Center of U(n), Cascade of Orthogonal Roots, and a Construction of Lipsman-Wolf

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 (see [K]) that S(\n)^{\n} is a polynomial ring $\Bbb C[\xi_1,...,\xi_m]$ where $m$ is the cardinality of ${\cal B}$. The authors in [LW] introduce a very nice representation-theoretic method for the construction of certain elements in S(\n)^{\n}. A key lemma in [LW] is incorrect but the idea is in fact valid. In our paper here we modify the construction so as to yield these elements in S(\n)^{\n} and use the [LW] result to prove a theorem of Tony Joseph.Comment: 10 pages, plain.tex; key words: cascade of orthogonal roots, Borel subgroups, nilpotent coadjoint action, dedicated to Joe Wolf; msc keywords: Representation theory, invariant theor

Experimental evidence for the occurrence of E8 in nature and the radii of the Gosset circles

A recent experimental discovery involving the spin structure of electrons in a cold one-dimensional magnet points to a validation of a Zamolodchikov model involving the exceptional Lie group $E_8$. The model predicts 8 particles and predicts the ratio of their masses. I.e., the vertices of the 8-dimensional Gosset polytope identifies with the 240 roots of $E_8$. Under the 2-D (Peter McMullen) projection of the polytope, the image of the vertices are arranged in 8 concentric circles, here referred to as the Gosset circles. The Gosset circles are understood to correspond to the 8 masses in the model, and it is understood that the ratio of their radii is the same as the ratio of the corres-ponding conjectural masses. A ratio of the two smallest circles (read 2 smallest masses) is the golden number. The conjectures have been now validated experimentally, at least for the first five masses. The McMullen projection generalizes to any complex simple Lie algebra whose rank is greater than 1. The Gosset circles also generalize, using orbits of the Coxeter element. Using results in a 1959 paper of mine, I found some time ago a very easily defined operator $A$ whose spectrum is exactly the squares of the radii $r_i$ of these generalized Gosset circles. As a confirmation, in the $E_8$ case, using only the eigenvalues of a suitable multiple of $A$, Vogan computed the ratio of the $r_i$. Happily these agree with the corresponding ratio of the Zamolodchikov masses. The operator $A$ is written as a sum of $\ell +1$ rank 1 operators, parameterized by the points in the extended Dynkin diagram. Involved in this expansion are the coefficients $n_i$ of the highest root. Recalling the McKay correspondence, in the $E_8$ case, the $n_i$, together with 1, are the dimensions of the irreducible representations of the binary icosahedral group.Comment: 19 pAGES, PLAIN TE

A branching law for subgroups fixed by an involution and a noncompact analogue of the Borel-Weil theorem

We give a branching law for subgroups fixed by an involution. As an application we give a generalization of the Cartan-Helgason theorem and a noncompact analogue of the Borel-Weil theorem.Comment: 60 pages, plain te

Minimal coadjoint orbits and symplectic induction

Let $(X,\omega)$ be an integral symplectic manifold and let $(L,\nabla)$ be a quantum line bundle, with connection, over $X$ having $\omega$ as curvature. With this data one can define an induced symplectic manifold $(\widetilde {X},\omega_{\widetilde {X}})$ where $dim \widetilde {X} = 2 + dim X$. It is then shown that prequantization on $X$ becomes classical Poisson bracket on $\widetilde {X}$. We consider the possibility that if $X$ is the coadjoint orbit of a Lie group $K$ then $\widetilde {X}$ is the coadjoint orbit of some larger Lie group $G$. We show that this is the case if $G$ is a non-compact simple Lie group with a finite center and $K$ is the maximal compact subgroup of $G$. The coadjoint orbit $X$ arises (Borel-Weil) from the action of $K$ on \p where \g= \k +\p is a Cartan decomposition. Using the Kostant-Sekiguchi correspondence and a diffeomorphism result of M. Vergne we establish a symplectic isomorphism $(\widetilde {X},\omega_{\widetilde {X}})\cong (Z,\omega_Z)$ where $Z$ is a non-zero minimal "nilpotent" coadjoint orbit of $G$. This is applied to show that the split forms of the 5 exceptional Lie groups arise symplectically from the symplectic induction of coadjoint orbits of certain classical groups.Comment: 38 pages, plain te

The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group

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_-. There exists a unique nonempty Zariski open orbit $X$ of $B$ on \n_-. Any $N$-orbit in $X$ is a maximal coadjoint orbit of $N$ in \n_-. The cascade of orthogonal roots defines a cross-section \r_-^{\times} of the set of such orbits leading to a decomposition X = N/R\times \r_-^{\times}. This decomposition, among other things, establishes the structure of S(\n)^{\n} as a polynomial ring generated by the prime polynomials of $H$-weight vectors in S(\n)^{\n}. It also leads tothe multiplicity 1 of $H$ weights in S(\n)^{\n}.Comment: 24 pages, plain te

Cent U(n) and a construction of Lipsman-Wolf

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 fact valid. In our paper here we modify the construction so as to yield the desired elements in \hbox{\rm cent}\,U(\n).Comment: 14 pages in plain te

The Coxeter element and the branching law for the finite subgroups of SU(2)

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 branching data in the formal power series, $m(t)_j = \sum_{n=0}^{\infty}m_{n,j} t^n$. One shows that there exists a polynomial $z(t)_j$ and known positive integers $a,b$ (independent of $j$) such that $m(t)_j = {z(t)_j \over (1-t^a)(1-t^b)}$. The problem is the determination of the polynomial $z(t)_j$. If $o\in J$ is such that $\gamma_o$ is the trivial representation, then it is classical that $z(t)_o = 1 +t^h$ for a known integer $h$. The problem reduces to case where $\gamma_j$ is nontrivial. The McKay correspondence associates to $\Gamma$ a complex simple Lie algebra \g of type A-D-E. We explicitly determine $z(t)_j$ for $j\in J-\{o\}$ using the orbits of a Coxeter element on the set of roots of $\frak{g}$. Mysteriously the polynomial $z(t)_j$ has arisen in a completely different context in some papers of Lusztig. Also Rossmann has recently shown that the polynomial $z(t)_j$ yields the character of $\gamma_j$.Comment: 13 pages, plain.te

Generalized Amitsur-Levitski theorem and equations for sheets in a reductive complex 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.Comment: 17 pages, key words: Amitsur-Levitski theorem, invariant theory, polynomial identities, sheets, abelian ideals and solvable Lie algebras, representation theory, enveloping algebras, nilpotent cone, Lie algebra cohomolog

Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra

If $\frak g$ is a complex simple Lie algebra, and $k$ does not exceed the dual Coxeter number of $\frak g$, then the k$^{th}$ coefficient of the $dim \frak g$ power of the Euler product may be given by the dimension of a subspace of $\wedge^k\frak g$ defined by all abelian subalgebras of $\frak g$ of dimension $k$. This has implications for all the coefficients of all the powers of the Euler product. Involved in the main results are Dale Peterson's $2^{rank}$ theorem on the number of abelian ideals in a Borel subalgebra of $\frak g$, an element of type $\rho$ and my heat kernel formulation of Macdonald's $\eta$-function theorem, a set $D_{alcove}$ of special highest weights parameterized by all the alcoves in a Weyl chamber (generalizing Young diagrams of null $m$-core when $\frak g= Lie Sl(m,\Bbb C)$), and the homology and cohomology of the nil radical of the standard maximal parabolic subalgebra of the affine Kac-Moody Lie algebra.Comment: 44 pages, plain te

Dirac Cohomology for the Cubic Dirac Operator

Using the cubic Dirac operator and an extended notion of Dirac cohomology we generalize results of Huang-Pandzic, which appeared in JAMS (Sept. 6, 2001 electronic) on a conjecture of D. VoganComment: 35 pages, plain te