哈里什-钱德拉Harish-Chandra

哈里什-钱德拉的母亲是Satyagati Seth，她是律师Ram Sanehi Seth的女儿，他的父亲是土木工程师Chandrakishore。Chandrakishore曾在鲁尔基的Thomason工程学院接受教育，随后进入印度工程服务处，他职业生涯的第一部分是在14：-

……在野外度过的，通常骑马巡视和维护北部平原广阔运河网络中的堤坝。

哈里什-钱德拉的父亲大部分时间都在全国各地巡视运河，哈里什-钱德拉童年大部分时间住在坎普尔的外祖父家中。Ram Sanehi Seth是个富有的人，他的大房子里住着许多亲戚。尽管哈里什-钱德拉是个相当虚弱的孩子，他有时也会随父亲出行。他的教育被家庭视为至关重要14：-

家里请了一位家庭教师，还有舞蹈老师和音乐老师来上课。九岁时他入学，比同学都小，读七年级。他十四岁在基督教会高中毕业，随后在坎普尔读中间学院，十六岁完成……

随后他进入了阿拉哈巴德大学。在这里他学习理论物理，这个方向是他在大学图书馆读到保罗·狄拉克的Principles of Quantum Mechanics的结果。他于1941年获得学士学位，1943年获得硕士学位。哈里什-钱德拉在印度南部班加罗尔的印度科学研究所，在Homi Bhabha指导下作为研究生研究员研究理论物理问题。Bhabha在1930年代曾是保罗·狄拉克的学生。哈里什-钱德拉在班加罗尔期间开始发表理论物理论文，并与Bhabha发表了几篇联合论文，扩展了保罗·狄拉克的一些结果。在班加罗尔的前六个月，他与来自阿拉哈巴德的植物学家Kale博士及其妻子同住，Kale夫人在哈里什-钱德拉本科时教过他法语。Kale夫妇有一个女儿Lalitha，当时还是个小女孩，但大约八年后，即1952年，他从美国回到印度，并在那里与Lalitha结婚。

Bhabha和哈里什-钱德拉在阿拉哈巴德大学的老师K S Krishnan，推荐他到剑桥保罗·狄拉克处从事研究工作，以获得博士学位。1945年，哈里什-钱德拉前往剑桥大学冈维尔与凯斯学院，在保罗·狄拉克指导下攻读博士学位。然而，哈里什-钱德拉与导师见面相对较少，当他意识到保罗·狄拉克基本上是在照本宣科读自己的一本书时，便放弃了听保罗·狄拉克的讲座课程。他写道，保罗·狄拉克是（例如见[14]）：-

……非常温和善良，却又相当超然疏远。……[我决定]我不该太多打扰他，大约每学期去见他一次。

在剑桥期间，他开始远离物理学，对数学更感兴趣，参加了利特尔伍德和菲利浦·霍尔的讲座课程。他还听了沃尔夫冈·泡利的一次讲座，并指出了沃尔夫冈·泡利工作中的一处错误。两人后来成为终身朋友。哈里什-钱德拉于1947年凭借其学位论文Infinite irreducible representations of the Lorentz group获得学位，在论文中他给出了$SL(2,\mathbb{C})$的不可约酉表示的完全分类。

保罗·狄拉克于1947-48学年访问普林斯顿，哈里什-钱德拉随他前往美国，在此期间担任他的助手。然而，他深受在那里工作的顶尖数学家赫尔曼·外尔、埃米尔·阿廷和谢瓦莱的影响。正是在普林斯顿的这一年，他最终认定自己是一名数学家，而不是物理学家。他后来写道（例如见[14]）：-

来到普林斯顿后不久，我就意识到我在亨德里克·洛伦兹群上的工作建立在相当不牢靠的论证之上。我曾天真地操作无界算子，却完全不注意它们的定义域。有一次我向保罗·狄拉克抱怨我的证明不严格，他回答说：“我对证明不感兴趣，只对自然如何运作感兴趣。”这句话证实了我日益增长的信念：我不具备在物理学中取得成功所需的那种神秘的第六感，于是我很快决定转向数学。

在保罗·狄拉克回到剑桥之后，哈里什-钱德拉在普林斯顿又待了一年。1949—50年他在哈佛度过，在那里受到奥斯卡·扎里斯基的影响。1950年至1963年在纽约哥伦比亚大学度过的时期是他最多产的时期。在此期间，他研究半单索菲斯·李群的表示。也在此期间，他与安德烈·韦伊有密切接触。当我们说他在1950—63年间在哥伦比亚大学度过时，我们实际上是指他在这段时期是该大学的教员，但可以说他也在其他机构度过了很长时间。1952—53年他在孟买的塔塔研究所度过，正是在这一年他与Lalitha Kale（常被称为Lily）结婚；他们有两个女儿[14]：-

Lily……以良好的精神、慷慨的感情、耐心和全面的能力，宠爱了他三十年。

1955—56年他在普林斯顿的爱德华·斯图迪高等研究院度过，1957—58年作为古根海姆研究员在巴黎，在那里他得以与安德烈·韦伊合作，两人常常一起散步。1961年至1963年，他是斯图迪高等研究院的斯隆研究员。

在[14]中，引述哈里什-钱德拉的话说，他相信自己在数学方面缺乏背景，在某种意义上是他工作新颖性的原因：-

我常常思考知识和经验一方面、想象或直觉另一方面在发现过程中的作用。我相信两者之间存在某种根本冲突，而知识由于提倡谨慎，往往会抑制想象的飞翔。因此，某种不受传统智慧束缚的天真，有时可以成为一种积极的长处。

阿曼德·博雷尔在[6]中描述了哈里什-钱德拉的一些贡献。Schwermer评述了那部著作以及[25]，以下内容取自这些评述：-

人们对哈里什-钱德拉全部工作的印象是令人惊讶的连贯性和一致性。他的主要兴趣围绕约化群的表示论以及这些群及其相关齐性空间上的调和分析。然而哈里什-钱德拉曾多次尝试进入代数几何或数论。

他分别建立了李群和李代数的表示的基本理论，以及这些群及其齐性空间上的调和分析。正是哈里什-钱德拉将半单Lie群有限维表示的特征标概念推广到无限维表示的情形；他证明了赫尔曼·外尔特征标公式的一个类似物。

哈里什-钱德拉工作的一些主要贡献可以特别指出：半单群的米歇尔·普朗歇尔测度的显式确定，离散系列表示的确定，他在费迪南·艾森斯坦级数以及自守形式理论中的结果，他的“尖点形式哲学”——他这样称呼它——作为指导原则，以便在相当广泛的意义上对约化群的表示论中某些现象持有共同看法，不仅包括实李群，还包括p进群或阿代尔环上的群。他的科学工作，作为分析、代数与几何的综合，仍然具有持久的影响。

哈里什-钱德拉从1963年起在普林斯顿的高等斯图迪研究所工作。他于1968年被任命为IBM-冯·诺伊曼讲席教授。

他在普林斯顿一次为期一周的会议结束时死于心脏病发作，此前他曾三次心脏病发作，第一次在1969年，第三次在1982年。1983年10月，为纪念博雷尔的会议在普林斯顿举行[14]：-

……在会议那一周，[哈里什-钱德拉的]活力和力量重新显现出来。普林斯顿温暖晴朗的秋日天气持续着，在会议讲座之间，在研究所的草坪或露台上，他是一群活跃人群的中心，就各种话题表达自己的观点。10月16日星期日，会议的最后一天，他和Lily邀请许多参会者到家中。他是一位光彩照人的主人。傍晚时分，客人离去后，他像往常一样出去散步，再也没有活着回来。

哈里什-钱德拉在其职业生涯中获得了许多奖项。他是皇家学会的会士（1973年），National Academy of Sciences（美国）的会士（1981年），Indian Academy of Sciences和印度国家科学院的会士（1975年）。他于1954年因关于半单李代数与群的表示论文，特别是因他于1951年在Transactions of the American Mathematical Society上发表的论文On some applications of the universal enveloping algebra of a semisimple Lie algebra，而获得美国数学会颁发的法兰克·尼尔森·寇尔奖。1974年，他获得印度国家科学院颁发的拉马努金奖章。他获得德里大学（1973年）和耶鲁大学（1981年）的荣誉学位。

作为既是物理学家又是数学家，看看哈里什-钱德拉对他们之间关系的看法是很有趣的。这在[14]中有很好的描述：-

尽管他确信数学家的思维方式本身使他无法理解理论物理学的本质——他认为在理论物理学中，深刻直觉而非逻辑占主导地位——并且对任何自以为能理解理论物理学的数学家持怀疑态度，但他对那些缺乏对理论物理学同情的数学家更加不耐烦，他将这一缺陷特别归咎于20世纪50年代的法国学派。

他的性格在[28]中被描述为：-

……他是一位高大英俊的男子，有些内向，但拥有一种正式的礼貌，这并未掩盖他感情和思想的深度。早年他喜欢绘画，后来表达了对法国印象派的喜爱。……他的一位同事提出，哈里什-钱德拉在他的工作中得以延续，这些工作忠实地反映了他的个性：强烈、高尚、不妥协。

Harish-Chandra's mother was Satyagati Seth, who was the daughter of the lawyer Ram Sanehi Seth, and his father was Chandrakishore who was a civil engineer. Chandrakishore had been educated at the Thomason Engineering College at Roorkee, then entered the Indian Service of Engineers and the first part of his career was [14]:-

... spent in the field, usually on horseback, inspecting and maintaining the dikes of the extensive network of canals in the northern plains.

Since Harish-Chandra's father spent most of his time travelling around the country inspecting canals, Harish-Chandra spent most of his childhood living in his maternal grandfather's home in Kanpur. Ram Sanehi Seth was a wealthy man with a large house which was home to many of his relatives. Harish-Chandra, despite being a rather weakly child, did sometimes spend time travelling with his father. His education was treated by the family as of the utmost importance [14]:-

A tutor was hired, and there were visits from a dancing master and a music master. At the age of nine he was enrolled, younger than his schoolmates, in the seventh class. He completed Christ Church High School at fourteen, and remained in Kanpur for intermediate college, which he finished at sixteen ...

He then attended the University of Allahabad. Here he studied theoretical physics, this direction being the result of reading Principles of Quantum Mechanics by Dirac which he found himself in the university library. He was awarded a B.Sc. in 1941, then a master's degree in 1943. Harish-Chandra worked as a postgraduate research fellow on problems in theoretical physics under Homi Bhabha, at the Indian Institute of Science at Bangalore in Southern India. Bhabha had been a student of Dirac in the 1930s. Harish-Chandra began publishing papers on theoretical physics while at Bangalore, and he published a couple of joint papers with Bhabha extending some of Dirac's results. For the first six months he spent in Bangalore he lived with Dr Kale, a botanist from Allahabad, and his wife who had taught Harish-Chandra French when he was an undergraduate. The Kales had a daughter Lalitha who at this time was a young girl, but about eight years later in 1952 he returned from the United States to India and there married Lalitha.

Bhabha and Harish-Chandra's teacher, K S Krishnan at Allahabad University, recommended him to Dirac for research work at Cambridge which would lead to Ph.D. degree. In 1945 Harish-Chandra went to Gonville and Caius College, Cambridge, where he studied for his doctorate under Dirac's supervision. However Harish-Chandra saw comparatively little of his supervisor, giving up attending Dirac's lecture course when he realised that Dirac was essentially reading from one of his books. He wrote that Dirac was (see for example [14]):-

... very gentle and kind and yet rather aloof and distant. ... [I decided] I should not bother him too much and went to see him about once each term.

During his time in Cambridge he began to move away from physics and became more interested in mathematics attending the lecture courses of Littlewood and Hall. He also attended a lecture by Pauli and pointed out a mistake in Pauli's work. The two were to become life long friends. Harish-Chandra obtained his degree in 1947 for his thesis Infinite irreducible representations of the Lorentz group in which he gives a complete classification of the irreducible unitary representations of $SL(2,\mathbb{C})$.

Dirac visited Princeton for the year 1947-48 and Harish-Chandra went to the United States with him, working as his assistant during this time. However he was greatly influenced by the leading mathematicians Weyl, Artin and Chevalley who were working there. It was during this year in Princeton that he finally decided that he was a mathematician and not a physicist. He later wrote (see for example [14]):-

Soon after coming to Princeton I became aware that my work on the Lorentz group was based on somewhat shaky arguments. I had naively manipulated unbounded operators without paying any attention to their domains of definition. I once complained to Dirac about the fact that my proofs were not rigorous and he replied, "I am not interested in proofs but only in what nature does." This remark confirmed my growing conviction that I did not have the mysterious sixth sense which one needs in order to succeed in physics and I soon decided to move over to mathematics.

After Dirac returned to Cambridge, Harish-Chandra remained at Princeton for a second year. He spent 1949-50 at Harvard where he was influenced by Zariski. The period 1950 to 1963, spent at Columbia University, New York, was his most productive. During this time he worked on representations of semisimple Lie groups. Also during this period he had close contact with Weil. When we say that he spent the years 1950-63 at Columbia University we really mean that he was on the faculty there for this period, but it is fair to say that he spent long periods in other institutions. He spent 1952-53 at the Tata Institute in Bombay and it was during this year that he married Lalitha Kale (often known as Lily); they had two daughters [14]:-

Lily ... with good spirits, generous affection, patience, and all-round competence was to pamper him for thirty years.

He spent 1955-56 at the Institute for Advanced Study at Princeton, 1957-58 as a Guggenheim Fellow in Paris where he was able to work with Weil and the two often went walking together. He was a Sloan Fellow at the Institute for Advanced Study from 1961 to 1963.

In [14] Harish-Chandra is quoted as saying that he believed that his lack of background in mathematics was in a way responsible for the novelty of his work:-

I have often pondered over the roles of knowledge or experience, on the one hand, and imagination or intuition, on the other, in the process of discovery. I believe that there is a certain fundamental conflict between the two, and knowledge, by advocating caution, tends to inhibit the flight of imagination. Therefore, a certain naiveté, unburdened by conventional wisdom, can sometimes be a positive asset.

Armand Borel describes some of Harish-Chandra's contributions in [6]. Schwermer reviews that work as well as [25] and what follows is taken from these reviews:-

The impression one has of Harish-Chandra's work as a whole is one of surprising cohesiveness and uniformity. His primary interests revolved about the representation theory of reductive groups and harmonic analysis on these groups and their related homogeneous spaces. However Harish-Chandra made several attempts to get into algebraic geometry or number theory.

He has built a fundamental theory of representations of Lie groups and Lie algebras, respectively of harmonic analysis on these groups and their homogeneous spaces. It was Harish-Chandra who extended the concept of a character of finite-dimensional representations of semisimple Lie groups to the case of infinite-dimensional representations; he proved an analogue of Weyl's character formula.

Some major contributions by Harish-Chandra's work may be singled out: the explicit determination of the Plancherel measure for semisimple groups, the determination of the discrete series representations, his results on Eisenstein series and in the theory of automorphic forms, his "philosophy of cusp forms", as he called it, as a guiding principle to have a common view of certain phenomena in the representation theory of reductive groups in a rather broad sense, including not only the real Lie groups, but p-adic groups or groups over adele rings. His scientific work, being a synthesis of analysis, algebra and geometry, is still of lasting influence.

Harish-Chandra worked at the Institute of Advanced Study at Princeton from 1963. He was appointed IBM-von Neumann Professor in 1968.

He died of a heart attack at the end of a week long conference in Princeton, having earlier suffered from three heart attacks, the first being in 1969 and the third in 1982. In October 1983 a conference in honour of Armand Borel was held in Princeton [14]:-

... for the week of the conference, [Harish-Chandra's] vigour and force reasserted themselves. Princeton's warm, clear autumn weather prevailed and between lectures at the conference, on a lawn or a terrace of the Institute, he was the centre of a lively crowd, expressing his views on a variety of topics. On Sunday 16 October, the last day of the conference he and Lily had many of the participants to their home. He was a sparkling host. In the late afternoon, after the guests had departed, he went for his customary walk, and never returned alive.

Harish-Chandra received many awards in his career. He was a Fellow of the Royal Society (1973), a Fellow of the National Academy of Sciences (United States) (1981), the Indian Academy of Sciences and the Indian National Science Academy (1975). He won the Cole prize from the American Mathematical Society in 1954 for his papers on representations of semisimple Lie algebras and groups, and particularly for his paper On some applications of the universal enveloping algebra of a semisimple Lie algebra which he had published in the Transactions of the American Mathematical Society in 1951. In 1974, he received the Ramanujan Medal from Indian National Science Academy. He was awarded honorary degrees by Delhi University (1973) and Yale University (1981).

Having been both a physicist and a mathematician it is interesting to see Harish-Chandra's view of their relationship. This is described nicely in [14]:-

Although he was convinced that the mathematician's very mode of thought prevented him from comprehending the essence of theoretical physics, where, he felt, deep intuition and not logic prevailed, and sceptical of any mathematician who presumed to attempt to understand it, he was even more impatient with those mathematicians in whom a sympathy for theoretical physics was lacking, a failing he attributed in particular to the French school of the 1950s.

His character is described in [28]:-

... he was a tall, handsome man, who was somewhat reserved, but who possessed a formal courtesy that did not conceal the depth of his feelings and thought. In his early years he liked to paint and later expressed a fondness for the French Impressionists. ... One of his colleagues suggested that Harish-Chandra survives in his work, which faithfully mirrored his personality: intense, lofty, and uncompromising.