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**Induced \(R\)-representations of \(p\)-adic reductive groups.**
*(English)*
Zbl 0943.22017

The author completes the classification of irreducible \(R\)-representations of \(GL(n,F)\). Here \(F\) is a \(p\)-adic field, or more generally, a nonarchimedean local field with finite residue field of characteristic \(p\) and \(R\) is an algebraically closed field of positive characteristic \(l\neq p\). All representations are assumed to be smooth. Quoting from the introduction: “We reduce the classification of all irreducible \(R\)-representations of \(GL(n,F)\) to the classification of the unipotent irreducible representations and of the supercuspidal irreducible representations.”

The irreducible cuspidal or supercuspidal \(R\)-representations of \(GL(n,F)\) were classified by the author [Prog. Math. 141, 415-452 (1997; Zbl 0893.11048)]. In the paper under review, the author classifies the unipotent or superunipotent representations by defining a map from the restricted modular Deligne-Langlands parameters \((s,N)\) to the set of unipotent irreducible \(R\)-representations of \(GL(n,F)\). Here \(s\) is a semisimple element in \(GL(n,F)\) and \(N\) is a nilpotent element in \(M(n,R)\) satisfying certain conditions. Two couples which are \(GL(n,R)\) conjugate are identified. One of the main results of this paper is that this map is injective. The proof involves a careful study of the supercuspidal support of induced representations. In a note added before publication the author proves that the map is also surjective and hence the set of unipotent or superunipotent representations is determined. The paper contains other interesting results on \(R\)-representations of \(p\)-adic reductive groups. We note that the classification results were used by the author in her recent proof of the local Langlands conjecture for \(R\)-representations of \(GL(n,F)\).

The irreducible cuspidal or supercuspidal \(R\)-representations of \(GL(n,F)\) were classified by the author [Prog. Math. 141, 415-452 (1997; Zbl 0893.11048)]. In the paper under review, the author classifies the unipotent or superunipotent representations by defining a map from the restricted modular Deligne-Langlands parameters \((s,N)\) to the set of unipotent irreducible \(R\)-representations of \(GL(n,F)\). Here \(s\) is a semisimple element in \(GL(n,F)\) and \(N\) is a nilpotent element in \(M(n,R)\) satisfying certain conditions. Two couples which are \(GL(n,R)\) conjugate are identified. One of the main results of this paper is that this map is injective. The proof involves a careful study of the supercuspidal support of induced representations. In a note added before publication the author proves that the map is also surjective and hence the set of unipotent or superunipotent representations is determined. The paper contains other interesting results on \(R\)-representations of \(p\)-adic reductive groups. We note that the classification results were used by the author in her recent proof of the local Langlands conjecture for \(R\)-representations of \(GL(n,F)\).

Reviewer: Ehud Moshe Baruch (Santa Cruz)

### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |

20G25 | Linear algebraic groups over local fields and their integers |