Let G be a connected reductive (real or complex) Lie group.
A continuous representation (π, V) of G on a complex Hilbert space V[1] is called admissible if π restricted to K is unitary and each irreducible unitary representation of K occurs in it with finite multiplicity.
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.
Let G be a locally compact totally disconnected group (such as a reductive algebraic group over a nonarchimedean local field or over the finite adeles of a global field).
If, in addition, the space of vectors fixed by any compact open subgroup is finite dimensional then π is called admissible.
Admissible representations of p-adic groups admit more algebraic description through the action of the Hecke algebra of locally constant functions on G. Deep studies of admissible representations of p-adic reductive groups were undertaken by Casselman and by Bernstein and Zelevinsky in the 1970s.
by Howe, Moy, Gopal Prasad and Bushnell and Kutzko, who developed a theory of types and classified the admissible dual (i.e. the set of equivalence classes of irreducible admissible representations) in many cases.