The Weyl character formula, the half-spin representations, and equal rank subgroups.

Let B be a reductive Lie subalgebra of a semi-simple Lie algebra F of the same rank both over the complex numbers. To each finite dimensional irreducible representation Vlambda of F we assign a multiplet of irreducible representations of B with m elements in each multiplet, where m is the index of the Weyl group of B in the Weyl group of F. We obtain a generalization of the Weyl character formula; our formula gives the character of Vlambda as a quotient whose numerator is an alternating sum of the characters in the multiplet associated to Vlambda and whose denominator is an alternating sum of the characters of the multiplet associated to the trivial representation of F.

Structure of the truncated icosahedron (such as fullerene or viral coatings) and a 60-element conjugacy class in PSl(2, 11).

The proper symmetry group of a truncated icosahedron P is the icosahedral group PSl(2, 5). However, knowing the symmetry group is not enough to specify the graph structure (e.g., the carbon bonds for fullerene, C60) of P. The group PSl(2, 5) is a subgroup of the 660-element group PSl(2, 11). The latter contains a 60-element conjugacy class, say M, of elements of order 11. I show here that M exhibits a model for P where the graph structure is expressed group-theoretically. For example, the 12 pentagons are the maximal commuting subsets of M. Such a model creates the opportunity of using group-based harmonic analysis (e.g., convolution calculus) to deal with problems concerning the truncated icosahedron.

Minimal representations, geometric quantization, and unitarity.

In the framework of geometric quantization we explicitly construct, in a uniform fashion, a unitary minimal representation pio of every simply-connected real Lie group Go such that the maximal compact subgroup of Go has finite center and Go admits some minimal representation. We obtain algebraic and analytic results about pio. We give several results on the algebraic and symplectic geometry of the minimal nilpotent orbits and then "quantize" these results to obtain the corresponding representations. We assume (Lie Go)C is simple.

Minimal representations of E6, E7, and E8 and the generalized Capelli identity.

We explicitly construct, in a uniform fashion, the (unique) minimal and spherical representation pi0 of the split real Lie group of exceptional type E6, E7, or E8. We obtain several algebraic and analytic results about pi0.

Douglas' solution of the Plateau problem.

Using ideas suggested by some recent developments in string theory, we give here an elementary demonstration of one of the key steps in Douglas' celebrated proof of the existence of solutions of the Plateau problem in n dimensions.

T-equivariant K-theory of generalized flag varieties.

Let G be a Kac-Moody group with Borel subgroup B and compact maximal torus T. Analogous to Kostant and Kumar [Kostant, B. & Kumar, S. (1986) Proc. Natl. Acad. Sci. USA 83, 1543-1545], we define a certain ring Y, purely in terms of the Weyl group W (associated to G) and its action on T. By dualizing Y we get another ring Psi, which, we prove, is "canonically" isomorphic with the T-equivariant K-theory K(T)(G/B) of G/B. Now K(T)(G/B), apart from being an algebra over K(T)(pt.) approximately A(T), also has a Weyl group action and, moreover, K(T)(G/B) admits certain operators {D(w)}w[unk]W similar to the Demazure operators defined on A(T). We prove that these structures on K(T)(G/B) come naturally from the ring Y. By "evaluating" the A(T)-module Psi at 1, we recover K(G/B) together with the above-mentioned structures. We believe that many of the results of this paper are new in the finite case (i.e., G is a finite-dimensional semisimple group over C) as well.

The nil Hecke ring and cohomology of G/P for a Kac-Moody group G.

Let G be the group with Borel subgroup B, associated to a Kac-Moody Lie algebra [unk] (with Weyl group W and Cartan subalgebra [unk]). Then H(*)(G/B) has, among others, four distinguished structures (i) an algebra structure, (ii) a distinguished basis, given by the Schubert cells, (iii) a module for W, and (iv) a module for Hecke-type operators A(w), for w [unk] W. We construct a ring R, which we refer to as the nil Hecke ring, which is very simply and explicitly defined as a functor of W together with the W-module [unk] alone and such that all these four structures on H(*)(G/B) arise naturally from the ring R.

On finite subgroups of SU(2), simple Lie algebras, and the McKay correspondence.

The question is considered of how the restriction pi(n)Gamma, where pi(n) is irreducible for all n and Gamma is a finite subgroup of SU(2), decomposes into Gamma-irreducibles for any arbitrary n in [unk](+). It is announced that, using the McKay correspondence, the problem has an elegant solution in terms of the Coxeter element for the associated Lie algebra and that the numbers involved come in a beautiful way from the root structure.