Narrow Results

In this paper, we focus on the Janus symplectic group. We explore its various symmetries and its action on the elements of the dual of its Lie algebra, called torsors. Special attention is given to the charge symmetry, which highlights the matter–antimatter duality within both sets of components.

A one-parameter symplectic group {e^tÂ}_t∈ℝ derives proper canonical transformations indexed by t on a Boson–Fock space. It has been known that the unitary operator U_t implementing such a proper canonical transformation gives a projective unitary representation of {e^tÂ}_t∈ℝ on the Boson–Fock space and that U_t can be expressed as a normal-ordered form. We rigorously derive the self-adjoint operator Δ(Â) and a local exponent with a real-valued function τ_Â(·) such that .

By a proper cover of a finite group G we mean an extension of a nontrivial finite group by G. We study element orders in proper covers of a finite simple group L of Lie type and prove that such a cover always contains an element whose order differs from the element orders of L provided that L is not L₄(q), U₃(q), U₄(q), U₅(2), or ³D₄(2).

Under a certain kind of similarity transformation, a parameter-dependent (abbreviated as PD) symplectic group chain Sp(2M) ⊃ Sp(2M - 2) ⊃ ⋯ ⊃ Sp(2) that is characterized by a set of pairing parameters is introduced to build up the PD antisymmetrized fermion states for molecules with symplectic symmetry, and these states will be useful in carrying out the optimization procedure in quantum chemistry. In order to make a complete classification of the states, a generalized branching rule associated with the symplectic group chain is proposed. Further, we are led to the result that the explicit form of the PD antisymmetrized fermion states is obtained in terms of M one-particle operators and M geminal operators.

We construct and study the holomorphic discrete series representations and the principal series representations of the symplectic group Sp(2n, F) over a p-adic field F as well as a duality between some sub-representations of these two representations. The constructions of these two representations generalize those defined in Morita and Murase's works. Moreover, Morita built a duality for SL(2, F) defined by residues. We view the duality we defined as an algebraic interpretation of Morita's duality in some extent and its generalization to the symplectic groups.

We give a full set of generators for the center of the universal enveloping Lie algebra of the symplectic group of arbitrary genus. They are of trace type and are given in terms of a basis chosen such that the action on representations of given K-type becomes transparent. We give examples for the latter.

We study the image of the $\ell$-adic Galois representations associated to the four vector valued Siegel modular forms appearing in the work of Chenevier and Lannes [3]. These representations are symplectic of dimension 4. Following methods used by Dieulefait in [4], we determine the primes $\ell$ for which these representations are absolutely irreducible. In addition, we show that their image is “full” for all primes $\ell$ such that the associated residual representation is absolutely irreducible, except in two cases.

We describe work of Faltings on the construction of étale cohomology classes associated to symplectic Shimura varieties and show that they satisfy certain trace compatibilities similar to the ones of Siegel units in the modular curve case. Starting from those, we construct a two-variable family of trace compatible classes in the cohomology of a unitary Shimura variety.