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Langlands’ conjecture on Plancherel measures for \(p\)-adic groups. (English) Zbl 0852.22017

Harmonic analysis on reductive groups, Proc. Conf., Brunswick/ME (USA) 1989, Prog. Math. 101, 277-295 (1991).
One of Harish-Chandra’s many major achievements was a derivation of the Plancherel formulas for real and \(p\)-adic groups. While this formula has been made explicit in the real case, until recently, little was known in any generality concerning the precise measures involved in the \(p\)-adic case. Langlands has, however, conjectured that every Plancherel measure must be a product of certain root numbers with the ratios of corresponding Langlands \(L\)-functions at \(s = 0\) and \(s = 1\). This fundamentally global conjecture can be phrased in terms of normalizing standard intertwining operators by the root numbers and ratios in question. Since these quantities can often be computed this leads to explicit formulas in many new cases. Moreover, the \(L\)-functions, which are easier to compute in general, determine then the poles and zeroes of the measures and thus answer reducibility questions.
In the first part of this excellent paper the author explains Langlands’ conjecture and his fundamental result concerning the conjecture (generic representations of Levi subgroups of quasi-split groups) and some of its consequences. In the second part of the paper he works through three examples. In the first two examples he completes the analysis of the unitary duals of all rank two split \(p\)-adic groups supported on their maximal parabolics. He concludes with explicit formulas for the Plancherel measures for \(\text{GL} (n)\) and \(\text{SL} (n)\).
For the entire collection see [Zbl 0742.00061].

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
22E35 Analysis on \(p\)-adic Lie groups
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