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This is a Quantitative Study study.

October 16, 2025Open Access

Unitary dual of p-adic split SO₂₍+₁ and Sp₂₍: The good parity case (and slightly beyond)

Key Points

A smooth irreducible representation of good parity is unitary if and only if it is of Arthur type.
The findings provide an explicit algorithm to check the unitarity of irreducible representations in these groups.
Key groups involved are the split special orthogonal group and the symplectic group in the context of p-adic fields.
The study also determines the unitary representations that may be part of the discrete automorphic spectrum.

Abstract

Let F be a p-adic field, and let G be either the split special orthogonal group SO₂₍+₁ (F) or the symplectic group Sp₂₍ (F), with n 0. We prove that a smooth irreducible representation of good parity of G is unitary if and only if it is of Arthur type. Combined with the algorithms of the first author or Hazeltine-Liu-Lo for detecting Arthur type representations, our result leads to an explicit algorithm for checking the unitarity of any given irreducible representation of good parity. Finally, we determine the set of unitary representations that may appear as local components of the discrete automorphic spectrum.

Unitary dual of p-adic split SO₂₍+₁ and Sp₂₍: The good parity case (and slightly beyond)

Key Points

Abstract

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