Affine Weyl groups and the geometry of the unitary dual

David Vogan (MIT)

“Geometry” in this talk will mean points, lines, triangles, and so on. A really important example of what I’ll discuss is the $n$ -dimensional simplex $F_n \subset {\mathbb R}^n$ with vertices

$(0,0,\cdots,0,0),\; (0,0,\cdots,0,1),\; (0,\cdots,1/2,1/2),\ldots,\; (0,1/2,\cdots,1/2,1/2),\; (1/2,1/2,\cdots,1/2,1/2).$

You can make infinitely many simplices like $F_n$ by permuting the coordinates, changing signs of coordinates, and adding an integer vector. All the simplices you get in that way are distinct (that’s a lie for an interesting reason) and the simplices cover ${\mathbb R}^n,$ overlapping only on some lower-dimensional faces (that part is true).

I will explain how to generalize this example to the dual of a Cartan subalgebra in any compact simple group. (That’s math from the 1950s, which you could read about in many places.) I’ll formulate the unitary dual problem for real reductive
groups; and explain why its (still unknown) solution must be expressed in terms of compact polyhedra of the sort just discussed. (Now we’re up to the 1980s.)

Finally, I’ll explain how all this makes the unitary dual computable. (And the computers are chugging away as we speak.)