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We compute the Hodge filtration on cohomology groups of complements of complex coordinate subspace arrangements. By means of this result we construct integral representations of holomorphic functions such that kernels of these…

Let $\A$ be an irreducible Coxeter arrangement and $\bfk$ be a multiplicity of $\A$. We study the derivation module $D(\A, \bfk)$. Any two-dimensional irreducible Coxeter arrangement with even number of lines is decomposed into two orbits…

We prove a conjecture of Schmid and the second named author that the unitarity of a representation of a real reductive Lie group with real infinitesimal character can be read off from a canonical filtration, the Hodge filtration. Our proof…

In this work we study representations of certain Coxeter groups to obtain some properties of the corresponding reflection groups.

We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the $2$-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show…

We develop a complex differential geometric approach to the theory of higher residues and primitive forms from the viewpoint of Kodaira-Spencer gauge theory, unifying the semi-infinite period maps for Calabi-Yau models and Landau-Ginzburg…

Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form (the associated bilinear form of the Coxeter group). If the Tits…

The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups $H_2$, $H_3$ and $H_4$. Using a…

Let $W$ be a finite Coxeter group. We classify the reflection subgroups of $W$ up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup $R$ of $W$ the conjugacy class of its Coxeter…

We study the canonical Hodge filtration on the sheaf $\mathscr{O}_X(*D)$ of meromorphic functions along a divisor. For a germ of an analytic function $f$ whose Bernstein-Sato's polynomial's roots are contained in $(-2,0)$, we: give a simple…

We enumerate factorizations of a Coxeter element in a well generated complex reflection group into arbitrary factors, keeping track of the fixed space dimension of each factor. In the infinite families of generalized permutations, our…

In this paper, we establish connections between the first extensions of simple modules and certain filtrations of of standard modules in the setting of graded Hecke algebras. The filtrations involved are radical filtrations and Jantzen…

We show that the Hodge filtration of a tempered Hodge module is generated by the lowest piece of its Hodge filtration. As a consequence, we prove the main conjecture of [SV] in the special case of tempered representations of real reductive…

Let $Q$ be an acyclic quiver and $\Lambda$ be the complete preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama, Reiten and Scott have introduced and studied in…

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection…

We construct real polarizable Hodge structures on the reduced leafwise cohomology of K\"ahler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's K\"ahlerian…

Let $\bH$ be the generic Iwahori--Hecke algebra associated with a finite Coxeter group $W$. Recently, we have shown that $\bH$ admits a natural cellular basis in the sense of Graham--Lehrer, provided that $W$ is a Weyl group and all…

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a…

In a recent paper we claimed that both the group algebra of a finite Coxeter group $W$ as well as the Orlik-Solomon algebra of $W$ can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each…

Every Coxeter group admits a geometric representation as a group generated by reflections in a real vector space. In the projective representation space, limit directions are limits of injective sequences in the orbit of some base point.…