Unitary Spherical Representations:Brief Explanation

See the paper by Dan Barbasch and notes by John Stembridge for more information. Some of the other expository papers also have some helpful information.

Describing the spherical unitary dual: Let G be a simple split group (real or p-adic). The irreducible admissible spherical representations of G are parametrized by a complex vector space VC module the Weyl group. The spherical unitary dual of G is a closed subset of V. The computation of the spherical unitary representations may be reduced to the case of real parameters, so V is a real vector space, and we need to describe a subset S of V.

Each positive root defines a 1-hyperplane. These hyperplanes cut V up into a finite set of facets. For each facet and each root we assign +,1,- depending on whether this root is >1, =1 or <1 on this root. Thus a facet is described by a collection of +,1,-, one for each root.

A basic fact is that a representation associated to a dominant point v is unitary if and only if this holds for every v in the same facet. Thus S is described by a finite set of data: a dominant point v in each facet of unitary parameters.

Coordinates: A natural basis of V is the set of fundamental weights. John Stembridge uses these coordinates. Another choice is the standard coordinates of Bourbaki, Groupes et Algebras de Lie, Chapter 6. Dan Barbasch and Dan Ciubotaru use these coordinates. Note that for E6 and E7 the nmber of coordinates is greater than the rank.

For every point v in V and every irreducible representation X of the Weyl group W there is a matrix real symmetric A(v,X). The spherical representation pi(v) of the p-adic exceptional group is unitary if and only if A(v,X) is positive semi--definite for all X. This calculation can be done by computer. It has been carried out for all exceptional groups except E8 by Stembridge. This is the source of the data in his files. Barbasch and Ciubotaru do something similar, with a combination of mathematical and computational techniques.

More Details: Here is an explanation of the files.

data:The data files give the Dynkin Diagram with numbering, and the positive roots written as sums of simple roots.

Facets: An entry
[1, 0, 0, 0, 0, 1] {} {1} {23}
in a facets file gives the point [1, 0, 0, 0, 0, 1] in fundamental weight coordinates. This is stembridge's notation. The other numbers are some other data.

An entry
{{0,1/2,1/2,1,2,3,-9/2,9/2}},
gives a point in standard coordinates (this is an example from E7). This is Barbasch and Ciubotaru's notation.

Reports: The reports files have a line
[number(line number)] dimension (standard){weight}
A line
[8(21)] 0 (-1,-2,3) {1,1} <1,1,+,+,+,+>
(this is a G2 example) thus means this is point number 8, from line 21, of dimension 0, with the standard coordinates (-1,2,3) and weight coordinates (1,1) The chamber is given by 1,1,+,+,+,+; this means the first two roots have inner product 1, and the final 4 have inner product > 1. This is the point rho, corresponding to the trivial representation.

Facet closures are sometimes given. A line
[8/4]:[8/4][13/3][14/3][27/2][29/2][30/2][35/1][36/1][40/1][46/0][60/0][61/0][66/0] (from E6) means that facet #8, of dimension 4, has facet #13 of dimension 3, in its closure, also facet #14,.... #66, of dimension 0.

The closure matrix gives the closure relations between facets.

Comparisons: A file such as compare compares the facets described by two different files. It contains the following:

In both files:
[number(line) file 1][number(line) file 2] dimension (standard file 1){weight file 1}(standard file 2){weight file 2}

So a line (this is an E6 example)

[62(13)][42(51)] 0 (0,0,0,0,1/2,-1/2,-1/2,1/2){1/2,0,0,0,0,1/2}(0,0,0,0,1/2,-1/2,-1/2,1/2){1/2,0,0,0,0,1/2}<-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,-,1,-,-,-,1,-,-,1,-,1,1,1,1,1>

means point #62, line 13 from file 1 is in the same facet as point #42, line 51 from file 2. The standard and weight coordinates of both points are given. The points are not necessarily the same (only the facets). The final entry is the facet.