Avraham (Rami) Aizenbud 
אברהם (רמי) איזנבוד 










GENERALIZED FUNCTIONSFall: Tuesday 15:0018:00, Room 261. Spring: Tuesday 14:3017:30, Room 208 Goldsmith building. If you would like to come to the oﬃce hours or to ﬁx some other time please email me. Yotam Hendel Contents
1. General information
2. Overview 3. Prerequisite 4. Chronological list of topics 5. Literature List 6. Email list 7. How to get credit for the course? 8. Lecture notes 9. Homework 10. See also 1. General informationThe course is planned as a year course. However, in the end of the ﬁrst semester we will discuss in what manner we would like to proceed. Therefore the topics that are mentioned at the end of the list below are subject to change. I believe that the course can be of interest to pure and applied math students and those studying theoretical physics. The ﬁrst semester is aimed at M.Sc. students and ﬁrst years Ph.D. students. If you are above this level, but still interested in part of the topics, you can join the email list and you will get an email before each lecture with the topics that will be discussed in the lecture, so you can decide whether you want to come. 2. OverviewWe will study the theory of generalized functions and distributions (which are almost the same thing) on various geometric objects, operations with distributions (like pushforward, pullback and Fourier transform), and invariants of distributions (like the support and the wave front set) The topic by its nature is analytic, but my point of view on this topic is oriented towards representation theory and algebraic geometry, so the course will have some algebraic and geometric ﬂavor. We will discuss both the Archimedean case (i.e. distributions on real geometric objects) and the nonArchimedean case (i.e. distributions on padic geometric objects). We will discuss the similarity and diﬀerence of both cases. During the later stages of the course, we will discuss distributions in the presence of a group action, the notion of an invariant distribution, and diﬀerent methods to prove vanishing of invariant distributions. Those topics are closely related to representation theory. In addition to the main topic of the course, we will have “digressions” (i.e. some lectures that are related to the main topic but not part of it) on: Functional analysis, padic numbers, Harmonic analysis on locally compact abelian groups, Diﬀerentiable manifolds, Nuclear spaces, algebraic and semialgebraic geometry, Dmodules, the Weil representation and geometric invariant theory. Those “digressions” will be done on a very basic level, with the aim of making the students familiar with the basic notions in this topics. In case some of these topics will turn out to be too complicated, we will exclude them together with the related parts of the main topic. I’ll try to include in the course discussion some open (or semiopen) questions, which might ﬁt M.Sc. or Ph.D. students. 3. PrerequisiteI expect the students to be familiar on a basic level with at least 80% of the following notions: Vector space, linear map, subspace, quotient space, dual space, Tensor product. Topological space, Locally compact space, metric space, Complete metric space, completion of a metric space. Diﬀerentiable manifold, tangent space, tangent bundle. Group, group action, abelian group, Hilbert space, Fourier series, measure, Fourier transform 4. Chronological list of topicsEach topic should correspond to approximately one week. Intuitive description, generalized functions on the real line (2 deﬁnitions), derivative of generalized functions, convolution, Green’s function (time independent case), the distribution x^{λ}. Overview. References: [GS, Volume 1, Chapter 1]. 4.2. Digression – Functional analysis. Topological vector space, locally convex vector space, Frechet space, completion of topological vector space, measure, review of generalized functions. References: [Tre, sections 310 and 18,19,21], [GS, Volume 2, Chapter 1, sections 15 and Chapter 2, section 1]. 4.3. Distributions on real vector spaces and its open subsets. Continuous dual, measure, Distributions, sheaf property, partition of unity, support, Distributions supported on subspace (intro.) – References: [GS, Volume 1, Chapter 1, Chapter 3, section 1], [Hör, Chapters 12]. 4.4. Generalized functions on open subsets of real vector spaces (algebraic aspects). distributions vs. generalized functions, top diﬀerential forms, generalized functions supported on subspace. References: [Hör, Chapters 12], [GS, Volume 1, Chapter 3, section 1]. 4.5. Digression – padic numbers and lspaces. Deﬁnitions, Haar mesure. – References: [Kob, chapter 1]. 4.6. Generalized functions on padic vector spaces and lspaces. Deﬁnitions. generalized functions on a product (the nonArchimedean case). Comparison with the Archimedean case. References: [BZ76, Chapter 1, section 1] 4.7. Digression – Diﬀerentiable manifolds and nonArchimedean Geometry. vector bundle, (co)tangent bundle, (co)normal bundle, diﬀerential forms, densities, sheaves, analytic padic manifolds (smooth padic manifolds). References: [GP74, Chapters 1.11.4, 3.2, 4.14.5], [BZ76, Chapter 1, section 1], [Ser64, Part II, Chapter III]. 4.8. Generalized functions on manifolds. Generalized functions on smooth real manifolds, Distributions on lspaces, vector bundles and sheaves, generalized sections of vector bundels and shifs. – References: [Hör, Chapter 6, section 3], [BZ76, Chapter 1, section 1]. 4.9. Operations with distributions. Generalized functions supported on a submanifold. pushforward, pullback, Analytic property of the pushforward. – References: [Hör, chapter 6, section 1]. 4.10. Digresion – Harmonic analysis on locally compact abelian group. Haar measure, Pontryagin dual, Fourier transform. – References: [Dei, Part 2], [HR]. 4.11. Schwartz functions and Fourier transform. Schwartz functions on a real vector space, Fourier transform of Schwartz functions, Schwartz distributions and Generalized Schwartz functions, the nonArchimedean case, Poisson summation formula. – References: [Hör, Chapter 7, sections 1,2], [GS, Volume 1, Chapter 2], [Dei, Chapter 4]. 4.12. The wave front set. Deﬁnition on open sets in vector space, basic properties, relation to pushforward and pullback, deﬁnition on general manifold. – References: [Hör, chapter 8, sections 12]. 4.13. Digression – Dmodules. Dmodules on an aﬃne space, Dmodules on smooth Aﬃne varieties, singular support (a.k.a. Characteristic variety), Bernstein inequality, integrability theorem, holonomicity. – References: [Cou, chapters 02.1, 3, 56.1, 711]. 4.14. Singular suport of distributions. Dmodule attached to a distribution, singular support and holonomicity of distributions, relation to the wave front set, applications. [Hör, chapter 8, section 3]. 4.15. Digression – Nuclear spaces. – References: [Tre, part 3]. 4.16. Schwartz kernel theorem. Generalized functions on a product (the Archimedean case). – References: [GS, Chapter 5], [Tre, part 3, section 51]. 4.17. Schwartz functions on manifolds – Introduction. Motivation (short exact sequence of spaces of distributions), problems, comparison with the nonArchimedean case, diﬀerent approaches. – References: [AG08, section 1]. 4.18. Digression – algebraic and semialgebraic geometry. Spaces with a sheaf of functions, complex algebraic variety, real algebraic variety, Nash manifolds. ZaidenbergTarski theorem. – References: [Kem, Chapter 1], [Shi, chapters 14], [AG08, section 23], [BCR]. 4.19. Schwartz functions on Nash manifolds. Deﬁnitions, basic properties, Schwartz distributions supported on a submanifold, Schwartz functions on a product, pushforward, pullback of Schwartz functions. – References: [AG08]. 4.20. Group actions distributions. Group action on manifolds, invariant distributions, equivariant distributions, group actions on bundles, invariant generalized sections. The wave front set and the singular support of invariant distributions. 4.21. Vanishing of invariant distributions. Stratiﬁcation and extension (higher cohomology), product, Schwartz vs. general distributions. 4.22. Localization principle. The nonArchimedean case, problems with the Archimedean case. – References: [Ber82, section 1.4]. 4.23. Frobenius descent (a.k.a. Frobenius reciprocity). – References: [Ber82, section 1.5], [AGS08, Appendix A]. Introduction, Heisenberg uncertainty principle, qualitative uncertainty principle – simple example. 4.25. The wave front set and uncertainty principle. – References: [AG09b, section 2.3], [Aiz, section 4]. 4.26. Homogeneity and uncertainty principle. Formulation, example of usage, comparison of the Archimedean and nonArchimedean case. – References: [AGS08, Section 4.3]. 4.27. Digression – The Weil representation. The one dimensional case: Weil representation of the Lie algebra sl_{2}, Weil representation of the Lie group SL_{2}, relation with Heisenberg representation. The general case. – References: [LV, Part 1]. 4.28. The Weil representation and uncertainty principle. The nonArchimedean case, the Archimedean case, example of usage. – References: [AGa, section 6], [AG09b, section 2.2.4]. 4.29. Digression – Luna’s slice theorem. – References: [Dre00]. 4.30. Speciﬁc examples of vanishing of invariant distributions. – References: [AG09a, AGRS10, AG09b] 4.31. Invariant distributions vs. coinvariant test functions. – References: [AGb, section Appendix(es)]. 5. Literature ListMost of the material of the ﬁrst semester is contained in
However the approach in those sources is slightly diﬀerent and they contain a lot of additional information. For some of the lectures I’ll need other references as mentioned in the topics list above. Note that the “digression” lectures are usually on a much more basic level than the corresponding source, so do not be surprised if it looks much longer than you expect from onelecture material. In the beginning of each lecture I’ll try to mention the sources that I will be using. I’ll also do it one lecture ahead. It is also a good idea to use less formal web resources (like the ones that are linked above), especially for the “digression” lectures. Below you can ﬁnd the full alphabetical reference list, but everything except [Hör, GS, BZ76] should be considered as supplementary material rather than the main source.
[AG08] A. Aizenbud, D. Gourevitch, Schwartz functions on Nash Manifolds, International Mathematic Research Notes 2008/5 (2008). See also arXiv:0704.2891 [math.AG]. [AG09a] Aizenbud, A.; Gourevitch, D.:Generalized HarishChandra descent, Gelfand pairs and an Archimedean analog of JacquetRallis’ Theorem. Duke Mathematical Journal, 149/3 (2009). See also arxiv:0812.5063[math.RT]. [AG09b] A. Aizenbud, D. Gourevitch, Multiplicity one theorem for (GL_{n+1}(R),GL_{n}(R)), Selecta Mathematica, 15/2,(2009). See also arXiv:0808.2729 [math.RT]. [AGa] A. Aizenbud and D. Gourevitch, A proof of the multiplicity one conjecture for GL(n) in GL(n + 1)., arXiv:0707.2363v2 [math.RT] (2007). [AGb] A. Aizenbud, D. Gourevitch, Smooth Transfer of Kloosterman Integrals (the Archimedean case), arxiv:1001.2490, to appear in the American Journal of Mathematics. [AGRS10] A. Aizenbud, D. Gourevitch, S. Rallis, G. Schiﬀmann, Multiplicity One Theorems, Annals of Mathematics 172/2 (2010), see also arXiv:0709.4215 [math.RT]. [AGS08] A. Aizenbud, D. Gourevitch, E. Sayag : (GL_{n+1}(F),GL_{n}(F)) is a Gelfand pair for any local ﬁeld F, Compositio Mathematica, 144 (2008), see postprint: arXiv:0709.1273[math.RT] [Aiz] A. Aizenbud, A partial analog of integrability theorem for distributions on padic spaces and applications, arXiv:0811.2768, To appear in the Israel Journal of Mathematics. [BCR] Bochnak, J.; Coste, M.; Roy, MF.: Real Algebraic Geometry Berlin: Springer, 1998. [Ber82] J. Bernstein, Pinvariant Distributions on GL(N) and the classiﬁcation of unitary representations of GL(N) (nonArchimedean case), Lie group representations, II (College Park, Md., 1982/1983), 50–102. [BZ76] J. Bernstein, A.V. Zelevinsky, Representations of the group GL(n,F), where F is a local nonArchimedean ﬁeld, Uspekhi Mat. Nauk. 10/3, (1976). [Cou] S. Coutinho, A Primer of Algebraic DModules Cambridge University Press. [Dei] Anton Deitmar, A First Course in Harmonic Analysis [Dre00] J.M.Drezet, Luna’s Slice Theorem And Applications, 23d Autumn School in Algebraic Geometry ”Algebraic group actions and quotients” Wykno (Poland), (2000). [GS] I.M. Gelfand, G. Shilov Generalized functions volumes I,II. [GP74] V. Guillemin, A. Pollack, Diﬀerential Topology Englewood Cliﬀs, N.J. : PrenticeHall, 1974. [Hör] L. Hörmander, The analysis of linear partial diﬀerential operators. I. Distribution theory and Fourier analysis. Grundlehren der Mathematischen Wissenschaften 256. SpringerVerlag, Berlin, 1990. [HR] E. Hewitt, K. Ross, Abstract harmonic analysis. Volume I [Kem] G. Kempf , Algebraic Varieties. Cambridge University Press, 1993. [Kob] N. Koblitz, padic Numbers, padic Analysis, and ZetaFunctions, Volume 58 of Graduate Texts in Mathematics Series Edition 2, SpringerVerlag, 1984 [LV] G. Lion, M. Vergne, The Weil representation, Maslov index and theta series, Progress in Mathematics, 6, Boston: Birkhauser. [Tre] F. Treves, Topological vector spaces, Distributions and Kernels. Academic Press, New York and London, 1967 [Ser64] J.P. Serre: Lie Algebras and Lie Groups Lecture Notes in Mathematics 1500, SpringerVerlag, New York, (1964). [Shi] Shiota, M: Nash Manifolds. Lecture Notes in Mathematics 1269 (1987) 6. Email listTo join/unjoin the course email list send me an email (from the address you wish to join/unjoin) with subject join/unjoin me to genfun5776”. To send a message to the course mailing list send me an email with subject “email to genfun5776 – the subject of your message”. 7. How to get credit for the course?All of the information below is conditional, it might change if it will turn out that my expectations are too high or too low. My aim is that any student who meets the prerequisites will be able to get at least 90 if he or she does a reasonable amount of work (34 hours a week in addition to the lectures and practice sections). The course grade will consist mainly of the following 3 equal parts.
Some remarks:
In addition, throughout the course, there will be given bonus assignments. Each assignment can give one student up to 15 points bonus. If 2 (or 3) students want to do one bonus assignment together, the points will be split. If more than one student wishes to do one bonus assignment (and they do not want to do it together) the priority will be given ﬁrst to those who did not do any bonus assignments (or did a smaller part) and then to M.Sc. students. Although the regular homework will not be graded, you are strongly encouraged to do it and write it down. It is crucial for the understanding of the material. It is a good idea to do it in groups of 23 students. Each group can meet once a week (preferably not on the day of the lecture or an adjacent day, so your study will have no large gaps). Please let me know about the membership of your group and when do you meet, so I can try to be around, in case you need me. 8. Lecture notes1Lectures 19 (by Itay Glazer; based on previous lecture notes by Noam Kahalon) Exercises sessions (by Yotam Hendel) 9. Homework and solutions1Homework 110 (by Yotam Hendel) HW 1 solutions (by Yotam Alexander) HW 2 solutions (by Dan Mikulincer) HW 3 solutions (by Dan Mikulincer) HW 4 solutions (by Yotam Alexander) HW 5 solutions (by Shai Keidar) HW 6 solutions (by Shai Keidar) HW 7 solutions (by Yotam Alexander) HW 8 solutions (by Dan Mikulincer) HW 9 solutions (by Shai Keidar) 10. See also1. The notes and the solutions are writen by the students, they contain some mistakes. Students that wish to upload corrected versions are wellcome to send them to me. 