In mathematics,admissible representations are a well-behaved class ofrepresentations used in therepresentation theory ofreductiveLie groups andlocally compacttotally disconnected groups. They were introduced byHarish-Chandra.
LetG be a connected reductive (real or complex) Lie group. LetK be a maximal compact subgroup. A continuous representation (π, V) ofG on a complexHilbert spaceV[1] is calledadmissible if π restricted toK isunitary and eachirreducible unitary representation ofK occurs in it with finite multiplicity. The prototypical example is that of an irreducible unitary representation ofG.
An admissible representation π induces a-module which is easier to deal with as it is an algebraic object. Two admissible representations are said to beinfinitesimally equivalent if their associated-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-modules. This reduces the study of the equivalence classes of irreducible unitary representations ofG 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 byRobert Langlands and is called theLanglands classification.
LetG be alocally compact totally disconnected group (such as a reductive algebraic group over a nonarchimedeanlocal field or over the finiteadeles of aglobal field). A representation (π, V) ofG on a complex vector spaceV is calledsmooth if the subgroup ofG fixing any vector ofV isopen. If, in addition, the space of vectors fixed by anycompact open subgroup is finite dimensional then π is calledadmissible. Admissible representations ofp-adic groups admit more algebraic description through the action of theHecke algebra of locally constant functions onG.
Deep studies of admissible representations ofp-adic reductive groups were undertaken byCasselman and byBernstein andZelevinsky in the 1970s. Progress was made more recently[when?] byHowe, Moy,Gopal Prasad and Bushnell and Kutzko, who developed atheory of types and classified the admissible dual (i.e. the set of equivalence classes of irreducible admissible representations) in many cases.[citation needed]