A Tribute to C. S. Seshadri Perspectives in Geometry and Representation Theory Volumes containing conference proceedings, lecture series, and collection of invited articles in any area of mathematics are published. The aim is to provide an overview of the development through surveys, recent trends, problems and their current status as well as historical background. All contributions for publication should be thoroughly reviewed before submitting to us. We need manuscripts ready for direct reproduction. All articles must be prepared in a uniform format. Necessary instructions for format etc., are available from the publisher. Already Published Volumes Algebra-Some Recent Advances Edited by I. B. S. Passi Number Theory Edited by R. P. Bambah, V. C. Dumir, R. J. Hans-Gill Number Theory and Discrete Mathematics Edited by A. K. Agarwal, Bruce C. Berndt, Christian F. Krattenthaler, Gray L. Mullen, K. Ramachandra and Michel Waldschmidt Current Trends in Number Theory Edited by S.D. Adhikari, S.A. Katre, B. Ramakrishnan Advances in Algebra and Geometry Edited by C. Musili A Tribute to C. S. Seshadri Perspectives in Geometry and Representation Theory Edited by V. Lakshmibai V. Balaji V. B. Mehta K. R. Nagarajan K. Pranjape P. Sankaran R. Sridharan ~HINDUSTAN U LQj UBOOKAGENCY Published by Hindustan Book Agency (India) Visit our home page at: http://www.hindbook.com VOLUME SPONSORED BY Chennai Mathematical Institute 92, G. N. Chetty Road, Chennai Copyright © 2003 by Hindustan Book Agency (India) No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner, who has also the sole right to grant licences for translation into other languages and publication thereof. All export rights for this edition vest exclusively with Hindustan Book Agency (India). Unauthorized export is a violation of Copyright Law and is subject to legal action. ISBN 978-81-85931-39-5 ISBN 978-93-86279-11-8 (eBook) DOI 10.1007/978-93-86279-11-8 Contents Preface ................................................................ ix C.S. Seshadri - A Biographical Sketch ................................. xi List of Publications of C.S. Seshadri .................................. xiii List of Participants at the Symposium ................................ xvi Excerpts from Birthday Messages .. .. ...... .. ..................... xviii Part I Mathematical Contributions of C.S. Seshadri Talks given at the Symposium The Early Work of C.S. Seshadri S. Ramanan .. .. ................. .. .... ..... ..... ...................... 1 Geometry of Moduli Spaces (The Work of C.S. Seshadri) V. Balaji ............................ .................................. 6 Seshadri's Contributions to Moduli and Geometric Invariant Theory V.B. Mehta ........................................................... 16 Seshadri's Work on Moduli Spaces - The Case of Singular Curves D.S. Nagaraj ........ ... ......... .. ... ......... ............. ....... ... 20 Parabolic Bundles N. Nitsure ............................................................ 28 Seshadri's Work on Flag and Schubert Varieties V. Lakshmibai and C. Musili .......................................... 34 Seshadri and the Chennai Mathematical Institute K.R. Nagarajan ....................................................... 42 vi C.S. Seshadri's Impact on Education: A Computer Scientist's· Perspective M. Mukund ........................................................... 47 CMI's Undergraduate Programme and NBHM M.S. Raghunathan .................................................... 49 Part II - Invited Articles Translates of Polynomials S.S. Abhyankar, w.J. Heinzer, A. Sathaye ............................ 51 Orbits of Certain Endomorphisms of Nilmanifolds and Haus­ dorff Dimension C.S. Amvinda and P. Sankamn ...................................... 125 A Family of Quantum Stabilizer Codes Based on the Weyl Commutation Relations over a Finite Field V. Arvind and K.R. Parthasamthy ................................... 133 Principal Bundles, Parabolic Bundles, and Holomorphic Con­ nection 1. Biswas ............................................................ 154 The Cone of Effective One-Cycles of Certain G-Varieties M. Brion ............................................................ 180 Representations of Double Affine Lie Algebras V. Chari and Thang Le .............................................. 199 Poincare Series of Line Bundles on Varieties S.D. Cutkosky ........................................ ............... 220 Triple Canonical Covers of Varieties of Minimal Degree F.J. Gallego and B.P. Purnapmjna .................................. 241 Semistability and Semi simplicity in Representations of Low Height in Positive Characteristic S. Ilangovan, V.B. Mehta and A.J. Pammeswamn ................... 271 The Development of Standard Monomial Theory-II V. Lakshmibai ........................................................ 283 vii Maximal Subbundles and Gromov-Witten Invariants H. Lange and P. Newstead ........................................... 310 Bases for Representations, LS-Paths and Verma Flags P. Littelmann ........................................................ 323 Invariant Tensor Fields and Orbit Varieties for Finite Algebraic Transformation Groups M. Losik, P. W. Michor, and V.L. Popov ............................. 346 Generators of a General Ideal M.P. Murthy ......................................................... 379 The Development of Standard Monomial Theory-I C. Musili ............................................................ 385 Measures on the Circle Invariant Under z -+ Z2 and z -+ z3 M.G. Nadkarni ....................................................... 421 Affine Hecke Algebras, Cyclotomic Hecke Algebras, and Clif­ ford Theory A. Ram and J. Rammage ............................................ 428 The Helgason Fourier Transform for Symmetric Spaces R.P. Sarkar and A. Sitaram .......................................... 467 On Representations of Special Orthogonal Groups over Fields of Positive Characteristics S. Senthamaraikannan and K. V. Subrahmanyam ..................... 474 Equivariant Analogue of Grothendieck's Theorem for Vector Bundles on pI S. Kumar ............................................................ 500 Vector Bundles and Connections in Physics and Mathematics: Some Historical Remarks V.S. Varadarajan .................................................... 502 C.S. Seshadri Preface Seshadri turned seventy on March 1st, 2002. Some of his colleagues, friends and students in India and abroad decided to celebrate this occasion by holding a symposium to highlight Seshadri's contributions to mathe­ matics and to felicitate him. A one-day symposium was organised by The Chennai Mathematical Institute and the Institute of Mathematical Sciences, Chennai on March 1st, 2002, and held at The Hotel Residency. Invitations were sent to a large number of friends and admirers of Se­ shadri. Many of them attended the function and those who could not, sent in their greetings. The proceedings began with an inauguration by Prof. R. Balasubrama­ niam, the director of the Institute of Mathematical Sciences, Chennai. The academic session started with some of Seshadri's colleagues and students giving talks on various aspects of his mathematical contributions spread over nearly five decades. The speakers were: S. Ramanan, V. Balaji, V. Mehta, N. Nitsure, D.S. Nagaraj, C. Musili and V. Lakshmibai. This was followed by talks by K.R. Nagarajan and M. Mukund on the educational programme at CMI, initiated by Seshadri. The session concluded with a talk by M.S. Raghunathan on the National Board for Higher Mathematics and the role of CMI in mathematics education. Late in the afternoon, the invitees gathered to felicitate Seshadri, his mother Chudamani and his wife Sundari. Messages of greetings from many of his friends were read out. Several distinguished guests also spoke bringing out the multi-faceted personality of Seshadri. Seshadri, in his reply, recalled among other things the genesis of CMI and the key role played by the late Dr. S. Parthasarathy and Shri A.C. Muthiah in its development. The day, which was marked by warmth and goodwill, ended with a banquet in honour of Seshadri. The help and cooperation extended by several individuals made the function a memorable one. Sripathy and his colleagues at the offfice of the CMI deserve special mention for their unstinted and enthusiastic support. About this volume: We thought that it would be appropriate to publish a volume in honour of Seshadri, containing the texts of the talks at the symposium as well as invited research and expository articles from some of his colleagues. The response to our invitation was spontaneous, and I want to thank all the authors who have contributed to this volume. The articles published here were refereed, and I am indebted to the referees for their gracious cooperation. I thank the Hindustan Book Agency, New Delhi for their immense help in bringing out this volume. Finally, I am x much obliged to my colleagues in the editorial board for their efforts which made this volume possible. v. Lakshmibai Chief Editor C.S. Seshadri A Biographical Sketch Conjeevaram Srirangachari SESHADRI was born on February 29th, 1932, in Kanchipuram (the "Golden City" turned into "Conjeevaram" dur­ ing the British Raj). He was the eldest among eleven children of his