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. VignĂ©ras).

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. VignĂ©ras).

Reviewer: S. Gelbart (Rehovot)

##### 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 |