Smooth representation

In mathematics, admissible representations are a well-behaved class of representations used in the representation theory of reductive Lie groups and locally compact totally disconnected groups. They were introduced by Harish-Chandra.

Let G be a connected reductive (real or complex) Lie group. Let K be a maximal compact subgroup. A continuous representation (π, V) of G on a complex Hilbert space V is called admissible if π restricted to K is unitary and each irreducible unitary representation of K occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation of G.

An admissible representation π induces a ${\displaystyle ({\mathfrak {g}},K)}$-module which is easier to deal with as it is an algebraic object. Two admissible representations are said to be infinitesimally equivalent if their associated ${\displaystyle ({\mathfrak {g}},K)}$-modules are isomorphic. Though for general admissible representations, this notion is different than the usual equivalence, it is an important result that the two notions of equivalence agree for unitary (admissible) representations. Additionally, there is a notion of unitarity of ${\displaystyle ({\mathfrak {g}},K)}$-modules. This reduces the study of the equivalence classes of irreducible unitary representations of G to the study of infinitesimal equivalence classes of admissible representations and the determination of which of these classes are infinitesimally unitary. The problem of parameterizing the infinitesimal equivalence classes of admissible representations was fully solved by Robert Langlands and is called the Langlands classification.

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