K-orbits in G⁄B

Last updated: October 15, 2005

The classification of K-orbits in G⁄B is the basis of the geometric picture of representation theory, where representations correspond to K-equivariant local systems on such orbits. Also, the corresponding orbits on the dual side provide the partition of representations into “L-packets” in Langlands' classification of representations.

It turns out that the main combinatorial structure that the program uses internally for the parametrization of blocks of representations is in natural (1,1) correspondence with the set of K-orbits, and moreover we can compute the conjugation action of W (a.k.a. the cross-action), and the Cayley transforms in this description. It would be possible to output actual orbit representatives as products of certain canonically defined elements of finite order in the group (or even as matrices, for classical groups), but this is not currently implemented.

In our language, K-orbits are in (1,1)-correspondence with strong involutions normalizing T, in the G-conjugacy class corresponding to a strong real form lying over the chosen real form of G, modulo T-conjugacy. It is easy to see that this is equivalent to the description in [1]. The enumeration of K-orbits and the various actions are output by the “kgb” command. For example, here is the output of “kgb” for Sp(4,R) (the split form of simply connected C2):

 0:     1   2     6   4    0
 1:     0   3     6   5    0
 2:     2   0     *   4    0
 3:     3   1     *   5    0
 4:     8   4     *   *    1  2
 5:     9   5     *   *    1  2
 6:     6   7     *   *    1  1
 7:     7   6    10   *    2  212
 8:     4   9     *  10    2  121
 9:     5   8     *  10    2  121
10:    10  10     *   *    3  1212

The first column is just the number of the orbit in the enumeration we are using. The next two are the cross-actions for the two simple reflections in the complex Weyl group; they are always defined. Then come two columns describing the Cayley transforms for the simple reflections: they are defined only if the simple root is imaginary noncompact for the corresponding involution. The next-to-last column is the value of the length function; in terms of orbits this is the difference in dimension between the current orbit and the closed orbits; combinatorially, it is the smallest number of operations (cross-actions or Cayley transforms) that will get you from a closed orbit to the current one. Finally, the last column lists the Weyl group element defining the root datum involution associated to the orbit; in our language, this is just the restriction of the involution to the torus. The Weyl group element is written as a product of simple reflections: for instance “121” means s1s2s1. The output is sorted by length, and then by Weyl group element; the orbits associated to the identity element in the Weyl group are the closed orbits—they are all isomorphic to the flag variety of K.

In the example, we are in the equal rank case, so root datum involutions are just involutions in the Weyl group. There are four W-conjugacy classes of those, viz. {e}, {1,212}, {2,121}, {1212}, corresponding to the four conjugacy classes of Cartan subgroups in Sp(4,R). They correspond to the four W-orbits for the cross-action: {0,1,2,3}, {4,5,8,9}, {6,7}, {10}. In this case, there is only one split strong real form, and it is a “packet” all by itself. Hence the number of orbits lying over any given root datum involution is the cardinality of the component group of the torus dual to the corresponding Cartan subgroup. So there are four for the fundamental Cartan (the dual torus is split), two for the intermediate Cartan that is U(1).R× (corresponding to reflections through long roots), one for the intermediate Cartan that is C× (corresponding to reflections through short roots), and one for the split Cartan (the dual torus is compact.)

[1] R.W Richardson and T.A. Springer, The Bruhat Order on Symmetric Varieties, Geom. Dedicata 35(1990), pp. 389-436.

Back to the introduction.
Back to the Atlas homepage.