The Work of Harish-Chandra

Description: This quiz is designed to assess your knowledge of the work of Harish-Chandra, a renowned Indian mathematician who made significant contributions to the field of representation theory.
Number of Questions: 14
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Tags: harish-chandra representation theory lie groups algebraic groups
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In which year did Harish-Chandra receive the prestigious Wolf Prize in Mathematics?

  1. 1974

  2. 1978

  3. 1982

  4. 1986


Correct Option: A
Explanation:

Harish-Chandra was awarded the Wolf Prize in Mathematics in 1974 for his groundbreaking work on representation theory.

What is the name of the theorem that Harish-Chandra proved, which characterizes the irreducible representations of a semisimple Lie group?

  1. Harish-Chandra's Theorem

  2. Weyl's Theorem

  3. Peter-Weyl Theorem

  4. Cartan-Weyl Theorem


Correct Option: A
Explanation:

Harish-Chandra's Theorem is a fundamental result in representation theory that provides a complete classification of the irreducible representations of a semisimple Lie group.

Which of the following is NOT a type of Lie group that Harish-Chandra extensively studied?

  1. Semisimple Lie Groups

  2. Compact Lie Groups

  3. Reductive Lie Groups

  4. Abelian Lie Groups


Correct Option: D
Explanation:

Harish-Chandra's work primarily focused on semisimple, compact, and reductive Lie groups, while Abelian Lie groups were not a major area of his research.

What is the name of the integral formula that Harish-Chandra developed for the characters of irreducible representations of a semisimple Lie group?

  1. Harish-Chandra's Integral Formula

  2. Weyl's Integral Formula

  3. Peter-Weyl Integral Formula

  4. Cartan-Weyl Integral Formula


Correct Option: A
Explanation:

Harish-Chandra's Integral Formula is a powerful tool for studying the characters of irreducible representations of semisimple Lie groups.

Which of the following is NOT a concept that Harish-Chandra introduced in his work on representation theory?

  1. Harish-Chandra Modules

  2. Discrete Series Representations

  3. Principal Series Representations

  4. Unitary Representations


Correct Option: D
Explanation:

While Harish-Chandra made significant contributions to the study of discrete series representations, principal series representations, and Harish-Chandra modules, he did not introduce the concept of unitary representations.

What is the name of the book that Harish-Chandra wrote, which is considered a classic in the field of representation theory?

  1. Harmonic Analysis on Reductive Lie Groups

  2. Representation Theory of Semisimple Lie Groups

  3. Lie Algebras and Lie Groups

  4. Automorphic Forms on Semisimple Lie Groups


Correct Option: B
Explanation:

Harish-Chandra's book 'Representation Theory of Semisimple Lie Groups' is a seminal work that provides a comprehensive treatment of the subject.

Which of the following mathematicians was NOT a student of Harish-Chandra?

  1. Roger Howe

  2. David Kazhdan

  3. George Mackey

  4. Robert Langlands


Correct Option: D
Explanation:

Roger Howe, David Kazhdan, and George Mackey were all students of Harish-Chandra, while Robert Langlands was not.

What is the name of the conjecture that Harish-Chandra formulated, which relates the characters of irreducible representations of a semisimple Lie group to the geometry of its flag variety?

  1. Harish-Chandra's Conjecture

  2. Weyl's Conjecture

  3. Peter-Weyl Conjecture

  4. Cartan-Weyl Conjecture


Correct Option: A
Explanation:

Harish-Chandra's Conjecture is a deep and challenging problem in representation theory that has attracted considerable attention from mathematicians.

Which of the following is NOT a topic that Harish-Chandra investigated in his research?

  1. Automorphic Forms

  2. Harmonic Analysis

  3. Invariant Theory

  4. Number Theory


Correct Option: D
Explanation:

Harish-Chandra's research primarily focused on representation theory, automorphic forms, and harmonic analysis, while number theory was not a major area of his work.

What is the name of the theorem that Harish-Chandra proved, which relates the characters of irreducible representations of a semisimple Lie group to the eigenvalues of its Casimir operators?

  1. Harish-Chandra's Character Formula

  2. Weyl's Character Formula

  3. Peter-Weyl Character Formula

  4. Cartan-Weyl Character Formula


Correct Option: A
Explanation:

Harish-Chandra's Character Formula is a fundamental result that provides a deep connection between the representation theory of semisimple Lie groups and their Casimir operators.

Which of the following is NOT a type of representation that Harish-Chandra studied in his work?

  1. Irreducible Representations

  2. Reducible Representations

  3. Unitary Representations

  4. Infinite-Dimensional Representations


Correct Option: B
Explanation:

Harish-Chandra primarily focused on irreducible representations and infinite-dimensional representations, while reducible representations were not a major area of his research.

What is the name of the theorem that Harish-Chandra proved, which relates the characters of irreducible representations of a semisimple Lie group to the geometry of its Weyl group?

  1. Harish-Chandra's Weyl Group Formula

  2. Weyl's Weyl Group Formula

  3. Peter-Weyl Weyl Group Formula

  4. Cartan-Weyl Weyl Group Formula


Correct Option: A
Explanation:

Harish-Chandra's Weyl Group Formula is a powerful tool for studying the characters of irreducible representations of semisimple Lie groups.

Which of the following is NOT a concept that Harish-Chandra introduced in his work on representation theory?

  1. Harish-Chandra Modules

  2. Discrete Series Representations

  3. Principal Series Representations

  4. Cuspidal Representations


Correct Option: D
Explanation:

While Harish-Chandra made significant contributions to the study of discrete series representations, principal series representations, and Harish-Chandra modules, he did not introduce the concept of cuspidal representations.

What is the name of the theorem that Harish-Chandra proved, which relates the characters of irreducible representations of a semisimple Lie group to the geometry of its flag manifold?

  1. Harish-Chandra's Flag Manifold Formula

  2. Weyl's Flag Manifold Formula

  3. Peter-Weyl Flag Manifold Formula

  4. Cartan-Weyl Flag Manifold Formula


Correct Option: A
Explanation:

Harish-Chandra's Flag Manifold Formula is a deep and beautiful result that connects the representation theory of semisimple Lie groups to the geometry of their flag manifolds.

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