In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in the case that G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.
The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book Gruppentheorie und Quantenmechanik.
The theory of unitary representations of topological groups is closely connected with harmonic analysis.
In the case of an abelian group G, a fairly complete picture of the representation theory of G is given by Pontryagin duality.
The general form of the Plancherel theorem tries to describe the regular representation of G on L2(G) using a measure on the unitary dual.
For G compact, this is done by the Peter–Weyl theorem; in that case, the unitary dual is a discrete space, and the measure attaches an atom to each point of mass equal to its degree.
A strongly continuous unitary representation of G on a Hilbert space H is a group homomorphism from G into the unitary group of H, such that g → π(g) ξ is a norm continuous function for every ξ ∈ H. Note that if G is a Lie group, the Hilbert space also admits underlying smooth and analytic structures.
Analytic vectors are dense by a classical argument of Edward Nelson, amplified by Roe Goodman, since vectors in the image of a heat operator e–tD, corresponding to an elliptic differential operator D in the universal enveloping algebra of G, are analytic.
Not only do smooth or analytic vectors form dense subspaces; but they also form common cores for the unbounded skew-adjoint operators corresponding to the elements of the Lie algebra, in the sense of spectral theory.
Since unitary representations are much easier to handle than the general case, it is natural to consider unitarizable representations, those that become unitary on the introduction of a suitable complex Hilbert space structure.
This works very well for finite groups, and more generally for compact groups, by an averaging argument applied to an arbitrary hermitian structure (more specifically, a new inner product defined by an averaging argument over the old one, w.r.t which the representation is unitary).
The problem is that it is in general hard to tell when the quadratic form is positive definite.