Admissible representation
- Admissible representation
In mathematics, admissible representations are a well behaved class of representations used in the representation theory of reductive Lie groups over real or "p"-adic fields. They were introduced by Harish-Chandra.
Real reductive groups
For real reductive Lie groups "G", an admissible representation is not quite a representation of "G" itself. Instead it is a representation "V" of the pair ("g","K"), where "g" is the Lie algebra of "G", and "K" is a maximal compact subgroup, such that all vectors of "V" are "K"-finite and any irreducible representation of "K" occurs only a finite number of times.
Harish-Chandra proved that the "K"-finite vectors of any irreducible unitary representation of "G" form an admissible representation of ("g", "K"). This reduced the analytic problem of studying unitary representations of "G" to the algebraic problem of studying admissible representations.
p-adic groups
Any "p"-adic Lie group "G" has a topology making it into a totally disconnected (t.d.) topological group. A representation "π" of "G" is called smooth if the subgroup of "G" fixing any vector of "π" 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 Bill 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.
References
*cite book|last = Knapp|first = Anthony W.|title = Representation Theory of Semisimple Groups: An Overview Based on Examples|publisher = Princeton University Press|date = 2001|isbn = 0691090890|url = http://books.google.com/books?id=QCcW1h835pwC
*cite book|last = Bushnell|first = Colin J.|coauthors = Philip C. Kutzko|title = The admissible dual of GL(N) via compact open subgroups|series = Annals of Mathematics Studies 129|publisher=Princeton University Press|date = 1993|isbn=0691021147
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