Last updated: October 16, 2005
Let G be a connected reductive complex group, with a fixed inner class of real forms. We are interested in parametrizing isomorphism classes of irreducible (g,K)-modules, for the various real forms in the given inner class. Let G∨ be the dual group of G.
We refer to Jeff Adams' 2004 paper for background. Fix a regular integral infinitesimal character χ, and fix a “packet” of strong real forms of G, corresponding to an element z ∈ Z(G). Then the problem that's stable under duality and affords a nice description is the following:
Classify isomorphism classes of irreducible g,K)-modules with infinitesimal character χ for the various strong real forms in the packet corresponding to z.
The answer is expressed in terms of (G×G∨)-conjugacy classes of “sextuples” in Jeff's paper. However, after reduction, one may formulate it as follows. Notice that the categories of g,K)-modules with infinitesimal character χ are equivalent for values of χ that differ by translation by the character lattice of T (also the cocharacter lattice of T∨); and translation classes of cocharacters of T∨; are in (1,1)-correspondence with elements of Z(G∨). So the translation class of χ corresponds to a central element dz in G∨, and therefore also to a packet of strong real forms for G∨. Denote \cal X(z) the set of strong involutions in N.δ with square z, up to T-conjugacy, and define \cal X∨(dz) analogously (using the dual inner class). Then the set of representations we're after is in natural (1,1)-correspondence with the set of pairs (ξ,η) in \cal X(z) × \cal X∨(dz) such that the restrictions σ, τ of ξ and η to T an T∨ respectively satisfy τ=-tσ.
This description has a number of nice features. First of all, since the conjugacy classes of Cartans are described in terms of conjugacy classes of twisted involutions, we have the familiar partition of representations in terms of Cartans, with discrete series corresponding to compact tori (if any.) Second, there is an obvious (and beautiful!) duality by reversing the roles of G and G∨: the same picture also describes the irreducible representations of the packet of strong real forms of G∨ defined by dz, for the translation class of regular integral infinitesimal characters corresponding to z. Third, the partition afforded by the various strong real forms of G and G∨ corresponds to the blocks in the categories of representations with infinitesimal character χ, for the strong real forms of G in the chosen packet (or dually, to blocks in the corresponding categories for strong real forms of G∨.)
If we are interested in the representations of a fixed real form of G, we should of course consider only the pairs (ξ,η) where ξ belongs to the G-conjugacy class corresponding to a fixed strong real form mapping to our chosen real form (after having chosen z so that such a strong real form exists.)