Unitary dual of an abelian Lie group - Imported Knowlpedia Content

Definition

Let $G$ be a connected abelian Lie group. Every irreducible strongly continuous unitary representation of $G$ on a complex Hilbert space is one-dimensional, hence is a continuous unitary character. Using

G\cong\mathbb R^n\times\mathbb T^m

from the structure theorem for connected abelian Lie groups, its unitary dual is

\widehat G\cong\mathbb R^n\times\mathbb Z^m .

The point $(\xi,k)$ corresponds to $\chi_{\xi,k}(x,z)=e^{i\langle\xi,x\rangle}z_1^{k_1}\cdots z_m^{k_m}$ . This identification carries the usual Euclidean topology on the first factor and the discrete topology on the second.

Why irreducibles are characters

All operators in a unitary representation of an abelian group commute. For an irreducible representation, the commutant consists only of scalars, so every group element acts by a scalar of modulus one. Strong continuity makes the resulting scalar map $G\to\mathbb T$ continuous. Thus the unitary dual agrees with the Pontryagin dual, not merely as a set but with its natural topology Folland, Chapter 4.

Basic factors

The characters of $\mathbb R^n$ are $x\mapsto e^{i\langle\xi,x\rangle}$ , parametrized by $\xi\in\mathbb R^n$ . The characters of $\mathbb T^m$ are the monomials $z\mapsto z^k$ , parametrized by $k\in\mathbb Z^m$ . Consequently, $\widehat{\mathbb R^n}\cong\mathbb R^n$ and $\widehat{\mathbb T^m}\cong\mathbb Z^m$ , while the dual of a product is the product of the duals.

Scope and Plancherel theory

Disconnected abelian Lie groups may have additional discrete components; the general statement remains that their irreducible unitary representations are characters, but their dual is obtained from the full locally compact abelian group rather than the displayed connected classification. Haar measure on $\widehat G$ combines Lebesgue measure on $\mathbb R^n$ with counting measure on $\mathbb Z^m$ , up to reciprocal normalization, and is the measure used in abelian Plancherel theory.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 4 on characters, Pontryagin duality, and Fourier analysis of locally compact abelian groups.
Walter Rudin, Fourier Analysis on Groups, Wiley-Interscience, 1962. Wiley DOI record. Relevant: Chapter 1 on locally compact abelian groups and their character groups.