Totally Disconnected Groups
Let G be a locally compact totally disconnected group (such as a reductive algebraic group over a local field or over the finite adeles of a global field). A representation (π, V) of G on a complex vector space V is called smooth if the subgroup of G fixing any vector of V is open. 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. Much progress has been made more recently by Howe and Moy 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.
Read more about this topic: Admissible Representation
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