Bertram Kostant

The nil Hecke ring and cohomology of GP 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.

Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras

Abstract This paper gives the mathematical foundations for the BRS quantization procedure. We first discuss the classical finite dimensional BRS procedure and relate it to Marsden-Weinstein reduction. This leads to interesting relations between Lie algebras and Clifford algebras and a novel way of computing Lie algebra cohomology in terms of the spin representation. We then discuss infinite-dimensional Clifford algebras and their spin representations. We find that in the infinite-dimensional case, the analog of the finite-dimensional construction of Lie algebra cohomology breaks down, the obstruction (anomaly) being the Kac-Peterson class which is the cohomology class associated to the representation of the Lie algebra on spinors which is now only a projective representation. Tensoring by a projective representation of opposite class kills the obstruction and gives rise to a cohomology theory and a quantization procedure. We discuss the gradings and Hermitian structures on the absolute and relative complexes.

A cubic Dirac operator and the emergence of Euler number multiplets of representations for equal rank subgroups

0. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447 1. A Clifford algebra criterion for(ν,Bg) to be of Lie type. . . . . . . . . . . . . . . . . . 455 2. The cubic Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 3. Tensoring with the spin representation and the emergence of d-multiplets . . . 475 4. Multiplets and the kernel of the Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . 483 5. Infinitesimal character values on multiplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 6. Multiplets and topological K-theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

A formula for the multiplicity of a weight

1. Introduction. 1. Let g be a complex semi-simple Lie algebra and f) a Cartan subalgebra of g. Let ir\ be an irreducible representation of g, with highest weight X, on a finite dimensional vector space V\. A well known theorem of E. Cartan asserts that the highest weight, X, of w\ occurs with multi-plicity one. It has been a question of long standing to determine, more generally , the multiplicity of an arbitrary weight of irx. Weyls formula (1.12) for the character of tt\ is an expression for the function xx(x) =tr exp ^x(x), x (Ef), on I) in terms of X and quantities independent of the representation. In the same spirit the author has always understood the multiplicity question to mean the following: Let I be the set of all integral linear forms on fi. Let m\ be the function in / which assigns to each integral linear form vE I the mul-tiplicity m\(v) of its occurrence as a weight of w\. Find a formula for the multiplicity function m\ in terms of X and quantities independent of the

On the tensor product of a finite and an infinite dimensional representation

Abstract There are two main results in the paper. The first gives the infinitesimal character that can occur in the tensor product V ⊗ V λ of an irreducible finite dimensional representation V λ and an irreducible infinite dimensional representation V of a semisimple Lie algebra g . The statement is that the infinitesimal characters are x v + μ i , i = 1, 2,…, k , where μ i are the weights of V λ and v is the “pseudo” highest weight of V . The second result proves that if V is a Harish-Chandra module (one which comes from a group representation), then V ⊗ V λ has a finite composition series. But then the irreducible components in the composition series have the infinitesimal characters given in the first results.

Gelfand-Zeitlin Theory from the Perspective of Classical Mechanics. II

Let M(n) be the algebra (both Lie and associative) of n × n matrices over ℂ. Then M(n) inherits a Poisson structure from its dual using the bilinear form (x, y) = −tr xy. The Gl(n) adjoint orbits are the symplectic leaves and the algebra, P(n), of polynomial functions on M(n) is a Poisson algebra. In particular, if f ∈ P(n), then there is a corresponding vector field ξ f on M(n). If m ≤ n, then M(m) embeds as a Lie subalgebra of M(n) (upper left hand block) and P(m) embeds as a Poisson subalgebra of P(n). Then, as an analogue of the Gelfand-Zeitlin algebra in the enveloping algebra of M(n), let J(n) be the subalgebra of P(n) generated by P(m)Gl(m) for m = 1, . . ., n. One observes that

Dirac Cohomology for the Cubic Dirac Operator

J\left( n \right) \cong P\left( 1 \right)^{Gl\left( 1 \right)} \otimes \cdots \otimes P\left( n \right)^{Gl\left( n \right)}

The Capelli identity, tube domains, and the generalized Laplace transform☆

. We prove that J(n) is a maximal Poisson commutative subalgebra of P(n) and that for any p ∈ J(n) the holomorphic vector field ξ p is integrable and generates a global one-parameter group σ p(z) of holomorphic transformations of M(n). If d(n) = n(n + 1)/2, then J(n) is a polynomial ring ℂ[p 1, . . ., p d(n)] and the vector fields \( \xi _{p_i } \) , i = 1, . . ., d(n − 1), span a commutative Lie algebra of dimension d(n − 1). Let A be a corresponding simply-connected Lie group so that A ≅ ℂd(n−1). Then A operates on M(n) by an action σ so that if a ∈ A, then