# Introductory Lectures on Automorphic Forms

Walter L. Baily
Pages: 279
https://www.jstor.org/stable/j.ctt130hkbh

1. Front Matter
(pp. i-iv)
2. INTRODUCTION
(pp. v-x)
W. L. Baily Jr.

This book is based on lectures that I gave in Tokyo University in 1970 and 1971. Those lectures were given to a group most of whose members were graduate students, and were based on what seemed to me to be a reasonable introduction to the subject of automorphic forms on (domains equivalent to) bounded domains inCn, the space ofncomplex variables. The content of the lectures was based on the assumption that the hearer would seek out many of the details of proofs for himself elsewhere, especially in related areas such as those of algebraic groups and functional...

(pp. xi-xiv)
4. Supplementary notational references
(pp. xv-xvi)
5. ### Part I Elementary theory of automorphic forms on a bounded domain

• CHAPTER 1 GENERAL NOTIONS AND EXAMPLES
(pp. 3-9)

We begin by introducing the general context in which we shall consider automorphic forms and functions.

LetDbe an open connected domain in the spaceCnofncomplex variables. LetG= Hol(D) be the group of all holomorphic one-to-one transformations ofDonto itself, acting on the right. Denote byΓa subgroup ofGoperating in properly discontinuous fashion onD(i.e., given two compact subsetsAandΒofD, the setΓA,B={γΓ|ΑγΒ≠0, the empty set} is finite). Ifgg∈G, ZD, letj(Z, g) denote the determinant of the functional (Jacobian) matrix ofg...

• CHAPTER 2 ANALYTIC FUNCTIONS AND ANALYTIC SPACES
(pp. 10-25)

By a formal power series over a field t (for our purposes usually the complex numbersC), we mean a formal expression of the form

(1)\sum _{_{k_{1}\geqq0,}\; \; \; ,k_{n}\geqq0}a_{k_{1}} _{k_{n}}Z_{1}^{k_{1}}\cdot \cdots \cdot Z_{n}^{k_{n}}=\sum_{(k)\geqq0}a_{(k)Z^{(k)}}

where the right side is a convenient abbreviation of the longer expression on the left and alla(k∈t. We may also write it as

\sum_{(k)\geqq0}a_{(k)Z^{(k)}}=\sum_{\iota\geqq0 }(_{k_{1}+}\sum_{+k_{n}=\iota }a_{(k)Z^{(k)})}=\sum_{\iota \geqq0}H_{\iota }(z),

whereHtdenotes the homogeneous terms of degreel. We also use the notation: Ifɑ=(ɑ1,…,ɑn), then(z-\alpha )^{(k)}=(z_{1}-\alpha _{1})^{k_{1}}\cdots (z_{n}-\alpha _{n})^{k_{n}}. When t is a complete, valued field, then we may speak of convergence. A convergent power series means a power series that converges for some positive range of...

• CHAPTER 3 HOLOMORPHIC FUNCTIONS AND MAPPINGS ON A BOUNDED DOMAIN
(pp. 26-33)

LetKbe a field which is eitherRorCsupplied with the usual absolute value. LetXbe a linear space overK. A real-valued functionponXis called a semi-norm if

(1)p(x+y)\leqq p(x)+p(y), and

(2)p(ax)=|\alpha |p(x),

for allx, yX,αK. These conditions imply, in addition, that

(3)p(x)\geqq0for allxX.

A subsetSofXis called [64: p. 24]:

a) convex if for any two pointsx, yS, the real straight line segment between them is contained inS.

b) balanced if for everyxSand anyαKsuch...

• CHAPTER 4 ANALYSIS ON DOMAINS IN Cn
(pp. 34-42)

In this section, we set down without proof some further definitions and facts from measure theory for our present and future needs. Our direct reference is [11a] where chapter, section, and subsection will be cited as [B, Chap., §, no.]; most of the main ideas may also be found in [41].

LetXbe a locally compact topological space, andF, a normed vector space overRwith norm | |.CF(X) will denote the space of continuousF-valued functions onX. IfAis a compact subset ofX, define the semi-normpAonCF(X) by:p_{A}(f)=\sup_{x\in A}|f(x)|; we...

• CHAPTER 5 AUTOMORPHIC FORMS ON BOUNDED DOMAINS
(pp. 43-50)

LetDbe a bounded domain inCnand letΓbe a discrete subgroup of Hol(D). ThenΓacts in properly discontinuous fashion onD, and the orbit spaceX=D/Γis a locally compact Hausdorff space. Letπ:DXbe the canonical mapping. We define a ringed structureRonXas follows: IfUis an open subset ofX, and iffis a continuous complex-valued function onU, thenfRU, by definition, iffoπis analytic onπ-1(UD.

Proof. The theorem is purely local in nature, so it will suffice to show that ifaX,...

6. ### Part II Automorphic forms on a bounded symmetric domain and analysis on a semi-simple Lie group

• CHAPTER 6 EXAMPLES FOR ALGEBRAIC GROUPS
(pp. 53-81)

IfSis a compact Riemann surface of genus >1, the universal covering surface ofSis complex-analytically isomorphic to the unit discσ, the fundamental groupΓofSacts in properly discontinuous fashion onσ, and the orbit spaceσ/Γis compact and isomorphic toS. Aside from groups that arise in this way and those derived from them in trivial fashion by taking products, etc., about the only known groupsΓwhich operate discontinuously on a bounded domainDsuch thatD/Γis compact, or else has a reasonable compactification, are obtained as arithmetic subgroups of algebraic...

• CHAPTER 7 ALGEBRAIC GROUPS
(pp. 82-101)

As in Chapter 6, section 1, we define an algebraic linear group as a subgroupGofGL(V) for some finite-dimensional vector spaceVsuch thatG=GL(V)∩W, whereWis an algebraic subset of End(V). IfKis a field of definition forW, thenKis called a field of definition forG. We set down here a number of results for the proofs of which we refer the reader to [6a]. Unless stated otherwise, we assume all fields considered are of characteristic zero. When we need to speak of an algebraic group variety (possibly not connected) in the...

• CHAPTER 8 REPRESENTATIONS OF COMPACT GROUPS
(pp. 102-129)

We continue here with the references, notation, and definitions of Chapter 4, section 1. However, in this chapter, and henceforth,M(X) will be used to denote thecomplexmeasures with compact support on the spaceX, hence is the same as what would be denoted byM_{c}(X)_{c}=M_{c}(X)+iM_{c}(X)in the previous notation. Let(X_{1}, \mu ),\cdots ,(X_{n},\mu _{n})be locally compact measure spaces, letXbe the Cartesian product ΠXt, and letμμ2(product measure); denote byφa mapping ofXinto a locally compact spaceY. Then\mu _{1},\cdots ,\mu _{n}are said to be convolvable with respect toφifφisμ-proper,...

• CHAPTER 9 SOME WORK OF HARISH-CHANDRA
(pp. 130-168)

LetVbe a finite-dimensional vector space over a fieldkof characteristic zero. LetT(V) be the tensor algebra ofV,

(1)T(V)=\sum_{n\geqq0}' T_{n}(V)

whereTn=Tn(V) is spanned by the productsv_{1}\otimes \cdots \otimes v_{n}of degreen,T_{o}(V)=kandT(V) is supplied with the obvious non-commutative multiplication such thatT_{m}\cdot T_{n}\subset T_{m}\otimes T_{n}=T_{m+n}. LetIbe the two-sided ideal inT(V) generated by all elements of the formX\otimes Y-Y\otimes X,\; X,\; Y\in T. The associative, commutative factor algebraS=S(V)=T(V)/Iis called the symmetric algebra ofV. The image inSofTnis denoted bySn=Sn(V) andS=\sum_{n\geqq0}'S_{n}. ThenSis the polynomial algebrak[x_{1},\cdots ,x_{\iota}], where...

• CHAPTER 10 FUNCTIONAL ANALYSIS FOR AUTOMORPHIC FORMS
(pp. 169-185)

We recall the definitions of algebraic irreducibility, complete irreducibility, and topological irreducibility from Chapter 8, section 2. The definitions there were for representations of an associative algebra\mathfrak{A}over the complex numbers. Ifρis a representation of a topological groupG, thenρwill be said to have a certain property of irreducibility if the naturally associated representation of the algebraK(G) (with convolution product) has that property.

Letx\rightarrow \rho (x)be a completely irreducible representation of a complex associative algebraUby an algebra of bounded operators on a Banach spaceH. It is easy to see thatρ...

• CHAPTER 11 CONSTRUCTION OF AUTOMORPHIC FORMS
(pp. 186-222)

LetGbe a reductive Lie group with finitely many components and having a finite-dimensional, faithful, linear representation. LetKbe a maximal compact subgroup ofG, and letΓbe a discrete subgroup ofG. Let g be the Lie algebra ofG, letU=U(g) be its universal enveloping algebra, let\mathfrak{k}be the Lie algebra ofK, and let\mathfrak{X}be the universal enveloping algebra of\mathfrak{k}

LetVbe a finite-dimensional complex vector space and letF=F(G, V) denote the vector space of measurable mappings fromGtoV. We assumeVto be supplied with a...

7. ### Part III Some special topics

• CHAPTER 12 FOURIER COEFFICIENTS OF EISENSTEIN SERIES
(pp. 225-242)

LetVbe the space ofn\times nreal symmetric matricesY=(y_{i j}), y_{ij}=y_{ji}\in R, and letBbe the cone of those which are positive-definite, i.e., thoseYVsuch thatY=X2for someXVwith detX≠0. In this chapter, we denote byHnthe space ofn\times ncomplex symmetric matricesZ=X+iY, X, Yreal,YB. Letdvbe the ordinary Euclidean measure onV,\; dv=dy_{11}dy_{12}\cdots dy_{nn}. We want to calculate the value of the integral

I=\int _{\mathfrak{B}}|Y|\rho^{-(n+1)/2}e^{-trY}dv,\; \rho \geqq(n+1)/2

where |Y|=detY. For this purpose, we make a change of variables as follows: EachY\in \mathfrak{B}can be written uniquely in the formY=^{t}TAT, where...

• CHAPTER 13 THETA FUNCTIONS AND AUTOMORPHIC FORMS
(pp. 243-252)

We first derive a simple classical version of the Poisson summation formula. Letfbe a function of differentiability classCvonRn. We consider the series

(1)g(x)=\sum_{m_{1},\; m_{n}\; \; z}f(x_{1}+m_{1},\cdots ,x_{n}+m_{n})

and assume that it and all of the series obtained by replacingfby any of its mixed partial derivatives of orders\leqq vconverge normally onRn, so that alsogCv. Clearlyg(x+m=g(x) for allxRn,mZn; then ifνis large enough, it is known (Chapter 9, section 6) thatghas a Fourier expansion

(2)g(x) = \sum_{l_{1},\; \; ,l_{n\; \; Z}}a_{l_{1}}\; _{l_{n}}e(l_{1}x_{1}+\cdots +l_{n}x_{n})

converging normally onRn, where...

8. BIBLIOGRAPHY
(pp. 253-258)
9. Index
(pp. 259-262)