Weyl Character Formula - Harish-Chandra Character Formula

Harish-Chandra Character Formula

Harish-Chandra showed that Weyl's character formula admits a generalization to representations of a real, reductive group. Suppose is an irreducible, admissible representation of a real, reductive group G with infinitesimal character . Let be the Harish-Chandra character of ; it is given by integration against an analytic function on the regular set. If H is a Cartan subgroup of G and H' is the set of regular elements in H, then

Here

• W is the complex Weyl group of with respect to
• is the stabilizer of in W

and the rest of the notation is as above.

The coefficients are still not well understood. Results on these coefficients may be found in papers of Herb, Adams, Schmid, and Schmid-Vilonen among others.