Representation theory of real reductive groups

ATLAS software

• http://www.aimath.org/E8/technicaldetails.html

Here is a little more information about what this software does. First of all it allows you to enter the data to describe an arbitrary, connected, reducted complex group G(C). Then you can define a particular real form G. The software computes structural information about G, in particular Cartan subgroups and their ("real") Weyl groups. It then computes the irreducible representations of G with a given regular integral infinitesimal character.

Now suppose we have fixed G(C), G, and a regular integral infinitesimal character for G. There is a parameter set P, which is a finite set. For each element x of P there are two natural admissible representations of G, with the given infinitesimal character. First of all there is a "standard" module I(x). I(x) is the full induced representation of a discrete series representation on a parabolic subgroup. Secondly there is an irreducible representation π(x), which is a distinguished subrepresentation of I(x). The representation π(x) is typically smaller than I(x), interesting and difficult to understand. The unitary dual problem is to determine which irreducible representations π(x) are unitary.

Every standard representation I(x) can be decomposed into a "sum" of irreducible representations. It is not the case that I(x) is completely reducible; it has a Jordan-Holder series in which each irreducible subquotient is irreducible. In other words there is an equality I(y)=Σxm(x,y)π(x) (the sum runs over x in P) in the Grothendieck group. This equality can be inverted: π(y)=ΣxM(x,y)I(x).

It is important to compute the non-negative integers m(x,y) and the integers M(x,y). The latter is what the Kazhdan-Lusztig-Vogan polynomials do. (See below for more on the Kazhdan-Lusztig-Vogan polynomials, and their relation to Kazhdan-Lusztig polynomials.) The atlas software computes the KLV polynomials for any real group. For any group of rank 7 or less the computation is very fast. E6 takes less than one second and E7 about 3 minutes. Computing KLV polynomials for the split real form of E8 naively requires a computer with about 256 gigabytes of memory (all accessible from a single processor). The number of representation (the size of the parameter set P) is 453,060, so we're computing a matrix of size 453,060x453,060 (this matrix has ones on the diagonal and is upper triangular).

• Atlas of Lie Groups and Representations project

memo

• Vogan duality
• David A. Vogan, Jr. Irreducible characters of semisimple Lie groups IV. character-multiplicity duality
• David A. Vogan, Jr. Representations of Real Reductive Lie Groups

expositions

• Kobayashi, Toshiyuki. “A Program for Branching Problems in the Representation Theory of Real Reductive Groups.” arXiv:1509.08861 [math-Ph], September 29, 2015. http://arxiv.org/abs/1509.08861.

articles

• Jing-Song Huang, Binyong Sun, Kazhdan's orthogonality conjecture for real reductive groups, arXiv:1509.01755 [math.RT], September 06 2015, http://arxiv.org/abs/1509.01755
• Adams, Jeffrey, Marc van Leeuwen, Peter Trapa, and David A. Vogan Jr. “Unitary Representations of Real Reductive Groups.” arXiv:1212.2192 [math], December 10, 2012. http://arxiv.org/abs/1212.2192.