The Admissible Dual of GL(N) via Compact Open Subgroups. (AM-129), Volume 129

This work gives a full description of a method for analyzing the admissible complex representations of the general linear group __G__ = __Gl(N,F)__ of a non-Archimedean local field __F__ in terms of the structure of these representations when they are restricted to certain compact open subgroups of __G__. The authors define a family of representations of these compact open subgroups, which they call __simple types.__ The first example of a simple type, the "trivial type," is the trivial character of an Iwahori subgroup of __G__. The irreducible representations of __G__ containing the trivial simple type are classified by the simple modules over a classical affine Hecke algebra. Via an isomorphism of Hecke algebras, this classification is transferred to the irreducible representations of __G__ containing a given simple type. This leads to a complete classification of the irreduc-ible smooth representations of __G__, including an explicit description of the supercuspidal representations as induced representations. A special feature of this work is its virtually complete reliance on algebraic methods of a ring-theoretic kind. A full and accessible account of these methods is given here. Comments For The Reader -- 1. Exactness And Intertwining. (1.1). Hereditary Orders. (1.2). Hereditary Orders Relative To Subfields. (1.3). Tame Corestriction. (1.4). Adjoint Maps. (1.5). Simple Strata And Intertwining. (1.6). The Simple Intersection Property -- 2. The Structure Of Simple Strata. (2.1). Equivalence Of Pure Strata. (2.2). Refinements Of Simple Strata. (2.3). Split Refinements. (2.4). Approximation Of Simple Strata. (2.5). Nonsplit Fundamental Strata. (2.6). Intertwining And Conjugacy -- 3. The Simple Characters Of A Simple Stratum. (3.1). The Rings Of A Simple Stratum. (3.2). Characters And Commutators. (3.3). Intertwining. (3.4). A Nondegeneracy Property. (3.5). Intertwining And Conjugacy. (3.6). Change Of Rings -- 4. Interlude With Hecke Algebras. (4.1). Induction And Intertwining. (4.2). Scalar Hecke Algebras. (4.3). Unitary Structures -- 5. Simple Types. (5.1). Heisenberg Representations. (5.2). Extending To Level Zero. (5.3). A Bound On Intertwining. (5.4). Affine Hecke Algebras And Weyl Groups. (5.5). Intertwining And Weyl Groups. (5.6). The Hecke Algebra Of A Simple Type. (5.7). Intertwining And Conjugacy For Simple Types -- 6. Maximal Types. (6.1). Extension By A Central Character. (6.2). Supercuspidal Representations -- 7. Typical Representations. (7.1). Some Iwahori Decompositions. (7.2). Iwahori Factorisation Of A Simple Type. (7.3). Main Theorems. (7.4). Proof Of The Principal Lemma. (7.5). The Strong Intertwining Property. (7.6). Jacquet Functors And Hecke Algebra Maps. (7.7). Discrete Series And Formal Degree -- 8. Atypical Representations. (8.1). Split Types. (8.2). Jacquet Module Of A Split Type I. (8.3). Jacquet Module Of A Split Type Ii. (8.4). The Main Theorems. (8.5). Classification -- Index Of Notation And Terminology. By Colin J. Bushnell & Philip C. Kutzko. Includes Bibliographical References And Index. Contents Introduction Comments for the reader 1. Exactness and intertwining (1.1) Hereditary orders (1.2) Hereditary orders relative to subfields (1.3) Tame corestriction (1.4) Adjoint maps (1.5) Simple strata and intertwining (1.6) The simple intersection property 2. The structure of simple strata (2.1) Equivalence of pure strata (2.2) Refinements of simple strata (2.3) Split refinements (2.4) Approximation of simple strata (2.5) Nonsplit fundamental strata (2.6) Intertwining and conjugacy 3. The simple characters of a simple stratum (3.1) The rings of a simple stratum (3.2) Characters and commutators (3.3) Intertwining (3.4) A nondegeneracy property (3.5) Intertwining and conjugacy (3.6) Change of rings 4. Interlude with Hecke algebras (4.1) Induction and intertwining (4.2) Scalar Hecke algebras (4.3) Unitary structures 5. Simple types (5.1) Heisenberg representations (5.2) Extending to level zero (5.3) A bound on intertwining (5.4) Affine Hecke algebras and Weyl groups (5.5) Intertwining and Weyl groups (5.6) The Hecke algebra of a simple type (5.7) Intertwining and conjugacy for simple types 6. Maximal types (6.1) Extension by a central character (6.2) Supercuspidal representations 7. Typical representations (7.1) Some Iwahori decompositions (7.2) Iwahori factorisation of a simple type (7.3) Main theorems (7.4) Proof of the principal lemma (7.5) The strong intertwining property (7.6) Jacquet functors and Hecke algebra maps (7.7) Discrete series and formal degree 8. Atypical representations (8.1) Split types (8.2) Jacquet module of a split type I (8.3) Jacquet module of a split type II (8.4) The main theorems (8.5) Classification References Index of notation and terminology This work gives a full description of a method for analyzing the admissible complex representations of the general linear group G = Gl(N, F) of a non-Archimedean local field F in terms of the structure of these representations when they are restricted to certain compact open subgroups of G. The authors define a family of representations of these compact open subgroups, which they call simple types. The first example of a simple type, the "trivial type," is the trivial character of an Iwahori subgroup of G. The irreducible representations of G containing the trivial simple type are classified by the simple modules over a classical affine Hecke algebra. Via an isomorphism of Hecke algebras, this classification is transferred to the irreducible representations of G containing a given simple type. This leads to a complete classification of the irreduc-ible smooth representations of G, including an explicit description of the supercuspidal representations as induced representations. A special feature of this work is its virtually complete reliance on algebraic methods of a ring-theoretic kind. A full and accessible account of these methods is given here A work that gives a full description of a method for analyzing the admissible complex representations of the general linear group G = Gl(N,F) of a non-Archimedean local field F in terms of the structure of these representations when they are restricted to certain compact open subgroups of G. This monograph describes a method for analyzing the admissable complex representations of the general linear group G = Gl(N,F) of a non-Archimedian local field F in terms of the structure of these representations when they are restricted to certain compact open subgroups of G.