Charles H. Conley

Conformal symbols and the action of contact vector fields over the superline

Abstract Let 𝒦 be the Lie superalgebra of contact vector fields on the supersymmetric line . We compute the action of 𝒦 on the modules of differential and pseudo-differential operators between spaces of tensor densities, in terms of their conformal symbols. As applications we deduce the geometric subsymbols, 1-cohomology, and various uniserial subquotients of these modules. We also outline the computation of the 𝒦-equivalences and symmetries of their subquotients.

A Family of Irreducible Representations of the Witt Lie Algebra with Infinite-Dimensional Weight Spaces

We define a 4-parameter family of generically irreducible and inequivalent representations of the Witt Lie algebra on which the infinitesimal rotation operator acts semisimply with infinite-dimensional eigenspaces. They are deformations of the (generically indecomposable) representations on spaces of polynomial differential operators between two spaces of tensor densities on S1, which are constructed by composing each such differential operator with the action of a rotation by a fixed angle.

Relative extremal projectors

Abstract This paper proves the existence of relative extremal projectors. An infinite factorization is given as well as a summation formula.

Rotundus: Triangulations, Chebyshev Polynomials, and Pfaffians

We introduce and study a cyclically invariant polynomial which is an analog of the classical tridiagonal determinant usually called the continuant. We prove that this polynomial can be calculated as the Pfaffian of a skew-symmetric matrix. We consider the corresponding Diophantine equation and prove an analog of a famous result due to Conway and Coxeter. We also observe that Chebyshev polynomials of the first kind arise as Pfaffians.

Jacobi forms over complex quadratic fields via the cubic Casimir operators

We prove that the center of the algebra of differential operators invariant under the action of the Jacobi group over a complex quadratic field is generated by two cubic Casimir operators, which we compute explicitly. In the spirit of Borel, we consider Jacobi forms over complex quadratic fields that are also eigenfunctions of these Casimir operators, a new approach in the complex case. Theta functions and Eisenstein series provide standard examples. In addition, we introduce an analog of Kohnens plus space for modular forms of half-integral weight over K D Q.i/, and provide a lift from it to the space of Jacobi forms over K D Q.i/. Mathematics Subject Classification (2010). Primary 11F50; Secondary 43A85.

Equivalence Classes of Subquotients of Supersymmetric Pseudodifferential Operator Modules

We study the equivalence classes of the non-resonant subquotients of spaces of pseudodifferential operators between tensor density modules over the superline ℝ1|1

Equivalence classes of subquotients of pseudodifferential operator modules

\mathbb {R}^{1|1}

Geometric Realizations of Representations of Finite Length

, as modules of the Lie superalgebra of contact vector fields. There is a 2-parameter family of subquotients with any given Jordan-Hölder composition series. We give a complete set of even equivalence invariants for subquotients of all lengths l. In the critical case l = 6, the even equivalence classes within each non-resonant 2-parameter family are specified by a pencil of conics. In lengths l ≥ 7 our invariants are not fully simplified: for l = 7 we expect that there are only finitely many equivalences other than conjugation, and for l ≥ 8 we expect that conjugation is the only equivalence. We prove this in lengths l ≥ 15. We also analyze certain lacunary subquotients.

Quantization and injective submodules of differential operator modules

Certain subquotients of Vec(R)-modules of pseudodifferential operators from one tensor density module to another are categorized, giving necessary and sufficient conditions under which two such subquotients are equivalent as Vec(R)-representations. These subquotients split under the projective subalgebra, a copy of ????2, when the members of their composition series have distinct Casimir eigenvalues. Results were obtained using the explicit description of the action of Vec(R) with respect to this splitting. In the length five case, the equivalence classes of the subquotients are determined by two invariants. In an appropriate coordinate system, the level curves of one of these invariants are a pencil of conics, and those of the other are a pencil of cubics.

Factorizations of relative extremal projectors

Let G=H×ℝn be a semidirect product Lie group. We reduce the problem of deciding which indecomposable representations of G may be realized in subquotients of spaces of sections of vector bundles over infinitesimal neighborhoods of orbits of H in the dual of ℝn to a problem involving only representations of the H-stabilizers of the orbits.