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The admissible dual of $$\text{GL}(N)$$ via compact open subgroups. (English) Zbl 0787.22016
Annals of Mathematics Studies. 129. Princeton, NJ: Princeton University Press,. ix, 313 pp. (1993).
In this work, the authors give a new and effectively complete classification of the irreducible smooth complex representations of the general linear group $$G=\text{GL}(N,F)$$ over a non-archimedean local field $$F$$. Their approach represents the sophisticated culmination of R. Howe’s earlier suggestion that the representations of $$G$$ can be studied by first considering restrictions to a suitable compact open subgroup (“essential $$K$$-types”), and then giving a classification in terms of Hecke algebra isomorphisms. On the other hand, their approach is also an extension of Kazhdan and Lusztig’s classification of the irreducible representations of $$G$$ (or more generally any $$p$$-adic reductive algebraic group) with an Iwahori-fixed vector in terms of $$L$$-group data attached to simple modules over certain Hecke algebras.
In more detail, the authors introduce the notation of simple and split “types” for an arbitrary smooth irreducible representation $$\pi$$ of $$G$$. A simple type is a pair $$(J,\lambda)$$ consisting of an open compact subgroup $$J$$ and smooth irreducible representation $$\lambda$$ of $$J$$; attached to $$(J,\lambda)$$ is an integer $$e\geq 1$$, and a field extension $$E/F$$ contained in $$A=M(N,F)$$ and of degree $$N/e$$, such that if $$C$$ denotes the centralizer in $$A$$ of $$E$$ (so $$C\cong M(e,E)$$), then $$J\cap C^ \times$$ is an Iwahori subgroup of $$C^ \times$$, and the Hecke algebra $${\mathcal H}(G,J,\lambda)$$ is isomorphic to $${\mathcal H}(C^ \times, J\cap C^ \times,1)$$ (as $$\mathbb C$$-algebras with 1). Thus there results a correspondence between the irreducible representations of $$G$$ which contain $$\lambda$$ (for some $$J$$) and the “arithmetically unramified” irreducible representations of $$C^ \times$$ (i.e. those with an Iwahori-fixed vector); moreover, this correspondence preserves supercuspidals (whose corresponding simple types $$(J,\lambda)$$ are called “maximal”, since $$E/F$$ is then a maximal subfield of $$A$$).
On the other hand, there is the complementary notion of a split type, again a pair $$(K,\vartheta)$$ consisting of an open compact subgroup $$K$$ and irreducible smooth representation of $$K$$. These split types are so constructed that any $$\pi$$ which contains no simple type must contain a split type $$(K,\vartheta)$$, and for such a split type, there is a proper parabolic subgroup $$P$$ of $$G$$, with unipotent radical $$U$$, such that for any representation $$(\pi,v)$$ of $$G$$ containing $$\vartheta$$, we have (the Jacquet module) $$V_ U\neq\{0\}$$. It follows that an irreducible supercuspidal representation of $$G$$ must contain some simple type. (This proves, “incidentally”, the long standing conjecture that any irreducible supercuspidal representation is induced from an open compact mod center subgroup of $$G$$!) It is shown that an irreducible representation of $$G$$ either contains a simple type (in which case its “supercuspidal support” consists of unramified twists of a single supercuspidal representation of some $$\text{GL}(N/e,F)$$, $$e\geq 1$$), or else $$\pi$$ contains a split type (in which case there exists an integer $$r\geq 2$$, and irreducible representations $$\pi_ i$$ of $$\text{GL}(N_ i,F)$$, with $$1\leq i\leq r$$ and $$\sum N_ i=N$$, such that each $$\pi_ i$$ contains a simple type, the supercuspidal support of these distinct $$\pi_ i$$ is “disjoint”, and $$\pi\approx \pi_ 1\times\cdots\times \pi_ r$$ ($$=\text{Ind }\pi_ 1\otimes\cdots \otimes\pi_ r)$$).
In conclusion: any irreducible smooth representation of $$G$$ is described by a prescribed set of simple types, which in turn can be described by Kazhdan-Lusztig type invariants. This classification of the admissible dual of $$\text{GL}(N,F)$$ is not only abstractly pleasing, but should also be especially well suited for proving the local Langlands conjecture for $$G$$ (and in particular for computing the Godement-Jacquet local constants $$\varepsilon(\pi,s)$$). It should also prove well-adapted to the newly developing theory of $$p$$-adic (versus complex) representations of $$\text{GL}(N,F)$$ (cf. the recent work of M.-F. Vignéras).

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 22-02 Research exposition (monographs, survey articles) pertaining to topological groups 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11S37 Langlands-Weil conjectures, nonabelian class field theory
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