Bertram Kostant

• Research affiliate MIT. Ann Kostant (wife) at Massachusetts Institute of Technology

This paper connects results on Amitsur–Levitski identities for simple Lie algebras, ideals in Borel subalgebras, commutative Lie subalgebras in simple Lie algebras, filtration of sheets, and recent work with Nolan Wallach on the variety of singular elements in a complex semisimple Lie algebra.

I connect an old result of mine on a Lie algebra generalization of the Amitsur-Levitski theorem with equations for sheets in a reductive Lie algebra and with recent results of Kostant-Wallach on the variety of singular elements in a reductive Lie algebra.

Let $G$ be a complex simply-connected semisimple Lie group and let $\frak{g}= Lie G$. Let $\frak{g} = \frak{n}_- +\frak{h} + \frak{n}$ be a triangular decomposition of $\frak{g}$. One readily has that $Cent\,U({\frak n})$ is isomorphic to the ring $S({\frak n})^{{\frak\n}}$ of symmetric invariants. Using the cascade ${\cal B}$ of strongly orthogona...

Let $G$ be a complex simply-connected semisimple Lie group and let $\g=\hbox{\rm Lie}\,G$. Let $\g = \n_- +\hh + \n$ be a triangular decomposition of $\g$. One readily has that $\hbox{\rm Cent}\,U(\n)$ is isomorphic to the ring $S(\n)^{\n}$ of symmetric invariants. Using the cascade ${\cal B}$ of strongly orthogonal roots, some time ago we proved (...

The representation of the conformal group (PSU(2,2)) on the space of solutions to Maxwell's equations on the conformal compactification of Minkowski space is shown to break up into four irreducible unitarizable smooth Fr\'echet representations of moderate growth. An explicit inner product is defined on each representation. The frequency spectrum of...

Let $G$ be a semisimple Lie group and let $\g =\n_- +\hh +\n$ be a triangular decomposition of $\g= \hbox{Lie}\,G$. Let $\b =\hh +\n$ and let $H,N,B$ be Lie subgroups of $G$ corresponding respectively to $\hh,\n$ and $\b$. We may identify $\n_-$ with the dual space to $\n$. The coadjoint action of $N$ on $\n_-$ extends to an action of $B$ on $\n_-$...

Let $G$ be a complex simply-connected semisimple Lie group and let $\g= \hbox{\rm Lie}\,G$. Let $\g = \n_- +\hh + \n$ be a triangular decomposition of $\g$. The authors in [LW] introduce a very nice representation theory idea for the construction of certain elements in $\hbox{\rm cent}\,U(n)$. A key lemma in [LW] is incorrect but the idea is in fac...

A proof (by Serre and by Cohen, Griess and Lisser) verified, in the special case of E 8, a conjecture of mine, that the finite projective group L 2(61) embeds in \( {E_8}\left( \mathbb{C} \right) \). Subsequently, Griess and Ryba have shown (using computers) that L 2(49) and, in addition, (established by Serre without computers) L 2(41) also embed...

Let $G$ be a complex simple Lie group and let $\g = \hbox{\rm Lie}\,G$. Let $S(\g)$ be the $G$-module of polynomial functions on $\g$ and let $\hbox{\rm Sing}\,\g$ be the closed algebraic cone of singular elements in $\g$. Let ${\cal L}\s S(\g)$ be the (graded) ideal defining $\hbox{\rm Sing}\,\g$ and let $2r$ be the dimension of a $G$-orbit of a r...

A recent experimental discovery involving the spin structure of electrons in a cold one-dimensional magnet points to a validation of a (1989) Zamolodchikov model involving the exceptional Lie group E 8. The model predicts 8 particles and predicts the ratio of their masses. The conjectures have now been validated experimentally, at least for the fir...

Let M be a Riemannian manifold with the corresponding affine connection, X an infinitesimal motion on M, and V 0 the tangent space at a point 0 ∈ M. Let S 0 (the holonomy algebra) be the Lie algebra of the restricted holonomy group at 0 ∈ M.

We retain the notation of our previous article. Numbered theorems quoted here are also to be found in that article.

Let G be a group of linear transformations on a finite dimensional real or complex vector space X. Assume X is completely reducible as a G-module. Let S be the ring of all complex-valued polynomials on X, regarded as a G-module in the obvious way, and let J ⊆ S be the subring of all G-invariant polynomials on X.

We retain the notation of the preceding paper.1 We will say that g is effective relative to t if t contains no ideal of g.

We introduce a general approach to unitary representations for all Lie groups. An underlying feature is a study of sympletic manifolds X 2n (i. e. there exists a closed non-singular 2-form on X). If [ω] ∈ H 2(X, R) is an integral class there is an associated affinely connected Hermitian line bundle L over X which is unique if X is simply connected.

This paper is referred to as Part II. Part I is [4], The numerical I used as a reference will refer to that paper. A third and final part, Clifford algebras and the intersection of Schubert cycles is also planned.

The present paper will be referred to as Part I. A subsequent paper entitled, “Lie algebra cohomology and generalized Schubert cells,” will be referred to as Part II.

In general a homogeneous space admits many invariant affine connections. Among these are certain connections which appear in many ways to be more natural than the others. We refer to the connections which K. Nomizu in [4] calls canonical affine connections of the first kind. When G is a compact connected Lie group and K a closed subgroup we called...

In [1] Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem—Theorem 1 (the proof of Theorem 1 became suggestive to us after we noted that the T x of [1] is just the a x of...

The symmetric algebra S(g) over a Lie algebra g has the structure of a Poisson algebra. Assume g is complex semisimple. Then results of Fomenko-Mischenko (translation of invariants) and Tarasov construct a polynomial subalgebra H = C[q(1), ... , q(b)] of S(g) which is maximally Poisson commutative. Here b is the dimension of a Borel subalgebra of g...

Let $\frak{m}$ be a Levi factor of a proper parabolic subalgebra $\frak{q}$ of a complex semisimple Lie algebra $\frak{g}$. Let $\frak{t} = cent \frak{m}$. A nonzero element $\nu \in \frak{t}^*$ is called a $\frak {t}$-root if the corresponding adjoint weight space $\frak{g}_{nu}$ is not zero. If $\nu$ is a $\frak{t}$-root, some time ago we proved...

Let g=k+p be a complexified Cartan decomposition of a complex semisimple Lie algebra g and let K be the subgroup of the adjoint group of g corresponding to k. If H is an irreducible Harish-Chandra module of U(g), then H is completely determined by the finite-dimensional action of the centralizer U(g)K on any one fixed primary k component in H. This...

Let $\frak{g} = \frak{k} +\frak{p}$ be a complexified Cartan decomposition of a complex semisimple Lie algebra $\frak{g}$ and let $K$ be the subgroup of the adjoint group of $\frak{g}$ corresponding to $\frak{k} $. If $H$ is an irreducible Harish-Chandra module of $U(\frak{g})$, then $H$ is completely determined by the finite-dimensional action of...

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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 corresp...

Let $\Gamma$ be a finite subgroup of SU(2) and let $\widetilde {\Gamma} = \{\gamma_i\mid i\in J\}$ be the unitary dual of $\Gamma$. The unitary dual of SU(2) may be written $\{\pi_n\mid n\in \Bbb Z_+\}$ where $dim \pi_n = n+1$. For $n\in \Bbb Z_+$ and $j\in J$ let $m_{n,j}$ be the multiplicity of $\gamma_j$ in $\pi_n|\Gamma$. Then we collect this b...

A commutative Poisson subalgebra of the Poisson algebra of polynomials on the Lie algebra of n x n matrices over ${\Bbb C}$ is introduced which is the Poisson analogue of the Gelfand-Zeitlin subalgebra of the universal enveloping algebra. As a commutative algebra it is a polynomial ring in $n(n+1)/2$ generators, $n$ of which can be taken to be basi...

Let (X, ω) be an integral symplectic manifold and let (L, ∇) be a quantum line bundle, with connection, over X having ω as curvature. With this data one can define an induced symplectic manifold \( (\tilde X,\omega _{\tilde X} ) \), where dim \( \tilde X = 2 + \dim X \). It is then shown that prequantization on X becomes classical Poisson bracket o...

If \(\mathfrak{g}\) is a complex simple Lie algebra, and k does not exceed the dual Coxeter number of \(\mathfrak{g}\), then the absolute value of the kth coefficient of the \(\dim\mathfrak{g}\) power of the Euler product may be given by the dimension of a subspace of \(\wedge^k\mathfrak{g}\) defined by all abelian subalgebras of \(\mathfrak{g}\) o...

Let \( \mathfrak{g} \) be a complex semisimple Lie algebra and let \( \mathfrak{r} \subset \mathfrak{g} \) be any reductive Lie subalgebra such that B∣ \( \mathfrak{r} \) is nonsingular where B is the Killing form of \( \mathfrak{g} \) . Let Z(r) and Z(g) be, respectively, the centers of the enveloping algebras of \( \mathfrak{r} \) and \( \mathfra...

Let g be a complex semisimple Lie algebra and let t be the subalgebra of fixed elements in g under the action of an involutory automorphism of g. Any such involution is the complexification of the Cartan involution of a real form of g. If V λ is an irreducible finite-dimensional representation of g, the Iwasawa decomposition implies that V λ is a c...

Each infinitesimally faithful representation of a reductive complex connected algebraic group Ginduces a dominant morphism Φ from the group to its Lie algebra g by orthogonal projection in the endomorphism ring of the representation space. The map Φ identifies the field Q(G)of rational functions on Gwith an algebraic extension of the field Q(g)of r...

A Lie superalgebrea of Riemannian type leads to a representation of a quadratic Lie algebra into a Weyl algebra. A necessary and sufficient condition that such a representation leads to a Lie superalgebra of Riemannian type is that the Casimir maps to a constant.

A Lie superalgebrea of Riemannian type leads to a representation of a quadratic Lie algebra into a Weyl algebra. A necessary and sufficient condition that such a representation leads to a Lie superalgebra of Riemannian type is that the Casimir maps to a constant.

Analogous to the holomorphic discrete series of Sl(2, ℝ) there is a continuous family (πr), − 1 r ∞, of irreducible unitary representations of G, the simply-connected covering group of Sl(2, ℝ). A construction of this series is given in this paper using classical function theory. For all r the Hilbert space is L2((0, ∞003B)). First of all one exhib...

The Bott–Borel–Weil theorem (BBW) is a statement about a complex homogeneous space X=G/R where G is a compact semisimple Lie group and R is the centralizer of a torus in G. One knows that BBW is equivalent to the determination of how R operates on the cohomology of a certain nilpotent Lie algebra of antiholomorphic tangent vectors operating on an a...

while taking an evening walk. He was seventy-nine and was vigorously engaged in research. Born on September 13, 1918, in the Bronx, he grew up in Trenton and received his A.B. from Princeton in 1937. What must it have been like to be a member of the Jewish quota at Princeton in the 1930s? He told me once that a fellow undergraduate offered him mone...

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 obta...

The truncated icosahedron has emerged in biology and chemistry as a remarkably stable structure. A new group theoretic approach is discussed here as a possible key to this stability.

A central problem in the theory of unitary representations of a real semisimple Lie group G R is the construction of explicit geometric models. In such models GR acts on a Hilbert space ℌ of (L 2 or holomorphic) sections of (usually) a line bundle and the unitary structure on ℌ pairs the sections in some geometric way (such as integration). Such mo...

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,...

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 resul...

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.

Let $M$ be a $G$-covering of a nilpotent orbit in $\g$ where $G$ is a complex semisimple Lie group and $\g=\text{Lie}(G)$. We prove that under Poisson bracket the space $R[2]$ of homogeneous functions on $M$ of degree 2 is the unique maximal semisimple Lie subalgebra of $R=R(M)$ containing $\g$. The action of $\g'\simeq R[2]$ exponentiates to an ac...

We explicitly construct, in a uniform fashion, the (unique) minimal and spherical representation pi_0 of the split real Lie group of exceptional type E_6, E_7, or E_8. We obtain several algebraic and analytic results about pi_0.

The purpose of this article is to collect a number of remarkable group theoretical facts having to do with icosahedral symmetry. Some of these have been already applied to the discovery and identification of the new C 60 carbon molecule, called the Buckminsterfullerene, or buckyball, for short, and we hope that other results described here will fin...

The suggestion that there is an operator to be obtained (and consequently a spectrum to be obtained) from a function on a symplectic manifold originated from the work of physicists, on atomic radiation spectra, in the early part of this century. It is without doubt a profound and extremely important mathematical question to determine under what cir...

The purpose of this paper is to establish a connection between semisimple Jordan algebras and certain invariant differential operators on symmetric spaces and to prove an identity for such operators which generalizes the classical Capelli identity.

Let $M$ be a $G$-covering of a nilpotent orbit in $\g$ where $G$ is a complex semisimple Lie group and $\g=\text{Lie}(G)$. We prove that under Poisson bracket the space $R[2]$ of homogeneous functions on $M$ of degree 2 is the unique maximal semisimple Lie subalgebra of $R=R(M)$ containing $\g$. The action of $\g'\simeq R[2]$ exponentiates to an ac...

Let G be a real semisimple Lie Group, which we may assume to be connected or even simply connected, and let g0 = Lie(G). The dual vector space of g0 will be denoted by g0*. Also, g and g* will denote the complexifications of g0 and g0*, respectively. We recall that the group G and the Lie algebra g operate on g by the adjoint representation and on...

In our opinion one of the most interesting aspects of the orbit method is the occurrence of “independence of polarization”. From the perspective of geometric quantization, given two transverse polarizations, one may set up a formal kernel operator which then intertwines the two corresponding quantizations. This has been referred to as the BKS kerne...

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.

The group SO(4,2) is a familiar symmetry group in physics since among other things it is the conformal group of compactified Minkowski space S 3 x S 1. The double covering SU(2,2) of its identity component SO(4,2)e appears as a subgroup of the symmetry group Sp(8, R). In fact, if one considers the real and imaginary part of the Hermitian form in C...

Let H denote the single sheeted hyperboloid with the unique (up to multiplicative constant) Lorentz metric, g, invariant under SL(2, ℝ), so g has constant non-zero curvature. We study the group Conf(H) of conformal diffeomorphisms of g, in particular the boundary behavior of the conformal factor fτ for τ ∈ Conf(H) and τ*g = fτg. We characterize tho...

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 "ca...

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 repres...

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...

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 L...

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Let H be a complex Hilbert space and P(H) the corresponding projective space, i.e., the space of all one dimensional subspaces of H. If v is a non-zero vector in H we shall denote the corresponding point in P(H), i.e., the line throughv by [v]. Let G be a compact Lie group which is unitarily represented on H so that we may consider the correspondin...

Lie groups and their representations occupy an important place in mathematics with applications in such diverse fields as differential geometry, number theory, differential equations and physics. In 1977 a symposium was held in Oxford to introduce this rapidly developing and expanding subject to non-specialists. This volume contains the lectures of...

There are two main results in the paper. The first gives the infinitesimal character that can occur in the tensor product V ⊗ Vλ of an irreducible finite dimensional representation Vλ and an irreducible infinite dimensional representation V of a semisimple Lie algebra . The statement is that the infinitesimal characters are xv + μi, i = 1, 2,…, k,...

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Let C be a commutative ring with 1. Let A be a coalgebra over C with diagonal map d : A ? A ? C A (it is assumed A has a counit ? : A ? C) and let R be an algebra over C with multiplication m: R ? C R ? R (it is assumed R has a unit p : C ? R).

In this paper, Part II, of a two-part paper we apply the results of [KW], Part I, to establish, with an explicit dual coordinate system, a commutative analogue of the Gelfand-Kirillov theorem for M(n), the algebra of n × n complex matrices. The function field F(n) of M(n) has a natural Poisson structure and an exact analogue would be to show that F...

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Throughout this note V will denote a real n-dimensional vector space with a positive-definite inner product, its value for u, v ∈ V being denoted by (u, v). Let Λ p V be the space of contravariant skew-tensors over V of degree p. The inner product in V induces a positive-definite inner product in Λ P V, where (u 1 Λ … Λ u p , v 1 Λ … Λ v p )...