Blocks

Last updated: October 16, 2005

Let G be a connected reductive complex group, with a fixed involution θ; let K be the group of θ-fixed points in G, and g=Lie(G).

Fix a regular integral infinitesimal character χ for G. The category of (finite-length) (g,K)-modules with generalized infinitesimal character χ naturally splits up as a finite direct sum of "indecomposables"; the summands are called the blocks of the category. A definition is as follows: consider the set of (classes of) irreducible (g,K)-modules with infinitesimal character χ, and put an edge between two irreducibles when there is a non-trivial Ext1 between them. Then the blocks correspond to the connected components of this graph; a module belongs to a block, if all its irreducible subquotients are in the corresponding connected component, and each (g,K)-module with generalized infinitesimal character χ splits up canonically as a direct sum of submodules lying in the various blocks.

As explained here, there is a nice description of the blocks in terms of the parameters we are using to describe representations. Each block is again partitioned over the set of conjugacy classes of Cartan subgroups of (G,θ); and for each Cartan, the resulting piece is an orbit of the natural action of W on the parameter space. This makes it rather easy to count the number of elements in the blocks, without having to fully construct the parameter space. The “blocksizes”command will print out the blocksizes as a matrix, where the rows are indexed by real forms of G, and the columns by real forms of G.

Example: SL(2). This is much too simple, but it will at least give an idea. For SL(2), up to equivalence there are three strong real forms: two compact ones corresponding to SU(2), and the split one SL(2,R). The dual group is PSL(2), and has two strong real forms, PSU(2) and PSL(2,R). For the compact real form, picking any of the two corresponding strong forms, there is a single parameter, corresponding to the split real form of PSL(2). For the split real form, there are four parameters, two for the compact Cartan, and two for the split Cartan. The two for the compact Cartan correspond to the split real form of PSL(2), because that is the only real form of the dual group that “lives” over that Cartan. The two parameters for the split Cartan correspond one to the split dual real form, and one to the compact one. So for SU(2) there is of course a single block of representations, consisting of a single element, whereas for SL(2,R) there are two: if we look, say, at the infinitesimal character of the trivial representation then the one-element block is the non-spherical principal series, the other is made up of the trivial representation (lying over the split torus), and the two discrete series (lying over the compact torus.) This would be the “(split,split)” block; the two one-element blocks are the “(split,compact)” for SL(2,R) and “(compact,split)” for SU(2), while the “(compact,compact)” block is empty. Notice that if we looked at an infinitesimal character where the finite-dimensional representation has even dimension, then it would be the spherical principal series that is in a block by itself, the equivalence of categories does not preserve the action of the center of the group.

Example: PSL(2). If we look now at PSL(2), there are two translation classes of infinitesimal characters to consider. One is dual to the picture obtained for the two compact strong real forms of SL(2), and will yield two one-element blocks (two irreducible principal series of PSL(2,R)); the other infinitesimal character will correspond to the picture dual to the one for SL(2,R): there will be two blocks, one one-element block for the compact real form PSU(2), and a three-element block containing the one discrete series representation and two finite-dimensional ones for PSL(2,R)).


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