Unitary spherical representations

The spherical unitary dual of a split p-adic group may be computed, at least in principle, by computer. See the paper by Dan Barbasch and notes by John Stembridge for more information.

This has been carried out for all exceptional groups except E₈ by John Stembridge, and independently by Dan Barbasch and Dan Ciubotaru. Jiu-Kang Yu has also done some of these computations.

These directories contain lists giving the answer, and files verifying that the answers obtained by Stembridge and Barbasch/Ciubotaru agree. Here is a brief explanation of how the data was generated, and how to read it.

The spherical unitary explorer is complementary to this page. The explorer gives Dan Barbasch's computation of the spherical unitary dual of a split classical group. This page is primarly intended for exceptional groups.

Each directory may contain files of the following types:

• data: Dynkin diagram and list of positive coroots, in simple coroots coordinates
• facets.*: list of representative points from the positive definite facets, given in standard or weight coordinates
• maxFacets*: list of maximal facets
• report.*: report on file facets.*, including points in standard and weight coordinates, facet (in terms of <0,=0,>0), dimension
• compare.*: comparison of two files, giving points in common, common facets, and disagreements (if any)

Data from John Stembridge is from his unitarity data, and he also has some maple software.

View data for: B₂  D₄  E₆  E₇  E₈  F₄  G₂ 

The report and comparison files were produced by the perl script sphericalReport.pl