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Discrete Subgroups of Lie Groups and Applications to Moduli

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Discrete Subgroups of Lie Groups and Applications to Moduli Discrete Subgroups of Lie Groups and Applications to Moduli DISCRETE SUBGROUPS OF LIE GROUPS AND APPLICATIONS TO MODULI Papers presented at the Bombay Colloquium 1973, by BAILY FREITAG GARLAND GRIFFITHS HARDER IHARA MOSTOW MUMFORD RAGHUNATHAN SCHMID VINBERG Published for the TATA INSTITUTE OF FUNDAMENTAL RESEARCH, BOMBAY OXFORD UNIVERSITY PRESS 1975 Oxford University Press OXFORD LONDON GLASGOW NEW YORK TORONTO MELBOURNE WELLINGTON CAPE TOWN DELHI BOMBAY CALCUTTA MADRAS KARACHI LAHORE DACCA KUALA LUMPUR SINGAPORE JAKARTA HONG KONG TOKYO NAIROBI DAR ES SALAAM LUSAKA ADDIS ABABA IBADAN ZARIA ACCRA BEIRUT c Tata Institute of Fundamental Research, 1975 Printed by V. B. Gharpure at Tata Press Limited, 414 Veer Savarkar Marg, Bombay 400 025, and Publish by C. H. Lewis, Oxford University Press, Apollo Bunder, Bombay 400 001 PRINTED IN INDIA International Colloquium on Discrete Subgroups of Lie Groups and Applications to Moduli Bombay, 8-15 January 1973 REPORT AN INTERNATIONAL COLLOQUIUM on ‘Discrete Subgroups of Lie Groups and Applications to Moduli’ was held at the Tata Institute of Fundamental Research, Bombay, from 8 to 15 January 1973. The purpose of the Colloquium was to discuss recent developments in some aspects of the following topics: (i) Lattices in Lie groups, (ii) Arithmetic groups, automorphic forms and related number-theoretic questions, (iii) Moduli problems and discrete groups. The Colloquium was a closed meeting of experts and of others specially interested in the subject. Th Colloquium was jointly sponsored by the International Mathe- matical Union and the Tata Institute of Fundamental Research, and was financially supported by them and the Sir Dorabji Tata Trust. An Organizing Committee consisting of Professors A. Borel. M. S. Narasimhan, M. S. Raghunathan, K. G. Ramananthan and E. Vesentini was in charge of the scientific programme. Professors A. Borel and E. Vesentini acted as representatives of the International Mathematical Union on the Organizing Committee. The following mathematicians gave invited addresses at the Col- loquium: W. L. Baily, Jr., E. Freitag, H. Garland, P. A. Griffiths, G. Harder, Y. Ihara, G. D. Mostow, D. Mumford, M. S. Raghunathan and W. Schmid. REPORT Professor E.` B. Vinberg, who was unable to attend the Colloquium, sent in a paper. The invited lectures were of fifty minutes’ duration. These were followed by discussions. In addition to the programme of invited ad- dresses, there we expository and survey lectures and lectures by some invited speakers giving more details of their work. The social programme during the Colloquium included a Tea Party on 8 January; a Violin recital (Classical Indian Music) on 9 January; a programme of Western Music on 10 January; a performance of Classical Indian Dances (Bharata Natyan) on 12 January; a Film Show (Pather Panchali) on 13 January; and a dinner at the Institute on 14 January. Contents 1. Walter L. Baily, Jr. : Fourier coefficients of 1–8 Eisenstein series on the Adele group 2. Eberhard Freitag : Automorphy factors of 9–20 Hilbert’s modular group 3. Howard Garland : On the cohomology of discrete 21–31 subgroups of semi-simple Lie groups 4. Phillip Griffiths and Wilfried Schmid : Recent 32–134 developments in Hodge theory: a discussion of techniques and results 5. G.Harder : On the cohomology of discrete 135–170 arithmetically defined groups 6. Yasutaka Ihara : On modular curves over finite 171–215 fields 7. G.D.Mostow : Strong rigidity of discrete 216–223 subgroups and quasi-conformal mappings over a division algebra 8. David Mumford : A new approach to compactifying 224–240 locally symmetric varieties 9. M.S.Raghunathan : Discrete groups and 241–343 Q-structures on semi-simple Lie groups 10. E.B.Vinberg : Some arithmetical discrete groups 344-372 in Lobaˆcevskiˆi spaces FOURIER COEFFICIENTS OF EISENSTEIN SERIES ON THE ADELE GROUP By WALTER L. BAILY, JR. Much of what I wish to present in this lecture will shortly appear 1 elsewhere [3], so for the published part of this presentation I shall con- fine myself to a restatement of certain definitions and results, concluding with a few remarks on an area that seems to hold some interest. As in [3], I wish to add here also that many of the actual proofs are to be found in the thesis of L. C. Tsao [8]. Let G be a connected, semi-simple, linear algebraic group defined over Q, which, for simplicity, we assume to be Q-simple (by which we mean G has no proper, connected, normal subgroups defined over Q). We assume G to be simply-connected, which implies in particular that GR is connected [2, Ch. 7, §5]. Assume that GR has no compact (connected) simple factors and that if K is a maximal compact subgroup of it, then X = K/GR has a GR-invariant complex structure, i.e., X is Hermitian symmetric. Then [6] strong approximation holds for G. We assume, finally, that rkQ(G) (the common dimension of all maximal, Q- split tori of G) is > 0 and that the Q-relative root system of G is Q P of type C (in the Cartan-Killing classification). Then there exists a to- tally real algebraic number field k and a connected, almost absolutely simple, simply-connected algebraic group G′ defined over k such that ′ G = Rk/QG ; therefore, if G is written as a direct product ΠGi of al- most absolutely simple factors Gi, then each Gi is defined over a totally real algebraic number field, each Gi is simply-connected, each GiR is connected and the relative root systems = (G ) are of type C R Pi R P i 1 2 WALTER L. BAILY, JR. [4]. Letting Ki denote a maximal compact subgroup of GiR, the Hermi- tian symmetric space Xi = Ki/GiR is isomorphic to a tube domain since is of type C [7], hence X = Π X is bi-holomorphically equivalent R Pi i i to a tube domain T = {Z = X + iY ∈ Cn|Y ∈ R}, 2 where R is a certain type of open, convex cone in Rn. Let H be the group of linear affine transformations of T of the form Z 7−→ AZ + B, where B ∈ Rn, and A is a linear transformation of Rn carrying R onto itself, and let H˜ be its complete pre-image in GR with respect to the natural homomorphism of GR into Hol(T), the group of biholomorphic automorphisms of T. Then H˜ = PR, where P is an R-parabolic sub- group of G, and from our assumption that is of type C, it follows Q P that we may assume P to be defined over Q (the reasons for which are somewhat technical, but may all be found in [4]). Assume G ⊂ GL(V), where V is a finite-dimensional, complex vec- tor space with a Q-structure. Let Λ be a lattice in VR, i.e., a discrete subgroup such that VR/Λ is compact, and suppose that Λ ⊂ VQ. Let Γ = {γ ∈ GQ | γ · Λ=Λ}, and for each finite prime p, let Λp = Λ = Λ = Λ Z Zp, Kp {γ ∈ GQp | γ · p p}. It may be seen, since strong Γ Γ approximation holds for G, that Kp is the closure p of in GQp (in the ordinary p-adic topology). Now the adele group GA of G is defined Π′ Π′ as GQp , where denotes restricted direct product with respect to ∗ the family {Kp} of compact sub-groups. Define K∞ = K, K = Π Kp p6∞ (Cartesian product). = For all but a finite number of finite p, we have GQp Kp · PQp , and by changing the lattice Λ at a finite number of places, we may assume = [5] that GQp Kp · PQp for all finite p. In addition, from the Iwasawa = 0 0 decomposition we have GR K∞ · PR, where PR denotes the identity component of PR. We may write the Lie algebra gC of G as the direct sum of kC, the complexification of the Lie algebra k of K, and of two Abelian subalge- bras p+ and p−, both normalized by k, such that p+ may be indentified FOURIER COEFFICIENTS OF EISENSTEIN SERIES ON THE ADELE GROUP 3 n with C ⊃ T. Let KC be the analytic subgroup of GC with Lie algebra ± ± + kC and let P = exp(p ); then KC · P is a parabolic subgroup of G which we may take to be the same as P, and P+ = U is its unipotent + radical. Now p has the structure of a Jordan algebra over C, supplied 3 with a homogeneous norm form N such that Ad KC is contained in the similarity group S = {g ∈ GL(n, C) = GL(p+) | N (gX) = v(g)N (X)} of N , where v : S → C× is a rational character [7], defined over Q if we arrange things such that KC = L is a Q-Levi subgroup of P. (Note that K and LR are, respectively, compact and non-compact real forms of KC.) Define v∞ as the character on KC given by v∞(k) = v(Adp+ k). Define v∞ as the character on KC given by v∞(k) = v(Adp +k). If p 6 ∞ let | |p be the “standard” p-adic norm, so that the product formula 6 = + holds. We define (for p ∞) χp on PQp by χp(ku) |v(Adp k)|p, = Π k ∈ LQp , u ∈ UQr and χA on PA by χA((pp)) pχp(pp), which is well defined since for (pp) ∈ PA, we have χp(pp) = 1 for all but a finite number of p. Now v∞ is bounded on K and v takes positive real 0 0 = values on PR, hence v∞(K ∩ PR) {1}. Moreover, Kp is compact and = therefore χp(Kp ∩ PQP ) {1}.
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