%% Papers Published
@article{fds320773,
Author = {Saper, L},
Title = {ℒmodules and microsupport},
Journal = {To Appear in Annals of Mathematics},
Year = {2018},
Key = {fds320773}
}
@article{fds320662,
Author = {Saper, L},
Title = {Perverse sheaves and the reductive BorelSerre
compactification},
Volume = {39},
Pages = {555581},
Booktitle = {Hodge Theory and L²analysis},
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 BorelSerre
compactification of a Hermitian locally symmetric space as a
tool to study perverse sheaves on the BailyBorel
compactification, a projective algebraic variety. We sketch
why the decomposition theorem holds for the natural map
between the reductive BorelSerre and the BailyBorel
compactifications. We demonstrate how to calculate
extensions of simple perverse sheaves on the reductive
BorelSerre 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 = {465486},
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 BruhatTits
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 BorelSerre
compactification of Γ\X, where Γ is an Sarithmetic
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 kparabolic 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 GoreskyMacPherson},
Volume = {298},
Series = {Astérisque},
Pages = {319334},
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_319334.html">click
here for abstract of published paper</a>},
Key = {fds37687}
}
@article{fds244104,
Author = {Saper, L},
Title = {ℒmodules and the conjecture of Rapoport and
GoreskyMacpherson},
Volume = {298},
Number = {298},
Pages = {319334},
Booktitle = {Formes Automorphes (I) — Actes du Semestre du Centre
Émile Borel, printemps 2000},
Editor = {Tilouine, J and Carayol, H and Harris, M and Vignéras,
MF},
Year = {2005},
ISBN = {2856291724},
ISSN = {03031179},
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 BorelSerre compactification and the
BailyBorelSatake compactification. This paper describes
the theory of ℒmodulcs and how it is used to solve the
conjecture. More generally we consider a Satake
compactification for which all real boundary components are
equalrank. Details will be given elsewhere, As another
application of ℒmodules, 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 = {L²cohomology of locally symmetric spaces.
I},
Journal = {Pure and Applied Mathematics Quarterly},
Volume = {1},
Number = {4},
Pages = {889937},
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. Lmodules are a
combinatorial analogue of constructible sheaves on the
reductive BorelSerre compactification of X; they were
introduced in [math.RT/0112251]. That paper also introduced
the microsupport of an Lmodule, 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
BorelSerre 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 L²cohomology.
In particular we construct an Lmodule whose cohomology is
the L²cohomology of X and we calculate its microsupport.
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 equalrank Satake
compactifications},
Journal = {Mathematical Research Letters},
Volume = {11},
Number = {5},
Pages = {653671},
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 BailyBorelSatake 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 equalrank
symmetric spaces are a natural generalization of the
BailyBorelSatake 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 equalrank 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 = {219289},
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, ST},
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 GoreskyMacPherson which
identifies the intersection cohomology of a real equalrank
Satake compactification of a locally symmetric space with
that of the reductive BorelSerre 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
Lmodules and microsupport (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 Lmodules 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 = {167201},
Publisher = {Research Institute for Mathematical Sciences, Kyoto
University},
Year = {1997},
Month = {December},
MRNUMBER = {99a:22019},
url = {http://dx.doi.org/10.1007/pl00000369},
Abstract = {Let Γ\X̄ be the BorelSerre 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
ArthurLanglands 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 = {L²cohomology of the WeilPeterson metric},
Volume = {150},
Pages = {345360},
Booktitle = {Mapping Class Groups and Moduli Spaces of Riemann Surfaces
Proceedings of Workshops Held June 2428, 1991, in
Göttingen, Germany, and August 610, 1991, in Seattle,
Washington},
Publisher = {Amer. Math. Soc.},
Editor = {Bödigheimer, CF and Hain, R},
Year = {1993},
ISBN = {0821851675},
MRNUMBER = {94j:32014},
url = {http://www.ams.org/mathscinetgetitem?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 Qrank 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/mathscinetgetitem?mr=93e:11070b},
Key = {fds9231}
}
@article{fds244107,
Author = {Saper, L},
Title = {L²cohomology of Kähler varieties with isolated
singularities},
Journal = {Journal of Differential Geometry},
Volume = {36},
Number = {1},
Pages = {89161},
Publisher = {International Press of Boston},
Year = {1992},
ISSN = {1945743X},
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 Qrank one, by M.
Rapoport},
Pages = {489491},
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 csspaces and Zeeman's
filtration},
Journal = {Inventiones Mathematicae},
Volume = {105},
Number = {1},
Pages = {247272},
Publisher = {Springer Nature},
Year = {1991},
Month = {December},
ISSN = {00209910},
MRNUMBER = {92k:55010},
url = {http://dx.doi.org/10.1007/BF01232267},
Doi = {10.1007/BF01232267},
Key = {fds244103}
}
@article{fds320486,
Author = {Saper, L},
Title = {L₂cohomology of algebraic varieties},
Volume = {1},
Pages = {735746},
Booktitle = {Proceedings of the International Congress of Mathematicians,
August 2129, 1990, Kyoto},
Publisher = {SpringerVerlag},
Editor = {Satake, I},
Year = {1991},
ISBN = {0387700471},
MRNUMBER = {93e:32037},
url = {http://www.ams.org/mathscinetgetitem?mr=93e:32037},
Key = {fds320486}
}
@article{fds320487,
Author = {Saper, L and Zucker, S},
Title = {An introduction to L²cohomology},
Volume = {52, Part 2},
Pages = {519534},
Booktitle = {Several Complex Variables and Complex Geometry},
Publisher = {Amer. Math. Soc.},
Year = {1991},
ISBN = {0821814907},
MRNUMBER = {92k:14023},
url = {http://www.ams.org/mathscinetgetitem?mr=92k:14023},
Key = {fds320487}
}
@article{fds320488,
Author = {SAPER, L and STERN, M},
Title = {L2COHOMOLOGY OF ARITHMETIC VARIETIES},
Journal = {Annals of Mathematics},
Volume = {132},
Number = {1},
Pages = {169},
Publisher = {JSTOR},
Year = {1990},
Month = {July},
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 = {L²cohomology of arithmetic varieties},
Journal = {Proceedings of the National Academy of Sciences of the
United States of America},
Volume = {84},
Number = {16},
Pages = {55165519},
Year = {1987},
Month = {August},
MRNUMBER = {89g:32052},
url = {http://dx.doi.org/10.1073/pnas.84.16.5516},
Abstract = {The L₂cohomology of arithmetic quotients of bounded
symmetric domains is studied. We establish the conjecture of
Zucker equating the L₂cohomology of these spaces to the
intersection cohomology of their BailyBorel
compactifications.},
Doi = {10.1073/pnas.84.16.5516},
Key = {fds320489}
}
@article{fds244102,
Author = {Saper, L},
Title = {L₂cohomology and intersection homology of certain
algebraic varieties with isolated singularities},
Journal = {Inventiones Mathematicae},
Volume = {82},
Number = {2},
Pages = {207255},
Publisher = {Springer Nature},
Year = {1985},
ISSN = {00209910},
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 BorelSerre 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 BruhatTits
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 BorelSerre
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
BorelSerre 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}
}
