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From Laplace to Langlands via representations of orthogonal groups. (English) Zbl 1159.11047

The main aim of this article is an exposition of the local Langlands correspondence, which classifies the complex irreducible representations of a reductive group \(G\) defined over a local field. Even though this correspondence is largely conjectural, it has significantly changed the understanding of representation theory and number theory, and it has shed new light on some traditional problems in representation theory.
In the first part of the article, the authors describe the Langlands correspondence both for real and \(p\)-adic reductive groups, giving many examples and assuming only a basic knowledge of representation theory. They then address the classical problem of restricting irreducible discrete series representations from \(\text{SO}_{2n+1}\) to \(\text{SO}_{2n}\). The Gross-Prasad conjectures describe these restrictions in terms of number-theoretic data attached to the Langlands parameters of the representations. Based on recent work, the authors verify these conjectures in some special cases. Their proof invokes a result of the second author concerning the restriction of Deligne-Lusztig characters for finite orthogonal groups.

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
22E50 Representations of Lie and linear algebraic groups over local fields
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