We continue studying compound Du Val singularities defined over an algebraically closed field $k$, and present concrete examples in characteristic $2$ which have one-dimensional singular loci but do not admit a description as a trivial product (a rational double point) $\times$ (a curve) up to analytic isomorphism at any point. Unlike in other characteristics, we find a large number of such examples whose general hyperplane sections have rational double points of type $D$. These compound Du Val singularities shall be viewed as a special class of canonical singularities. In the previous work with Ito and Saito, we classified such singularities in $p\ge 3$,

and I intend to complete our classification in arbitrary characteristic, reinforcing Reid’s result in characteristic $0$.

We explore the question of when the Rees algebra $\mathcal{R}(I)={\oplus }_{n\ge 0}{I}^n$ of $I$ is an almost Gorenstein graded ring, where $R$ is a 2-dimensional regular local ring and $I$ is a contracted ideal of $R$. Recently, we showed that $\mathcal{R}(I)$ is an almost Gorenstein graded ring for every integrally closed ideal $I$ of $R$. The main results of the present article show that if $I$ is a contracted ideal with $\mathrm{o}(I)\le 2$, then $\mathcal{R}(I)$ is an almost Gorenstein graded ring, while if $\mathrm{o}(I)\ge 3$, then $\mathcal{R}(I)$ is not necessarily an almost Gorenstein graded ring, even though $I$ is a contracted stable ideal. Thus both affirmative and negative answers are given.

In this article, we explicitly list all reducible scalar-type generalized Verma modules for all maximal parabolic subalgebras of the simple complex Lie algebras.

It is a wide open problem to give an intrinsic criterion for a ${\mathrm{II}}_1$ factor $M$ to admit a Cartan subalgebra $A$. When $A\subset M$ is a Cartan subalgebra, the $A$-bimodule $L^2(M)$ is simple in the sense that the left and right actions of $A$ generate a maximal abelian subalgebra of $B(L^2(M))$. A ${\mathrm{II}}_1$ factor $M$ that admits such a subalgebra $A$ is said to be $s$-thin. Very recently, Popa discovered an intrinsic local criterion for a ${\mathrm{II}}_1$ factor $M$ to be $s$-thin and left open the question whether all $s$-thin ${\mathrm{II}}_1$ factors admit a Cartan subalgebra. We answer this question negatively by constructing $s$-thin ${\mathrm{II}}_1$ factors without Cartan subalgebras.

We develop a nonlinear theory for infrahyperfunctions (also referred to as quasianalytic (ultra)distributions by L. Hörmander). In the hyperfunction case, our work can be summarized as follows. We construct a differential algebra that contains the space of hyperfunctions as a linear differential subspace and in which the multiplication of real analytic functions coincides with their ordinary product. Moreover, by proving an analogue of Schwartz’s impossibility result for hyperfunctions, we show that this embedding is optimal. Our results fully solve an earlier question raised by M. Oberguggenberger.

Beauville and Donagi proved in 1985 that the primitive middle cohomology of a smooth complex cubic $4$-fold and the primitive second cohomology of its variety of lines, a smooth hyper-Kähler $4$-fold, are isomorphic as polarized integral Hodge structures. We prove analogous statements for smooth complex Gushel–Mukai varieties of dimension $4$ (resp., $6$), that is, smooth dimensionally transverse intersections of the cone over the Grassmannian $\mathsf{Gr}(2,5)$, a quadric, and two hyperplanes (resp., of the cone over $\mathsf{Gr}(2,5)$ and a quadric). The associated hyper-Kähler $4$-fold is in both cases a smooth double cover of a hypersurface in ${\mathbf{P}}^5$ called an Eisenbud–Popescu–Walter sextic.

In this paper we construct an abstract Fock space for general Lie types that serves as a generalization of the infinite wedge $q$-Fock space familiar in type A. Specifically, for each positive integer $\ell$, we define a $\mathbb{Z}[q,q^{-1}]$-module ${\mathcal{F}}_{\ell }$ with bar involution by specifying generators and straightening relations adapted from those appearing in the Kashiwara–Miwa–Stern formulation of the $q$-Fock space. By relating ${\mathcal{F}}_{\ell }$ to the corresponding affine Hecke algebra, we show that the abstract Fock space has standard and canonical bases for which the transition matrix produces parabolic affine Kazhdan–Lusztig polynomials. This property and the convenient combinatorial labeling of bases of ${\mathcal{F}}_{\ell }$ by dominant integral weights makes ${\mathcal{F}}_{\ell }$ a useful combinatorial tool for determining decomposition numbers of Weyl modules for quantum groups at roots of unity.

We present a multidimensional continued fraction of Hirzebruch–Jung type which controls a certain resolution of an isolated cyclic quotient singularity, and we study its geometric properties. As an application, we explicitly express a certain $3$-dimensional Fourier–Dedekind sum in terms of our continued fraction.

We classify del Pezzo surfaces of Picard number $1$ with log-canonical singularities admitting $\mathbb{Q}$-Gorenstein smoothings.

In this paper we describe the local theta lift for $p$-adic unitary dual pairs $\mathrm{U}\left(2\right)\times \mathrm{U}\left(1\right)$ and $\mathrm{U}\left(2\right)\times \mathrm{U}\left(3\right)$. We also describe the local theta lift for a pair of $p$-adic quaternionic unitary groups of rank $1$.