Spherical Unitary Explorer

The simplest operation is to give a type A, B, C, D and a list λ of n real numbers. Associated to this is an irreducible spherical representation X(λ) of the corresponding split real or p-adic group. (The rank is n, except in type A it is n−1.) The program will determine whether X(λ) is unitary, and some supplementary data about the representation. This implements the algorithm due to Dan Barbasch.

Here is an example of how to read the results.

The type is C, and λ has 14 entries, so G=Sp(28), of type C₁₄. The dual group is SO(29), of type B₁₄.

Associated to λ is a nilpotent orbit O, given as a partition of 29.

The (reductive part of the) centralizer of the orbit is denoted Z. The 5 factors of Z in this case correspond to the 5 distinct entries occurring in the partition corresponding to O. For example the row of length 1 occurs 9 times in O, giving a factor of O(9) in Z.

Associated to the orbit O is a Levi factor M of G, and a semisimple parameter h, which may be decomposed according to the factors of M. The intersection of O with M is an orbit O_M. The rows of O_M are the same as those of O, with different rows corresponding to different factors of M. In this example the rows 5,3,1 correspond to the orbit 5,3,1 of Sp(6).

Each factor of M has a parameter nu (giving a one-dimensional representation of M), in such a way that λ=h+nu. The factor of O in the first factor of M is distinguished, meaning its centralizer is finite, so there is no nu parameter for this factor.

The unitarity of the representation is determined by whether nu, interpreted as a parameter for the dual group of Z, is unitary.

The second table has one row for each row length in the partition of O, i.e. each factor of Z. The multiplicity is the number of times the row occurs. The column labelled Z gives the corresponding factor of Z. The column labelled nu gives nu on this factor of Z. The unitary column indicates whether this representation (of the dual of this factor of Z) is unitary.

Note: The standard form of the coordinates are the “standard” coordinates for classical groups. You may also give fundamental weight coordinates by choosing that option in the Coordinates menu. See below.

Note: In types B, C, D, after acting by the Weyl group (and an outer automorphism in type D) we may take all coordinates to be positive. Negative signs in λ are ignored.

In type A for the parameter to be Hermitian each positive entry must occur in a pair with its negative. For convenience in type A you may enter all non-negative entries; the negative of each positive entry will be adjoined to λ. For example (2,1,1,0) will be replaced by (2,1,1,0,−1,−1,−2).

Note that in type D_odd λ must have a 0 entry to be Hermitian.

Coordinates

The standard coordinates are those for classical groups; λ is a decreasing sequence of real numbers, the number of coordinates is the rank, except in type A_n there are n+1 coordinates (summing to 0). There are also two versions of fundamental weight coordinates available. These correspond to the same choice of dominant Weyl chamber, but different ordering of the simple roots (they agree in type A). In types B, C, D: in Bourbaki coordinates the double bond or fork is near the node labelled n; the Gap labelling is the opposite. For example in type B₃: Bourbaki 1--2=>=3, Gap 3--2=>=1.

To view the Dynkin Diagram and the matrices relating the standard and weight coordinates, open Dynkin Diagram and coordinates in the results, or use the root systems section, which has much more information on each root system, including the Cartan matrix, the list of all positive roots, etc.

Examples, unipotent representations, lists

The Examples menu gives some illustrative examples, together with comments explaining their notable features.

Choosing File examples will test all of the points in the given files. These include the representatives of all facets, or representatives of just the facets which correspond to unitary representations. Choose High level of detail for full details, or Low for one line per example. Tick the box to see only the points representing unitary representations.

Choose a type and rank to view all Unipotent representations. These are all unitary. Set the level of detail for full listing, or one line per point.

You may also test a list of points, pasted into the box or read from a file. Choose a type with the menu. Each line should contain a single parameter. Everything after a # is ignored, as are all characters except numbers, ., /, commas and spaces.