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Types and unitary representations of reductive p-adic groups.

Dan Ciubotaru1

  • 1Mathematical Institute, University of Oxford, Oxford, OX2 6GG UK.

Inventiones Mathematicae
|April 19, 2019
PubMed
Summary

We prove that irreducible unitary representations correspond between Bernstein components and Hecke algebra modules for rigid types. Every irreducible smooth G-representation contains a rigid type, generalizing prior unitarity criteria.

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.
Keywords:
22E50

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  • 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.