MRNUMBER = {92k:55010},
url = {http://dx.doi.org/10.1007/BF01232267},
Doi = {10.1007/BF01232267},
Key = {fds244103}
}
@article{fds320486,
Author = {Saper, L},
Title = {L₂-cohomology of algebraic varieties},
Volume = {1},
Pages = {735-746},
Booktitle = {Proceedings of the International Congress of Mathematicians,
August 21-29, 1990, Kyoto},
Publisher = {Springer-Verlag},
Editor = {Satake, I},
Year = {1991},
ISBN = {9780387700472},
MRNUMBER = {93e:32037},
url = {http://www.ams.org/mathscinet-getitem?mr=93e:32037},
Key = {fds320486}
}
@article{fds320487,
Author = {Saper, L and Zucker, S},
Title = {An introduction to L²-cohomology},
Volume = {52, Part 2},
Pages = {519-534},
Booktitle = {Several Complex Variables and Complex Geometry},
Publisher = {Amer. Math. Soc.},
Year = {1991},
ISBN = {0821814907},
MRNUMBER = {92k:14023},
url = {http://www.ams.org/mathscinet-getitem?mr=92k:14023},
Key = {fds320487}
}
@article{fds320488,
Author = {Stern, M and Saper, L},
Title = {L²-cohomology of arithmetic varieties},
Journal = {Annals of Mathematics},
Volume = {132},
Number = {1},
Pages = {1-69},
Publisher = {JSTOR},
Year = {1990},
MRNUMBER = {91m:14027},
url = {http://dx.doi.org/10.2307/1971500},
Doi = {10.2307/1971500},
Key = {fds320488}
}
@article{fds320489,
Author = {Saper, L and Stern, M},
Title = {L²-cohomology of arithmetic varieties},
Journal = {Proc Natl Acad Sci U.S.A.},
Volume = {84},
Number = {16},
Pages = {5516-5519},
Year = {1987},
Month = {August},
MRNUMBER = {89g:32052},
url = {http://dx.doi.org/10.1073/pnas.84.16.5516},
Abstract = {The L₂-cohomology of arithmetic quotients of bounded
symmetric domains is studied. We establish the conjecture of
Zucker equating the L₂-cohomology of these spaces to the
intersection cohomology of their Baily-Borel
compactifications.},
Doi = {10.1073/pnas.84.16.5516},
Key = {fds320489}
}
@article{fds244102,
Author = {Saper, L},
Title = {L₂-cohomology and intersection homology of certain
algebraic varieties with isolated singularities},
Journal = {Inventiones Mathematicae},
Volume = {82},
Number = {2},
Pages = {207-255},
Publisher = {Springer Nature},
Year = {1985},
ISSN = {0020-9910},
MRNUMBER = {87h:32029},
url = {http://dx.doi.org/10.1007/BF01388801},
Doi = {10.1007/BF01388801},
Key = {fds244102}
}
%% Preprints
@article{fds213234,
Author = {L. Ji and K. Murty and L. Saper and J. Scherk},
Title = {The Congruence Subgroup Kernel and the Fundamental Group of
the Reductive Borel-Serre Compactification},
Year = {2011},
Month = {June},
url = {http://arxiv.org/abs/1106.4810},
Abstract = {Let <b>G</b> be an almost simple, simply connected algebraic
group defined over a number field <i>k</i>, and let <i>S</i>
be a finite set of places of <i>k</i> including all infinite
places. Let <i>X</i> be the product over
<nobr><i>v</i> ∈ <i>S</i></nobr> of the
symmetric spaces associated to <b>G</b>(<i>k<sub>v</sub></i>),
when <i>v</i> is an infinite place, and the Bruhat-Tits
buildings associated to <b>G</b>(<i>k<sub>v</sub></i>), when
<i>v</i> is a finite place. The main result of this paper is
to identify the congruence subgroup kernel with the
fundamental group of the reductive Borel-Serre
compactification of <nobr>Γ \ <i>X</i></nobr>
for certain sufficiently small <i>S</i>-arithmetic subgroups
Γ of <b>G</b>. Our result follows from explicit
computations of the fundamental group of the reductive
Borel-Serre compactifications of <nobr>Γ \ <i>X</i></nobr>.
In the case that Γ is neat, we show that this
fundamental group is isomorphic to <nobr>Γ / <i>E</i>Γ</nobr>,
where <i>E</i>Γ is the subgroup generated by the
elements of Γ belonging to unipotent radicals of
parabolic <i>k</i>-subgroups. Similar computations of the
fundamental group of the Satake compactifications are
made.},
Key = {fds213234}
}
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Mathematics Department