Objective 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. Fields of science natural sciencesmathematicspure mathematicsalgebra Programme(s) FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Topic(s) ERC-AG-PE1 - ERC Advanced Grant - Mathematical foundations Call for proposal ERC-2010-AdG_20100224 See other projects for this call Funding Scheme ERC-AG - ERC Advanced Grant Host institution UNIVERSITEIT VAN AMSTERDAM EU contribution € 960 017,60 Address SPUI 21 1012WX Amsterdam Netherlands See on map Region West-Nederland Noord-Holland Groot-Amsterdam Activity type Higher or Secondary Education Establishments Principal investigator Eric Marcus Opdam (Prof.) Administrative Contact J.C.M. Lansbergen (Ms.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data Beneficiaries (3) Sort alphabetically Sort by EU Contribution Expand all Collapse all UNIVERSITEIT VAN AMSTERDAM Netherlands EU contribution € 960 017,60 Address SPUI 21 1012WX Amsterdam See on map Region West-Nederland Noord-Holland Groot-Amsterdam Activity type Higher or Secondary Education Establishments Principal investigator Eric Marcus Opdam (Prof.) Administrative Contact J.C.M. Lansbergen (Ms.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data GOTTFRIED WILHELM LEIBNIZ UNIVERSITAET HANNOVER Participation ended Germany EU contribution € 128 480,23 Address WELFENGARTEN 1 30167 Hannover See on map Region Niedersachsen Hannover Region Hannover Activity type Higher or Secondary Education Establishments Administrative Contact Bernhard Kroetz (Prof.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data UNIVERSITAET PADERBORN Germany EU contribution € 680 502,17 Address WARBURGER STRASSE 100 33098 Paderborn See on map Region Nordrhein-Westfalen Detmold Paderborn Activity type Higher or Secondary Education Establishments Administrative Contact Daniela Gerdes (Ms.) Links Contact the organisation Opens in new window Website Opens in new window Total cost No data
