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Harmonic analysis on reductive groups

We propose to attack a variety of fundamental open problems in harmonic analysis on $p$-adic and real reductive groups. Specifically we seek solutions to the local Langlands conjectures and various normalization problems of discr... ver más

29/02/2016

UvA

2M€

Presupuesto del proyecto: 2M€

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UNIVERSITEIT VAN AMSTERDAM No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

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Líder del proyecto

UNIVERSITEIT VAN AMSTERDAM No se ha especificado una descripción o un objeto social para esta compañía.

TRL 4-5

Presupuesto del proyecto 2M€

Fecha límite de participación Sin fecha límite de participación.

Descripción del proyecto We propose to attack a variety of fundamental open problems in harmonic analysis on $p$-adic and real reductive groups. Specifically we seek solutions to the local Langlands conjectures and 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 has advanced recently and allows us to address these problems. We will compute the Plancherel measure on the Bernstein components explicitly. Using a new transfer principle of Plancherel measures between 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 of Reeder-Debacker. We will also explore these methods for $L$-packets of positive depth, using recent progress due to Yu and others. Furthermore we intend to study non-tempered unitary representations via affine Hecke algebras, extending the work of Barbasch-Moy on the Iwahori spherical unitary dual. As for real reductive groups we intend to address essential questions on the convergence of the Fourier-transform. This theory is widely developed for functions which transform finitely under a maximal compact subgroup. We wish to drop this condition in order to obtain global final statements for various classes of rapidly decreasing functions. We intend to extend our results to certain types of homogeneous spaces, e.g symmetric and multiplicity one spaces. For doing so we will embark to develop a suitable spherical character theory for discrete series representations and solve the corresponding normalization problems. The analytic nature of the Plancherel measure and the correct interpretation thereof is the underlying theme which connects the various parts of this proposal.

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