P-adic group representation theory

Definition

p-adic group representation theory studies representations of p-adic groups, which are groups defined over p-adic numbers. These representations play a crucial role in number theory and algebraic geometry, providing insights into the structure and properties of these groups. The connections between representations and various mathematical concepts, including Mackey's theorem, reveal how they interact with different subgroups and lead to decompositions that help understand the overall structure of the group.

5 Must Know Facts For Your Next Test

1. p-adic groups are often used in the study of local fields, which provide a framework for understanding number-theoretic phenomena.
2. Mackey's theorem plays a significant role in understanding the restriction and induction of representations from subgroups to the entire group in p-adic representation theory.
3. The classification of irreducible representations of p-adic groups is central to the field and often relies on deep results from harmonic analysis and number theory.
4. The representation theory of p-adic groups has connections to automorphic forms, which are functions defined on p-adic groups with important implications in number theory.
5. The study of characters of p-adic representations helps reveal symmetry properties and provides a tool for understanding more complex algebraic structures.

Review Questions

• How does Mackey's theorem apply to the representations of p-adic groups and what are its implications for subgroup relationships?
  • Mackey's theorem provides a framework for understanding how representations of a p-adic group can be decomposed into representations of its subgroups. This means when you take a representation of the whole group and restrict it to a subgroup, it can break down into simpler parts, revealing insights about the interactions between the group and its subgroups. This is crucial for analyzing the structure of p-adic groups, as it helps connect different levels of representation and indicates how they influence one another.
• Discuss the significance of irreducible representations in the context of p-adic group representation theory and their relation to Mackey's theorem.
  • Irreducible representations are fundamental in p-adic group representation theory because they cannot be further decomposed into simpler forms. Their significance lies in their ability to capture the essence of how a group acts on a vector space without any distractions from reducible components. Mackey's theorem highlights how these irreducible pieces can be analyzed through their relationships with various subgroups, providing deeper insights into the structure of the larger group by focusing on these indivisible building blocks.
• Evaluate how p-adic group representation theory can influence broader areas such as number theory or algebraic geometry through its key concepts like characters and Mackey's theorem.
  • p-adic group representation theory significantly impacts number theory and algebraic geometry by offering tools for understanding symmetries and structures within these fields. For instance, characters of p-adic representations allow mathematicians to explore deep relationships within number-theoretic objects, while Mackey's theorem helps to systematically analyze how these relationships change when considering subgroups. Together, they provide a bridge connecting abstract representation theory to concrete applications in solving problems related to automorphic forms and Diophantine equations, ultimately enriching our understanding of both areas.

Related terms

p-adic numbers: A system of numbers that extends the ordinary arithmetic of rational numbers to include solutions to equations that have no rational solutions, characterized by their unique topology.

Mackey's theorem: A result that describes how representations of a group can be decomposed into representations of its subgroups, providing insight into the relationship between different representations.

irreducible representation: A representation of a group that has no proper invariant subspaces, meaning it cannot be decomposed into simpler representations.