The Universal Cover of SL(2,R)
And Its Representations

The Solution

Representations of the universal cover of SL(2,R) can be realized as global sections of standard weakened Harish-Chandra sheaves on the flag variety of the group PSL(2,C).

For more details on representations of the universal cover of SL(2,R) you can download a short expository paper on the subject [1]. However, before doing so you might want to experiment with the Java applet below to pique your interest. This applet lets you explore the reducibility of standard modules and the geometry of representations. Below the applet graph are some remarks to help you understand what it means. Some abstract general remarks are necessary to frame the ideas, but after that some specific options in the applet are discussed.

You need a Java enabled browser to interact with this applet! You do not have such a browser, but this is what you would see for one of the 48 possible combinations:

General Remarks

Each red point with coordinates (t,e) on the graph above may have several standard modules attached to it. Nevertheless, each red point corresponds to at least one irreducible representation of the universal cover of SL(2,R). In general (t,e) can be a pair of complex numbers, but the graph above is only for the real parts of t and e.

Let G denote the universal covering group of SL(2,R). An irreducible (not necessarily unitary) representation of G has three parameters:

A complex number t corresponding to a strongly antidominant linear form on the abstract Cartan of PSL(2,C) coming from the infinitesimal character of the representation.
A complex number e coming from the action of the center Z of G on the representation. Note that since Z is infinite there is no restriction on e.
A K-orbit Q in the flag variety of PSL(2,C).

Here K is the group of diagonal matrices in PSL(2,C). There are three K-orbits in the flag variety: an open orbit and two closed orbits consisting of the point zero in the complex plane and the point at infinity respectively. The three parameters (Q,t,e) are called "standard data" for the irreducible representation of G. In fact, each distinct triple of standard data (Q,t,e) corresponds to a distinct irreducible representation of G.

An irreducible representation of G corresponds to the unique irreducible submodule of a standard module I(Q,t,e) for some standard data (Q,t,e), where I(Q,t,e) is a certain sheaf of D-modules on the flag variety of PSL(2,C). Here D is a naturally defined homogeneous sheaf of associative algebras which is locally isomorphic to the sheaf of local differential operators on the flag variety of PSL(2,C). By its construction the support of the sheaf I(Q,t,e) is the closure of the K-orbit Q.

For a given pair of complex numbers (t,e) one can ask whether there is a K-orbit Q which is "compatible" with the point (t,e) in the sense that the global sections of I(Q,t,e) give a representation of G. This is not always the case. A standard module sometimes does not exist when there is no K-orbit Q compatible with the point (t,e), and you will see when this happens if you experiment with the graph above.

If you experiment with the various options you may also discover when the standard module I(Q,t,e) is reducible or irreducible. A module is irreducible when it is nonzero and has no proper submodules; otherwise the module is trivial (0) or reducible.

Specific Graph Options

Below is a sequence of combinations which illustrates how the graphs can tell you something about the geometry of representations of G.

Both + Some Orbit + All Types: Every pair of complex numbers (t,e) corresponds to some irreducible representation of the universal cover G of SL(2,R). In other words, every irreducible representation of G corresponds to the unique irreducible submodule of a standard module I(Q,t,e) for some point (t,e) and some K-orbit Q.
Reducible + Some Orbit + All Types: Here you see that the lines with slopes +1 and -1 are the reducible standard modules. Are there standard modules support on the open orbit which are reducible? Try the next combination.
Reducible + Open Orbit + All Types: This combination shows that the answer is: Yes, there are reducible standard modules supported on the open orbit. However, not all standard modules are reducible.
Reducible + Zero Orbit + All Types: This is impossible because the zero orbit is closed. A standard module I(Q,t,e) is irreducible whenever the K-orbit Q is closed. Nevertheless, there are irreducible representations of G supported on closed orbits.
Irreducible + Zero Orbit + All Types: This combinations shows the irreducible representations supported on the zero orbit. All of these representations are tempered, but the set of global sections of the standard module I({0},0,1) is not a discrete series representation. You will discover this if you experiment with options for the representation type.

Further Remarks

It is easy to see the irreducible representations of SL(2,R) in the picture. In fact, a representation of its universal cover descends to a representation of SL(2,R) if, and only if, the parameter e is 0 or 1. Thus, the standard modules I(Q,t,0) and I(Q,t,1) in the picture correspond to the classical representations of SL(2,R).

References