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Harmonic analysis on reductive groups (HARG)
Date du début: 1 mars 2011, Date de fin: 29 févr. 2016 PROJET  TERMINÉ 

"We propose to attack a variety of fundamental open problems inharmonic analysis on $p$-adic and real reductive groups.Specifically we seek solutions to the local Langlands conjecturesand various normalization problems of discrete series representations.For $p$-adic groups, affine Hecke algebras are a major technical tool.Our understanding of these algebras with unequal parameters hasadvanced recently and allows us to address these problems.We will compute the Plancherel measure on the Bernstein componentsexplicitly. Using a new transfer principle of Plancherel measuresbetween Hecke algebras we will combine Bernstein components to form$L$-packets, following earlier work of Reeder in small rank.We start with the tamely ramified case, building on work ofReeder-Debacker. We will also explore these methods for $L$-packetsof positive depth, using recent progress due to Yu and others.Furthermore we intend to study non-temperedunitary representations via affine Hecke algebras, extending thework of Barbasch-Moy on the Iwahori spherical unitary dual.As for real reductive groups we intend to address essentialquestions on the convergence of the Fourier-transform. This theoryis widely developed for functions which transform finitely under amaximal compact subgroup. We wish to drop this condition in orderto obtain global final statements for various classes of rapidlydecreasing functions. We intend to extend our results to certain types ofhomogeneous spaces, e.g symmetric and multiplicity one spaces. For doingso we will embark to develop a suitable spherical character theory fordiscrete series representations and solve the corresponding normalizationproblems.The analytic nature of the Plancherel measure and the correct interpretationthereof is the underlying theme which connects the various parts ofthis proposal."



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