SL2(R)(英文影印版)
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- 原书名： SL2 (R) (Graduate Texts in Mathematics)
- 原出版社： Springer
- 作者： Serge Lang
- 出版社：世界图书出版公司
- ISBN：9787510004544
- 上架时间：2009-9-11
- 出版日期：2009 年8月
- 开本：24开
- 页码：428
- 版次：1-1
- 所属分类：
数学 > 代数，数论及组合理论 > 群表示论
内容简介回到顶部↑
Starting with Bargmann's paper on the infinite dimensional representations of SL2(R), the theory of representations of semisimple Lie groups has evolved to a rather extensive production. Some of the main contributors have been:Gelfand-Naimark and Harish-Chandra, who considered the Lorentz group in the late forties; Geifand-Naimark, who dealt with the classical complex groups, while Harish-Chandra worked out the general real case, especially through the derived representation of the Lie algebra, establishing the Plancherel formula (Gelfand-Graev also contributed to the real case); Car-tan, Gelfand-Naimark, Godement, Harish-Chandra, who developed the theory of spherical functions (Godement gave several Bourbaki seminar reports giving proofs for a number of spectral results not accessible other-wise); Selberg, who took the group modulo a discrete subgroup and obtained the trace formula; Gelfand, Fomin, Pjateckii-Shapiro, and Harish-Chandra,who established connections with automorphic forms; Jacquet-Langlands,who pushed through the connection with L-series and Hecke theory. This history is so involved and so extensive that I am incompetent to give a really good account, and I refer the reader to bibliographies in the books by Warner, Gelfand-Graev-Pjateckii-Shapiro, and Helgason for further infor-mation. A few more historical comments will be made in the appropriate places in the book. ...
目录回到顶部↑
notation .
chapter i general results
1 the representation on cc(g)
2 a criterion for complete reducibility
3 l2 kernels and operators
4 plancherel measures
chapter ii compact groups
1 decomposition over k for sl2(r)
2 compact groups in general
chapter iii induced representations
1 integration on coset spaces
2 induced representations
3 associated spherical functions
4 the kernel defining the induced representation
chapter iv spherical functions
1 bi-invariance
2 irreducibility
3 the spherical property
4 connection with unitary representations
5 positive v the spherical transform
chapter i general results
1 the representation on cc(g)
2 a criterion for complete reducibility
3 l2 kernels and operators
4 plancherel measures
chapter ii compact groups
1 decomposition over k for sl2(r)
2 compact groups in general
chapter iii induced representations
1 integration on coset spaces
2 induced representations
3 associated spherical functions
4 the kernel defining the induced representation
chapter iv spherical functions
1 bi-invariance
2 irreducibility
3 the spherical property
4 connection with unitary representations
5 positive v the spherical transform
序言回到顶部↑
Starting with Bargmann's paper on the infinite dimensional representations of SL2(R), the theory of representations of semisimple Lie groups has evolved to a rather extensive production. Some of the main contributors have been:Gelfand-Naimark and Harish-Chandra, who considered the Lorentz group in the late forties; Geifand-Naimark, who dealt with the classical complex groups, while Harish-Chandra worked out the general real case, especially through the derived representation of the Lie algebra, establishing the Plancherel formula (Gelfand-Graev also contributed to the real case); Car-tan, Gelfand-Naimark, Godement, Harish-Chandra, who developed the theory of spherical functions (Godement gave several Bourbaki seminar reports giving proofs for a number of spectral results not accessible other-wise); Selberg, who took the group modulo a discrete subgroup and obtained the trace formula; Gelfand, Fomin, Pjateckii-Shapiro, and Harish-Chandra,who established connections with automorphic forms; Jacquet-Langlands,who pushed through the connection with L-series and Hecke theory. This history is so involved and so extensive that I am incompetent to give a really good account, and I refer the reader to bibliographies in the books by Warner, Gelfand-Graev-Pjateckii-Shapiro, and Helgason for further infor-mation. A few more historical comments will be made in the appropriate places in the book. .
it is not easy to get into representation theory, especially for someone interested in number theory, for a number of reasons. First, the general theorems on higher dimensional groups require massive doses of Lie theory.Second, one needs a good background in standard and not so standard analysis on a fairly broad scale. Third, the experts have been writing for each other for so long that the literature is somewhat labyrinthine.
I got interested because of the obvious connections with number theory,principally through Langlands' conjecture relating representation theory to elliptic curves [La 2]. This is a global conjecture, in the adelic theory. I realized soon enough that it was best to acquire a good understanding of the real theory before getting everything on the adeles. I think most people who have worked in representations have looked at SL2(R) first, and I know this is the case for both Harish and Langlands.
Therefore, as I learned the theory myself it seemed a good idea to write up SL2(R). The topics are as follows:
1. We first show how a representation decomposes over the maximal compact subgroup K consisting of all matrices and see that an irreducible representation decomposes in such a way that each character of K (indexed by an integer) occurs at most once.
2. We describe the lwasawa decomposition G = ANK, from which most of the structure and theorems on G follow. In particular, we obtain represen-tations of G induced by characters of A.
3. We discuss in detail the case when the trivial representation of K occurs. This is the theory of spherical functions. We need only Haar measure for this, thereby making it much more accessible than in other presentations using Lie theory, structure theory, and differential equations.
4. We describe a continuous series of representations, the induced ones,some of which are unitary.
5. We discuss the derived representation on the Lie algebra, getting into the infinitesimal theory, and proving the uniqueness of any possible unitariza-sion. We also characterize the cases when a unitarization is possible, thereby obtaining the classification of Bargmann. Although not needed for the Plancherel formula, it is satisfying to know that any unitary irreducible representation is infinitesimally isomorphic to a subrepresentation of an induced one from a quasicharacter of the diagonal group. The derived representation of the Lie algebra on the algebraic space of K-finite vectors plays a crucial role, essentially algebraicizing the situation.
6. The various representations are related by the Plancherei inversion formula by Harish-Chandra's method of integrating over conjugacy classes.
7. We give a method of Harish-Chandra to unitarize the "discrete series," i.e. those representations admitting a highest and lowest weight vector in the space of K-finite vectors.
8. We discuss the structure of the algebra of differential operators, with special cases of Harish-Chandra's results on SL2(R) giving the center of the universal enveloping algebra and the commutator of K. At this point, we have enough information on differential equations to get the one fact about spherical functions which we could not prove before, namely that there are no other examples besides those exhibited in Chapter IV. ..
The above topics in a sense conclude a first part of the book. The second part deals with the case when we take the group modulo a discrete subgroup.The classical case is SL2(Z). This leads to inversion formulas and spectral decomposition theorems on L2(F\ G), which constitute the remaining chap-ters.
I had originally intended to include the Selberg trace formula over the reals, but in the case of non-compact quotient this addition would have been sizable, and the book was already getting big. ! therefore decided to omit it,hoping to return to the matter at a later date.
A good portion of the first part of the book depends only on playing with Haar measure and the lwasawa decomposition, without infinitesimal con-siderations. Even when we use these, we are able to carry out the Plancherel formula and the discussion of the various representations without caring whether we have "all" irreducible unitary representations, or "al!" spherical functions (although we prove incidentally that we do). A separate chapter deals with those theorems directly involving partial differential equations via the Casimir operator, and analytical considerations using the regularity theorem for elliptic differential equations. The organization of the book is therefore designed for maximal flexibility and minimal a priori knowledge.The methods used and the notation are carefully chosen to suggest the approach which works in the higher dimensional case.
Since I address this book to those who, like me before I wrote it, don't know anything, I have made considerable efforts to keep it self-contained, I reproduce the proofs of a lot of facts from advanced calculus, and also several appendices on various parts of analysis (spectral theorem for bounded and unbounded hermitian operators, elliptic differential equations, etc.) for the convenience of the reader. These and my Real Analysis form a sufficient background.
The Faddeev paper on the spectral decomposition of the Laplace opera-tor on the upper half-plane is an exceedingly good introduction to analysis,placing the latter in a nice geometric framework. Any good senior under-graduate or first year graduate student should be able to read most of it, and I have reproduced it (with the addition of many details left out to more expert readers by Faddeev) as Chapter XIV. Faddeev's method comes from pertur-bation theory and scattering theory, and as such is interesting for its own sake, as well as to analysts who may know the analytic part and may want to see how it applies in the group theoretic context. Kubota's recent book on Eisenstein series (which appeared while the present book was in production)uses a different method (Selberg-Langlands), and assumes most of the details of functional analysis as known. Therefore, neither Kubota's book nor mine makes the other unnecessary.
It would have been incoherent to expand the present book to a global context with adeles. I hope nevertheless that the reader will be well prepared to move in that direction after having gotten acquainted with SL2(R). The book by Gelfand-Graev-Pjateckii-Shapiro is quite useful in that respect.
I have profited from discussions with many people during the last two years, some of them at the Williamstown conference on representation theory in 1972. Among them i wish to thank specifically Godement, Harish-Chandra, Helgason, Labesse, Lachaud, Langlands, C. Moore, Sally, Wilfried Schmid, Stein. Peter Lax and Ralph Phillips were of great help in teaching me some PDE. I also thank those who went through the class at Yale and made helpful contributions during the time this book was evolving. I am especially grateful to R. Bruggeman for his careful reading of the manuscript. I also want to thank Joe Repka for helping me with the proofreading.
New Haven, Connecticut Serge Lang
it is not easy to get into representation theory, especially for someone interested in number theory, for a number of reasons. First, the general theorems on higher dimensional groups require massive doses of Lie theory.Second, one needs a good background in standard and not so standard analysis on a fairly broad scale. Third, the experts have been writing for each other for so long that the literature is somewhat labyrinthine.
I got interested because of the obvious connections with number theory,principally through Langlands' conjecture relating representation theory to elliptic curves [La 2]. This is a global conjecture, in the adelic theory. I realized soon enough that it was best to acquire a good understanding of the real theory before getting everything on the adeles. I think most people who have worked in representations have looked at SL2(R) first, and I know this is the case for both Harish and Langlands.
Therefore, as I learned the theory myself it seemed a good idea to write up SL2(R). The topics are as follows:
1. We first show how a representation decomposes over the maximal compact subgroup K consisting of all matrices and see that an irreducible representation decomposes in such a way that each character of K (indexed by an integer) occurs at most once.
2. We describe the lwasawa decomposition G = ANK, from which most of the structure and theorems on G follow. In particular, we obtain represen-tations of G induced by characters of A.
3. We discuss in detail the case when the trivial representation of K occurs. This is the theory of spherical functions. We need only Haar measure for this, thereby making it much more accessible than in other presentations using Lie theory, structure theory, and differential equations.
4. We describe a continuous series of representations, the induced ones,some of which are unitary.
5. We discuss the derived representation on the Lie algebra, getting into the infinitesimal theory, and proving the uniqueness of any possible unitariza-sion. We also characterize the cases when a unitarization is possible, thereby obtaining the classification of Bargmann. Although not needed for the Plancherel formula, it is satisfying to know that any unitary irreducible representation is infinitesimally isomorphic to a subrepresentation of an induced one from a quasicharacter of the diagonal group. The derived representation of the Lie algebra on the algebraic space of K-finite vectors plays a crucial role, essentially algebraicizing the situation.
6. The various representations are related by the Plancherei inversion formula by Harish-Chandra's method of integrating over conjugacy classes.
7. We give a method of Harish-Chandra to unitarize the "discrete series," i.e. those representations admitting a highest and lowest weight vector in the space of K-finite vectors.
8. We discuss the structure of the algebra of differential operators, with special cases of Harish-Chandra's results on SL2(R) giving the center of the universal enveloping algebra and the commutator of K. At this point, we have enough information on differential equations to get the one fact about spherical functions which we could not prove before, namely that there are no other examples besides those exhibited in Chapter IV. ..
The above topics in a sense conclude a first part of the book. The second part deals with the case when we take the group modulo a discrete subgroup.The classical case is SL2(Z). This leads to inversion formulas and spectral decomposition theorems on L2(F\ G), which constitute the remaining chap-ters.
I had originally intended to include the Selberg trace formula over the reals, but in the case of non-compact quotient this addition would have been sizable, and the book was already getting big. ! therefore decided to omit it,hoping to return to the matter at a later date.
A good portion of the first part of the book depends only on playing with Haar measure and the lwasawa decomposition, without infinitesimal con-siderations. Even when we use these, we are able to carry out the Plancherel formula and the discussion of the various representations without caring whether we have "all" irreducible unitary representations, or "al!" spherical functions (although we prove incidentally that we do). A separate chapter deals with those theorems directly involving partial differential equations via the Casimir operator, and analytical considerations using the regularity theorem for elliptic differential equations. The organization of the book is therefore designed for maximal flexibility and minimal a priori knowledge.The methods used and the notation are carefully chosen to suggest the approach which works in the higher dimensional case.
Since I address this book to those who, like me before I wrote it, don't know anything, I have made considerable efforts to keep it self-contained, I reproduce the proofs of a lot of facts from advanced calculus, and also several appendices on various parts of analysis (spectral theorem for bounded and unbounded hermitian operators, elliptic differential equations, etc.) for the convenience of the reader. These and my Real Analysis form a sufficient background.
The Faddeev paper on the spectral decomposition of the Laplace opera-tor on the upper half-plane is an exceedingly good introduction to analysis,placing the latter in a nice geometric framework. Any good senior under-graduate or first year graduate student should be able to read most of it, and I have reproduced it (with the addition of many details left out to more expert readers by Faddeev) as Chapter XIV. Faddeev's method comes from pertur-bation theory and scattering theory, and as such is interesting for its own sake, as well as to analysts who may know the analytic part and may want to see how it applies in the group theoretic context. Kubota's recent book on Eisenstein series (which appeared while the present book was in production)uses a different method (Selberg-Langlands), and assumes most of the details of functional analysis as known. Therefore, neither Kubota's book nor mine makes the other unnecessary.
It would have been incoherent to expand the present book to a global context with adeles. I hope nevertheless that the reader will be well prepared to move in that direction after having gotten acquainted with SL2(R). The book by Gelfand-Graev-Pjateckii-Shapiro is quite useful in that respect.
I have profited from discussions with many people during the last two years, some of them at the Williamstown conference on representation theory in 1972. Among them i wish to thank specifically Godement, Harish-Chandra, Helgason, Labesse, Lachaud, Langlands, C. Moore, Sally, Wilfried Schmid, Stein. Peter Lax and Ralph Phillips were of great help in teaching me some PDE. I also thank those who went through the class at Yale and made helpful contributions during the time this book was evolving. I am especially grateful to R. Bruggeman for his careful reading of the manuscript. I also want to thank Joe Repka for helping me with the proofreading.
New Haven, Connecticut Serge Lang
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