1.
Admissible representation generally represents an organized form of representation used in which theory?
Correct Answer
A. Representation theory of reductive Lie groups
Explanation
Admissible representation is generally used in the representation theory of reductive Lie groups. This theory focuses on studying the ways in which these groups can be represented through linear transformations. Admissible representations are those that satisfy certain conditions, such as being irreducible and unitary. By studying these representations, mathematicians can gain insights into the structure and properties of reductive Lie groups.
2.
An admissible representation π generally induces a?
Correct Answer
A. (g,K)-module
Explanation
An admissible representation π generally induces a (g,K)-module. This means that the representation π is compatible with the action of the group g and the subgroup K. The (g,K)-module structure allows for the representation to be decomposed into irreducible subrepresentations, which helps in studying the representation theory of the group.
3.
Admissible representation in which (E, G)-ring B can be regarded as being regular if?
Correct Answer
A. B is reduced
Explanation
The correct answer is B is reduced. In order for (E, G)-ring B to be regarded as regular, it needs to be reduced. This means that the coefficient in the bracket is equal to B. If B is increased or if E is equal to the sum of the ring B, the representation would not be admissible.
4.
The representation (π, V) of G on a complex vector space V is called _____if the subgroup of G fixing any vector of V is open.
Correct Answer
C. Smooth
Explanation
In the context of group representations on a complex vector space V, a representation (π, V) of G is called "smooth" if the subgroup of G fixing any vector of V is open. This means that for any vector in V, there exists an open neighborhood of G that fixes that vector. This property of smoothness ensures that the action of G on V is continuous and well-behaved, allowing for smooth transformations and operations within the representation. Therefore, the correct answer is "Smooth".
5.
When the space of vectors fixed by any compact open subgroup is finite dimensional then π is said to be?
Correct Answer
A. Admissible
Explanation
An π is said to be admissible when the space of vectors fixed by any compact open subgroup is finite dimensional. This means that for any compact open subgroup, there is a finite number of vectors that remain unchanged under the action of the subgroup. This property is important in various areas of mathematics, such as representation theory and harmonic analysis.
6.
What period of time was the admissible representations of p-adic reductive group researched?
Correct Answer
A. 1970s
Explanation
During the 1970s, the admissible representations of p-adic reductive groups were extensively researched.
7.
Langlands classification is typically used to solve which admissible representation problem?
Correct Answer
C. The problem of parameterizing the infinitesimal equivalence classes
Explanation
Langlands classification is typically used to solve the problem of parameterizing the infinitesimal equivalence classes. This means that Langlands classification provides a method or framework for organizing and categorizing these classes, allowing for a systematic approach to understanding and studying them.
8.
The study of the equivalence classes of irreducible unitary representations can be reduced by employing?
Correct Answer
B. The notion of unitarity of (g,M)-modules
Explanation
The notion of unitarity of (g,M)-modules can be employed to reduce the study of the equivalence classes of irreducible unitary representations. This notion helps in classifying the representations based on their unitarity properties, which simplifies the analysis and understanding of the equivalence classes. By considering the unitarity of (g,M)-modules, one can focus on specific properties and characteristics of the representations, leading to a more manageable and organized study of the equivalence classes.
9.
A sub division of Admissible representation used is known as?
Correct Answer
A. B-admissible representation
Explanation
A sub division of Admissible representation used is known as B-admissible representation. This implies that there are different types or categories within the broader concept of Admissible representation, and B-admissible representation is one of them. The other options mentioned (Linear admissible representation, Alpha admissible representation, and Polar admissible representation) are not specifically mentioned as sub divisions of Admissible representation, hence they are not the correct answer.
10.
The concept of Admissible representation was postulated by?
Correct Answer
A. Harish-Chandra
Explanation
Harish-Chandra is the correct answer because he was the mathematician who introduced the concept of Admissible representation. His work in representation theory, particularly in the field of Lie groups and Lie algebras, led to the development of this concept. Admissible representations are a class of representations that satisfy certain conditions and have important applications in various areas of mathematics, including harmonic analysis and number theory.