Modular representations of \(p\)-adic groups and of affine Hecke algebras. (English) Zbl 1151.11325
Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20–28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor (ISBN 7-04-008690-5/3-vol. set). 667-677 (2002).
From the text: The author surveys some results in the theory of modular representations of a reductive \(p\)-adic group, in positive characteristic \(\ell\neq p\) and \(\ell=p\).
The congruences between automorphic forms and their applications to number theory are a motivation to study the smooth representations of a reductive \(p\)-adic group \(G\) over an algebraically closed field \(R\) of any characteristic. The purpose of the talk is to give a survey of some aspects of the theory of representations of \(G\). In positive characteristic, most results are due to the author; when proofs are available in the literature (some of them are not !), references will be given.
A prominent role is played by the unipotent block which contains the trivial representation. There is a finite list of types, such that the irreducible representations of the unipotent block are characterized by the property that they contain a unique type of the list. The types define functors from the \(R\)-representations of \(G\) to the right modules over generalized affine Hecke algebras over \(R\) with different parameters; in positive characteristic \(\ell\), the parameters are 0 when \(\ell=p\), and roots of unity when \(\ell\neq p\).
In characteristic 0 or \(\ell\neq p\), for a \(p\)-adic linear group, there is a Deligne-Langlands correspondence for irreducible representations; the irreducible in the unipotent block are annihilated by a canonical ideal \(J\); the category of representations annihilated by \(J\) is Morita equivalent to the affine Schur algebra, and the unipotent block is annihilated by a finite power \(J^k\).
New phenomena appear when \(\ell=p\), as the supersingular representations discovered by Barthel-Livné and classified by Ch. Breuil for \(\text{GL}(2,\mathbb Q_p)\). The modules for the affine Hecke algebras of parameter 0 and over \(R\) of characteristic \(p\), are more tractable than the \(R\)-representations of the group, using that the center \(Z\) of a \(\mathbb Z[q]\)-affine Hecke algebra \(H\) of parameter \(q\) is a finitely algebra and \(H\) is a generated \(Z\)-module. The classification of the simple modules of the pro-\(p\)-Iwahori Hecke algebra of \(\text{GL}(2,\mathbb F)\) suggests the possibility of a Deligne-Langlands correspondence in characteristic \(p\).
For the entire collection see [Zbl 0993.00022].
The congruences between automorphic forms and their applications to number theory are a motivation to study the smooth representations of a reductive \(p\)-adic group \(G\) over an algebraically closed field \(R\) of any characteristic. The purpose of the talk is to give a survey of some aspects of the theory of representations of \(G\). In positive characteristic, most results are due to the author; when proofs are available in the literature (some of them are not !), references will be given.
A prominent role is played by the unipotent block which contains the trivial representation. There is a finite list of types, such that the irreducible representations of the unipotent block are characterized by the property that they contain a unique type of the list. The types define functors from the \(R\)-representations of \(G\) to the right modules over generalized affine Hecke algebras over \(R\) with different parameters; in positive characteristic \(\ell\), the parameters are 0 when \(\ell=p\), and roots of unity when \(\ell\neq p\).
In characteristic 0 or \(\ell\neq p\), for a \(p\)-adic linear group, there is a Deligne-Langlands correspondence for irreducible representations; the irreducible in the unipotent block are annihilated by a canonical ideal \(J\); the category of representations annihilated by \(J\) is Morita equivalent to the affine Schur algebra, and the unipotent block is annihilated by a finite power \(J^k\).
New phenomena appear when \(\ell=p\), as the supersingular representations discovered by Barthel-Livné and classified by Ch. Breuil for \(\text{GL}(2,\mathbb Q_p)\). The modules for the affine Hecke algebras of parameter 0 and over \(R\) of characteristic \(p\), are more tractable than the \(R\)-representations of the group, using that the center \(Z\) of a \(\mathbb Z[q]\)-affine Hecke algebra \(H\) of parameter \(q\) is a finitely algebra and \(H\) is a generated \(Z\)-module. The classification of the simple modules of the pro-\(p\)-Iwahori Hecke algebra of \(\text{GL}(2,\mathbb F)\) suggests the possibility of a Deligne-Langlands correspondence in characteristic \(p\).
For the entire collection see [Zbl 0993.00022].
MSC:
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 11S37 | Langlands-Weil conjectures, nonabelian class field theory |
| 20C08 | Hecke algebras and their representations |
| 20G05 | Representation theory for linear algebraic groups |
