Information AboutUnitary Dual |
| CATEGORIES ABOUT UNITARY REPRESENTATION | |
| representation theory of groups | |
| unsolved problems in mathematics | |
|
The theory has been widely applied in Quantum Mechanics since the 1920s , particularly influenced by Hermann Weyl 's 1928 book ''Gruppentheorie und Quantenmechanik''. One of the pioneers in constructing a general theory of unitary representations, for any group ''G'' rather than just for particular groups useful in applications, was George Mackey . CONTEXT IN HARMONIC ANALYSIS
The general form of the Plancherel Theorem tries to describe the regular representation of ''G'' on ''L''2(''G'') by means of a Measure on the unitary dual. For ''G'' abelian this is given by the Pontryagin duality theory. 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. FORMAL DEFINITIONS Let ''G'' be a topological group. 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 , this representation is necessarily Smooth (respectively Real Analytic ) with respect to the Differentiable Structure (respectively real analytic structure) of the Lie group. COMPLETE REDUCIBILITY A unitary representation is Completely Reducible , in the sense that for any closed Invariant Subspace , the Orthogonal Complement is again a closed invariant subspace. This is at the level of an observation, but is a fundamental property. For example, it implies that finite dimensional unitary representations are always a direct sum of irreducible representations, in the algebraic sense. 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 Group s, and more generally for Compact Group s, by an averaging argument applied to an arbitrary hermitian structure. For example, a natural proof of Maschke's Theorem is by this route. UNITARIZABILITY AND THE UNITARY DUAL QUESTION In general, for non-compact groups, it is a more serious question which representations are unitarizable. One of the important unsolved problems in mathematics is the description of the unitary dual, the effective classification of irreducible unitary representations of all real Reductive Lie Group s. All irreducible unitary representations are admissible (or rather their Harish-Chandra Module s are), and the admissible representations are given by the Langlands Classification , and it is easy to tell which of them have a non-trivial invariant sesquilinear form. The problem is that it is in general hard to tell when this form is positive definite. For many reductive Lie groups this has been solved; see Representation Theory Of SL2(R) and Representation Theory Of The Lorentz Group for examples. SEE ALSO |
|
|