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Harmonic analysis on reductive groups

Objective

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.

Call for proposal

ERC-2010-AdG_20100224
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Host institution

UNIVERSITEIT VAN AMSTERDAM
Address
Spui 21
1012WX Amsterdam
Netherlands

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Activity type
Higher or Secondary Education Establishments
Principal investigator
Eric Marcus Opdam (Prof.)
Administrative Contact
J.C.M. Lansbergen (Ms.)
EU contribution
€ 960 017,60

Beneficiaries (3)

UNIVERSITEIT VAN AMSTERDAM
Netherlands
EU contribution
€ 960 017,60
Address
Spui 21
1012WX Amsterdam

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Activity type
Higher or Secondary Education Establishments
Principal investigator
Eric Marcus Opdam (Prof.)
Administrative Contact
J.C.M. Lansbergen (Ms.)
GOTTFRIED WILHELM LEIBNIZ UNIVERSITAET HANNOVER

Participation ended

Germany
EU contribution
€ 128 480,23
Address
Welfengarten 1
30167 Hannover
Activity type
Higher or Secondary Education Establishments
Administrative Contact
Bernhard Kroetz (Prof.)
UNIVERSITAET PADERBORN
Germany
EU contribution
€ 680 502,17
Address
Warburger Strasse 100
33098 Paderborn

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Activity type
Higher or Secondary Education Establishments
Administrative Contact
Daniela Gerdes (Ms.)