Research in the Stieltjes institute

For general p-adic reductive groups the local Langlands correspondence is much more complicated than the case $GL(n)$. We have been working on the verification of the Langlands conjecture for so called level 1 representations of simple split forms $G(F)$. Lusztig [5] has given a conjectural form of the Langlands conjecture at this level for the most complicated class of singular representations, the so called unipotent representations. It is not clear at present whether Lusztig's version of the correspondence has the expected behavior with respect to natural partitions of the set of unipotent representations. To understand such questions we have to study the properties of unipotent representations. The methods that we use are based on Lusztig's theorem that the centralizer algebras of representations that are induced from cuspidal unipotent representations are affine Hecke algebras. We can therefore deduce many properties of unipotent representations by the analytic study of affine Hecke algebras. We study the Hecke algebras from the point of view of $C^*$-algebras, and this seems to be a fruitful approach. Some of the ideas leading to this approach come from new constructions of Hecke algebras, due to Ivan Cherednik (North Carolina). The origin of these constructions can be traced back to conformal field theory.

This work has led to a satisfactory description of the L-packets of unipotent representations for exceptional groups, and of the formal degrees of their cuspidal unipotent representations. These developments also have an impact on the theory of representations finite simple groups of Lie type. For example, it was shown that the dimension of a unipotent cuspidal representation of a simple group of Lie type is given by a formula resembling the famous Weyl character formula.

Part of this work was done in collaboration with Gerrit Heckman (Nijmegen) [3] and Mark Reeder (Boston college) [7].

The research group of Van Dijk (Leiden) studies the harmonic analysis on homogeneous spaces of real reductive groups. This topic is closely related to the research described above. There is also a natural connection with the research in quantum groups and special functions by Koornwinder (UVA) and his group, and the research in operator algebra theory by Landsman (UVA).

The references [1], [2], [4] may serve as a good starting point for further reading.