Modular forms for the Weil representation of ${SL}_2\left(\mathbb{Z}\right)$ play an important role in the theory of automorphic forms on orthogonal groups. In this paper we give some explicit constructions of these functions. As an application, we construct new examples of generalized Kac–Moody algebras whose denominator identities are holomorphic automorphic products of singular weight. They correspond naturally to the Niemeier lattices with root systems $D_{12}^2$, $E_8^3$ and to the Leech lattice.

The present notes provide a proof of $i^{\ast }(\mathbb{C}P+\mathbb{C}(I-P);\mathbb{C}Q+\mathbb{C}(I-Q\left)\right)=-{\chi }_{\mathrm{orb}}(P,Q)$ for any pair of projections $P,Q$ with $\tau \left(P\right)=\tau \left(Q\right)=1/2$. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.

A uniform upper bound for the Diederich–Fornaess index is given for weakly pseudoconvex domains whose Levi form of the boundary vanishes in $\ell$-directions everywhere.

As an application of a sharp $L^2$ extension theorem for holomorphic functions in Guan and Zhou, a stability theorem for the boundary asymptotics of the Bergman kernel is proved. An alternate proof of the extension theorem is given, too. It is a simplified proof in the sense that it is free from ordinary differential equations.

We develop stochastic calculus for symmetric Markov processes in terms of time reversal operators. For this, we introduce the notion of the progressively additive functional in the strong sense with time-reversible defining sets. Most additive functionals can be regarded as such functionals. We obtain a refined formula between stochastic integrals by martingale additive functionals and those by Nakao’s divergence-like continuous additive functionals of zero energy. As an application, we give a stochastic characterization of harmonic functions on a domain with respect to the infinitesimal generator of semigroup on $L^2$-space obtained by lower-order perturbations.