The Differential Geometry of the Orbit Space of Extended Affine Jacobi Group A1

We define certain extensions of Jacobi groups of A₁, prove an analogue of Chevalley theorem for their invariants, and construct a Dubrovin-Frobenius structure on its orbit space.

Algebraicity of special L-values attached to Siegel–Jacobi modular forms

In this work we obtain algebraicity results on special L-values attached to Siegel–Jacobi modular forms. Our method relies on a generalization of the doubling method to the Jacobi group obtained in our previous work, and on introducing a notion of near holomorphy for Siegel–Jacobi modular forms. Some of our results involve also holomorphic projection, which we obtain by using Siegel–Jacobi Poincaré series of exponential type.

The Jacobi group and its holomorphic discrete series representations on Siegel–Jacobi domains

This is the summary of a part of the talk delivered at the workshop held at the Tambov University in September 2012, reporting several results on Jacobi groups and its holomorphic representations published by the authors.

Harmonic Maaß–Jacobi forms of degree 1 with higher rank indices

We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.

Hamiltonian approach to Ehrenfest expectation values and Gaussian quantum states

The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical observables are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest’s theorem is shown to be Lie–Poisson for a semidirect-product Lie group, named the Ehrenfest group . The underlying Poisson structure produces classical and quantum mechanics as special limit cases. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie–Poisson structure associated with another semidirect-product group, which is called the Jacobi group. This structure produces the energy-conserving variant of a class of Gaussian moment models that have previously appeared in the chemical physics literature.

The Maaß space and Hecke operators

We give a new proof of the fact that the Maaß space is invariant under all Hecke operators. It is based on the characterization of the Maaß space by a symmetry relation and certain commutation relations of the Hecke algebra for the Jacobi group.