The Dixmier Conjecture says that every endomorphism of the (first) Weyl algebra $A_{1}$ (over a field of characteristic zero) is an automorphism, i.e., if $PQ-QP=1$ for some $P,Q\in A_{1}$, then $A_{1}=K\langle P,Q\rangle$. The Weyl algebra $A_{1}$ is a $\mathbb{Z}$-graded algebra. We prove that the Dixmier Conjecture holds if the elements $P$ and $Q$ are sums of no more than two homogeneous elements of $A_{1}$ (there is no restriction on the total degrees of $P$ and $Q$).

We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space of quasi-invariants of a given multiplicity is not, in general, an algebra but a module Qk over the coordinate ring of a (singular) affine variety Xk. We extend the main results of Berest et al. [Cherednik algebras and differential operators on quasi-invariants, Duke Math. J. 118 (2003), 279–337] to this setting: in particular, we show that the variety Xk and the module Qk are Cohen–Macaulay, and the rings of differential operators on Xk and Qk are simple rings, Morita equivalent to the Weyl algebra An(ℂ) , where n=dim Xk. Our approach relies on representation theory of complex Cherednik algebras introduced by Dunkl and Opdam [Dunkl operators for complex reflection groups, Proc. London Math. Soc. (3) 86 (2003), 70–108] and is parallel to that of Berest et al. As an application, we prove the existence of shift operators for an arbitrary complex reflection group, confirming a conjecture of Dunkl and Opdam. Another result is a proof of a conjecture of Opdam, concerning certain operations (KZ twists) on the set of irreducible representations of W.

Using the Wiener–Poisson isomorphism, we show that if $(F_t)_{0 \leq t \leq 1}$ is a real, bounded, predictable process adapted to the filtration of a compensated Poisson process $(X_t)_{0 \leq t \leq 1}$, and if $\hat{M}_t$ is the operator corresponding to multiplication by $M_t = \int_0^t F_s dX_s$, then for any regular self-adjoint quantum semimartingale $J = (J_t)_{0 \leq t \leq 1}$, the essentially self-adjoint quantum semimartingale $(\hat{M}_t + J_t)_{0 \leq t \leq 1}$ satisfies the quantum Ito formula.

We also introduce a generalisation of the Poisson process to a measure space $(M, \mathcal{M} , \mu)$ as an isometry $I : L^2 (M, \mathcal{M}, \mu) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ and give a new construction of the generalised Wiener–Poisson isomorphism ${\mathcal{W}_I} : \mathfrak{F}_+ (L^2(M)) \to L^2 (\Omega, \mathcal{F}, {\mathbb{P}} )$ using exponential vectors. Using C*-algebra theory, given any measure space we construct a canonical generalised Poisson process. Unlike other constructions, we make no a priori use of Poisson measures.

A symplectic fibration is a fibre bundle in the symplectic category (a bundle of symplectic fibres over a symplectic base with a symplectic structure group). We find the relation between the deformation quantization of the base and the fibre, and that of the total space. We consider Fedosov's construction of deformation quantization. We generalize the Fedosov construction to the quantization with values in a bundle of algebras. We find that the characteristic class of deformation of a symplectic fibration is the weak coupling form of Guillemin, Lerman, and Sternberg. We also prove that the classical moment map could be quantized if there exists an equivariant connection.