What Do You Know About Satake Isomorphism?

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".

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.

The geometric version of the Satake isomorphism was introduced in 2007.

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.

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.

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.

Mirković and Vilonen introduced the geometric version of the Satake isomorphism.

The Satake isomorphism was introduced in 1963.

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.

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.