On Discrete Series (表現論と大域解析学)
On Discrete Series (表現論と大域解析学)
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论离散级数（表示论与全局分析）
DOI：
10.1007/978-0-8176-4493-2_7
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发表时间：
1972
期刊：
影响因子：
3.7
通讯作者：
岡本 清郷
中科院分区：
文献类型：
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作者：
岡本 清郷
Harish-Chandra对半单李群的离散级数表示的分类是20世纪数学最伟大的成就之一。设G是具有极大紧子群K的非紧半单李群。离散级数表示是G的不可约酉表示，它们作为子表示出现在L2（G）的Plancherel分解中。Harish-Chandra证明了G有离散级数的一个充要条件是G有紧的Cartan子群。他建造的字符的所有离散系列表示。谈到哈里什-钱德拉的工作离散系列，我们引用瓦拉德拉扬在他的文章“哈里什-钱德拉，他的工作，及其遗产”[弗吉尼亚州]：“在我看来，字符的问题和问题的建设离散系列是那些定义他，通过拉伸他的强大的权力，以他们的极限。离散级数特征的哈里什-钱德拉公式是无限维酉表示理论中最美丽的公式。哈里什-钱德拉“实际上写下了所有的证明在一个非凡的序列的8篇论文[1964 a]-[1966 b]，总共461期刊页构成了一个最显着的系列论文在科学研究的编年史在我们的时代-显着的，因为它花了多长时间，他达到了他的目标，显着的旅程是多么困难，它是如何被疾病打断，他的成就是多么的独立，最后，他的定理的美丽和必然性是非凡的。
One of the greatest achievements of mathematics in the 20th century is Harish-Chandra’s classification of discrete series representations of semisimple Lie groups. LetGbe a noncompact semisimple Lie group with a maximal compact subgroupK. Discrete series representations are those irreducible unitary representations ofGwhich occur as subrepresentations in the Plancherel decomposition ofL2(G). Harish-Chandra proved that a necessary and sufficient condition forGto have a discrete series is to have a compact Cartan subgroup. He constructed the characters of all discrete series representations. Speaking of Harish-Chandra’s work on discrete series, we quote Varadarajan in his article “Harish-Chandra, His Work, and its Legacy” [Va]: “In my opinion the character problem and the problem of constructing the discrete series were the ones that defined him, by stretching his formidable powers to their limit. The Harish-Chandra formula for the characters of discrete series is the single most beautiful formula in the theory of infinite-dimensional unitary representations.” Harish-Chandra “actually wrote down all the proofs in an extraordinary sequence of 8 papers [1964a]–[1966b], totaling 461 journal pages constituting one of the most remarkable series of papers in the annals of scientific research in our times—remarkable because of how long it took him to reach his goal, remarkable for how difficult the journey was and how it was punctuated by illness, remarkable for how unaided his achievement was, and finally, remarkable for the beauty and inevitability of his theorems.”