Types and unitary representations of reductive p-adic groups

Area of Science:

• Representation theory
• Algebraic groups
• Harmonic analysis

Background:

• The study builds upon the theory of reductive groups and their representations.
• It extends concepts related to Bernstein components and Hecke algebras.
• Prior work by Barbasch and Moy on unitarity criteria for specific representations is a key reference.

Purpose of the Study:

• To establish a bijection between irreducible unitary representations in Bernstein components and modules of Hecke algebras for rigid types.
• To demonstrate that every irreducible smooth G-representation contains a rigid type.
• To generalize existing unitarity criteria for group representations.

Main Methods:

• Utilizing the concept of Bushnell-Kutzko types with a rigidity assumption.
• Leveraging category equivalences between Bernstein components and Hecke algebra module categories.
• Employing techniques from the representation theory of p-adic reductive groups.

Main Results:

• A proven bijection exists between irreducible unitary representations of Bernstein components and Hecke algebra modules for rigid types.
• It is shown that every irreducible smooth G-representation incorporates a rigid type.
• The findings generalize the unitarity criterion of Barbasch and Moy.

Conclusions:

• The established bijection provides a powerful tool for understanding unitary representations.
• The presence of rigid types in all irreducible smooth G-representations simplifies the study of their properties.
• This work offers a significant advancement in the representation theory of reductive groups.