Fomenko-Mischenko Theory, Hessenberg Varieties, and Polarizations

The symmetric algebra g (denoted S(\g)) over a Lie algebra \g (frak g) has the structure of a Poisson algebra. Assume \g is complex semi-simple. Then results of Fomenko- Mischenko (translation of invariants) and A.Tarasev construct a polynomial subalgebra \cal H = \bf 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 G be the adjoint group of \g and let \ell = rank \g. Identify \g with its dual so that any G-orbit O in \g has the structure (KKS) of a symplectic manifold and S(\g) can be identified with the affine algebra of \g. An element x \in \g is strongly regular if \{(dq_i)_x\}, i=1,...,b, are linearly independent. Then the set \g^{sreg} of all strongly regular elements is Zariski open and dense in \g, and also \g^{sreg \subset \g^{reg} where \g^{reg} is the set of all regular elements in \g. A Hessenberg variety is the b-dimensional affine plane in \g, obtained by translating a Borel subalgebra by a suitable principal nilpotent element. This variety was introduced in [K2]. Defining Hess to be a particular Hessenberg variety, Tarasev has shown that Hess \subset \g^sreg. Let R be the set of all regular G-orbits in \g. Thus if O \in R, then O is a symplectic manifold of dim 2n where n= b-\ell. For any O\in R let O^{sreg} = \g^{sreg}\cap O. We show that O^{sreg} is Zariski open and dense in O so that O^{sreg} is again a symplectic manifold of dim 2n. For any O \in R let Hess (O) = Hess \cap O. We prove that Hess(O) is a Lagrangian submanifold of O^{sreg} and Hess =\sqcup_{O \in R} Hess(O). The main result here shows that there exists, simultaneously over all O \in R, an explicit polarization (i.e., a "fibration" by Lagrangian submanifolds) of O^{sreg} which makes O^{sreg} simulate, in some sense, the cotangent bundle of Hess(O).Comment: 36 pages, plain te

A predictive phenomenological tool at small Bjorken-x

We present the results from global fits of inclusive DIS experimental data using the Balitsky-Kovchegov equation with running coupling.Comment: 5 pages, 2 figures, prepared for the Proceedings of 'Hot Quarks 2010

Quantum Mechanics as a Gauge Theory of Metaplectic Spinor Fields

A hidden gauge theory structure of quantum mechanics which is invisible in its conventional formulation is uncovered. Quantum mechanics is shown to be equivalent to a certain Yang-Mills theory with an infinite-dimensional gauge group and a nondynamical connection. It is defined over an arbitrary symplectic manifold which constitutes the phase-space of the system under consideration. The ''matter fields'' are local generalizations of states and observables; they assume values in a family of local Hilbert spaces (and their tensor products) which are attached to the points of phase-space. Under local frame rotations they transform in the spinor representation of the metaplectic group Mp(2N), the double covering of Sp(2N). The rules of canonical quantization are replaced by two independent postulates with a simple group theoretical and differential geometrical interpretation. A novel background-quantum split symmetry plays a central role.Comment: 61 pages, late

On Kostant's partial order on hyperbolic elements

We study Kostant's partial order on the elements of a semisimple Lie group in relations with the finite dimensional representations. In particular, we prove the converse statement of [3, Theorem 6.1] on hyperbolic elements.Comment: 7 page

Domains of holomorphy for irreducible unitary representations of simple Lie groups

We classify the domains of holomorphy of all Harish-Chandra modules of irreducible unitary representations of simple non-compact Lie groups.Comment: revised version, to appear in Invent. math., 14 page

Invariant four-forms and symmetric pairs

We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations whose second exterior power is irreducible or has an irreducible summand of co-dimension one, and we give a conceptual computation-free argument for the construction of the exceptional Lie algebras of compact type.Comment: 16 pages [v2: references added, last section expanded

On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces

Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, Suter.Comment: Latex file, 35 pages, minor corrections, some examples added. To appear in Selecta Mathematic

Shintani functions, real spherical manifolds, and symmetry breaking operators

For a pair of reductive groups $G \supset G'$, we prove a geometric criterion for the space $Sh(\lambda, \nu)$ of Shintani functions to be finite-dimensional in the Archimedean case. This criterion leads us to a complete classification of the symmetric pairs $(G,G')$ having finite-dimensional Shintani spaces. A geometric criterion for uniform boundedness of $dim Sh(\lambda, \nu)$ is also obtained. Furthermore, we prove that symmetry breaking operators of the restriction of smooth admissible representations yield Shintani functions of moderate growth, of which the dimension is determined for $(G, G') = (O(n+1,1), O(n,1))$.Comment: to appear in Progress in Mathematics, Birkhause

On the endomorphisms of Weyl modules over affine Kac-Moody algebras at the critical level

We present an independent short proof of the main result of arXiv:0706.3725 that the algebra of endomorphisms of a Weyl module of critical level is isomorphic to the algebra of functions on the space of monodromy-free opers on the disc with regular singularity and residue determined by the highest weight of the Weyl module. We derive this from the results of arXiv:0712.1183 about the shift of argument subalgebras.Comment: 9 page