Let G be a noncompact connected real semisimple Lie group with finite center and let K be a maximal compact subgroup. In 1963, Harish-Chandra announced a complete classification of the square-integrable irreducible unitary
representations of G. Such representations are necessarily infinite dimensional. A necessary and sufficient condition for the existence of such “discrete series” representations is that rank K = rank G. (A good example to
think about is SU(p,q), for positive natural numbers p and q.)
A general theorem of Harish-Chandra implies that an irreducible unitary representation of G decomposes with finite multiplicities over K. A precise
branching law for the restriction of a discrete series representation from G to K was conjectured by Robert Blattner in 1966. After ten years, Wilfried Schmid and Henryk Hecht gave a complete proof of Blattner’s conjecture.
However, very few numerical results were available until recently. We report on joint work with Jeb Willenbring on some new and very simple combinatorial properties of Blattner’s branching formula.
A key idea is to extend Blattner’s formula to a larger class of objects, so-called virtual Harish-Chandra modules. The extension can be constructed via sheaf cohomology on the flag variety of the complexification of G. This
geometrical picture is a counterpart for a noncompact group of the celebrated Bott-Borel-Weil Theorem for finite dimensional irreducible representations of
compact groups. (The extension to virtual modules can also be constructed by the derived Zuckerman functor, but we will not deal with that purely algebraic construction in this seminar.)
After discussing the some background for this talk, we will present some transparencies illustrating Blattner’s formula for the example G = split G_2, K
= SO(4). We will discuss some new theorems about Blattner’s formula and state some open problems about virtual Harish-Chandra modules. (A good introduction to infinite dimensional representations of real semisimple
groups can be found at the website for the Atlas of Lie Groups, which is the brainchild of Yale PhD Jeffrey Adams.)