1.
When was Satake isomorphism introduced?
Correct Answer
A. 1963
Explanation
The Satake isomorphism was introduced in 1963.
2.
What does G represent?
Correct Answer
B. Chevalley group
Explanation
G represents the Chevalley group. Chevalley groups are a class of algebraic groups introduced by Claude Chevalley. They are defined over a field and have a specific structure that makes them important in the study of algebraic groups and Lie algebras. The other options, Trojan group, Chivallas group, and Martins theorem, are unrelated to the concept of Chevalley groups.
3.
What does K represent?
Correct Answer
A. Non-archimedean local field
Explanation
K represents a non-archimedean local field. This type of field is a complete field with a non-archimedean absolute value, meaning that it does not satisfy the archimedean property. In a non-archimedean local field, the absolute value of a nonzero element is a positive real number, and the field is equipped with a topology induced by this absolute value. This is in contrast to an archimedean local field, where the absolute value satisfies the archimedean property. A local field is a field that is complete with respect to a non-trivial absolute value, and a global field is a field that is finite-dimensional over its prime field.
4.
What does O denote?
Correct Answer
D. Ring of integers
Explanation
The letter "O" is commonly used to denote the set or ring of integers. In mathematics, a ring is a set equipped with two operations, addition and multiplication, that satisfy certain properties. The set of integers, denoted by the symbol "Z", is a well-known example of a ring. Therefore, the correct answer is "Ring of integers".
5.
Over a local field, which group does it identifies?
Correct Answer
A. Reductive group
Explanation
Over a local field, the group that is identified is a reductive group. A reductive group is a type of algebraic group that is defined over a field and has no nontrivial connected normal unipotent subgroup. In the context of a local field, which is a field that is complete with respect to a non-Archimedean absolute value, a reductive group is a natural choice as it allows for the study of various properties and structures of the field.
6.
When was the geometric version of the Satake isomorphism introduced?
Correct Answer
B. 2007
Explanation
The geometric version of the Satake isomorphism was introduced in 2007.
7.
The Satake isomorphism identifies the Grothendieck group of complex representations of which of these?
Correct Answer
B. Langlands dual
Explanation
The Satake isomorphism identifies the Grothendieck group of complex representations of the Langlands dual. The Langlands dual is a mathematical concept that relates the representation theory of a group to the representation theory of its dual group. The Satake isomorphism is a result in algebraic geometry that establishes an isomorphism between the cohomology ring of a compact connected Lie group and the ring of symmetric polynomials. Therefore, the Satake isomorphism is used to study the representation theory of the Langlands dual group.
8.
For a Satake isomorphism in characteristic p, what is K?
Correct Answer
B. Hyperspecial maximal compact subgroup of G (F)
Explanation
The correct answer is the hyperspecial maximal compact subgroup of G (F). In the context of a Satake isomorphism in characteristic p, K refers to the hyperspecial maximal compact subgroup of G (F). This subgroup plays a crucial role in the theory of reductive algebraic groups and is used to study the structure and representation theory of these groups.
9.
For a Satake isomorphism in characteristic p, what is V?
Correct Answer
A. Irreducible representation of K
Explanation
The correct answer is "Irreducible representation of K". In the context of a Satake isomorphism in characteristic p, V refers to the irreducible representation of K. This means that the representation cannot be further decomposed into smaller subrepresentations.
10.
Who introduced the geometric version of the Satake isomorphism?
Correct Answer
A. Mirković and Vilonen
Explanation
Mirković and Vilonen introduced the geometric version of the Satake isomorphism.