%% Papers Published @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}, 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}, 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 = {December}, MRNUMBER = {99a:22019}, 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 = {0821851675}, MRNUMBER = {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 ofQ-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}, 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}, 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}, Month = {December}, ISSN = {0020-9910}, MRNUMBER = {92k:55010}, 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 = {0387700471}, MRNUMBER = {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}, Key = {fds320487} } @article{fds320488, Author = {SAPER, L and STERN, M}, Title = {L2-COHOMOLOGY OF ARITHMETIC VARIETIES}, Journal = {Annals of Mathematics}, Volume = {132}, Number = {1}, Pages = {1-69}, Publisher = {JSTOR}, Year = {1990}, Month = {July}, MRNUMBER = {91m:14027}, Doi = {10.2307/1971500}, Key = {fds320488} } @article{fds320489, Author = {Saper, L and Stern, M}, Title = {L²-cohomology of arithmetic varieties}, Journal = {Proceedings of the National Academy of Sciences of the United States of America}, Volume = {84}, Number = {16}, Pages = {5516-5519}, Year = {1987}, Month = {August}, MRNUMBER = {89g:32052}, 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}, 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}, 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} }