Let $D$ be a bounded homogeneous domain in $\mathbb{C}^{n}$ and let $\varphi$ be an automorphism of $D$ which generates a discrete subgroup $\Gamma$ of $\Aut_{\mathcal{O}}(D)$. It is shown that the complex space $D/\Gamma$ is Stein.

Let $(M,I,J,K,g)$ be a hyperkähler manifold, $\dim_{\mathbb{R}} M =4n$. We study positive, $\partial$-closed $(2p,0)$-forms on $(M,I)$. These forms are quaternionic analogues of the positive $(p,p)$-forms, well-known in complex geometry. We construct a monomorphism $\mathcal{V}_{p,p}\colon \Lambda^{2p,0}_{I}(M)\to\Lambda^{n+p,n+p}_{I}(M)$, which maps $\partial$-closed $(2p,0)$-forms to closed $(n+p,n+p)$-forms, and positive $(2p,0)$-forms to positive $(n+p,n+p)$-forms. This construction is used to prove a hyperkähler version of the classical Skoda--El Mir theorem, which says that a trivial extension of a closed, positive current over a pluripolar set is again closed. We also prove the hyperkähler version of the Sibony's lemma, showing that a closed, positive $(2p,0)$-form defined outside of a compact complex subvariety $Z\subset (M,I)$, $\codim Z > 2p$ is locally integrable in a neighbourhood of $Z$. These results are used to prove polystability of derived direct images of certain coherent sheaves.

For oriented closed surfaces of genera up to 4, we list presentations of periodic maps by Dehn twists. As an application of these presentations, we provide examples of non-holomorphic Lefschetz fibrations.

We consider hyperbolic Cauchy problems with characteristics of variable multiplicity and coefficients of polynomial growth in the space variables; we focus on second order equations and admit finite order intersections between the characteristics. We obtain well posedness results in $\mathcal{S}(\mathbb{R}^{n})$, $\mathcal{S}'(\mathbb{R}^{n})$ by imposing suitable Levi conditions on the lower order terms. By an energy estimate in weighted Sobolev spaces we show that regularity and behavior at infinity of the solution are different from the ones of the data.

The based loop space homology of a special family of homogeneous spaces, flag manifolds of connected compact Lie groups is studied. First, the rational homology of the based loop space on a complete flag manifold is calculated together with its Pontrjagin structure. Second, it is shown that the integral homology of the based loop space on a flag manifold is torsion free. This results in a description of the integral homology. In addition, the integral Pontrjagin structure is determined.

O. Chalykh, A.P. Veselov and M. Feigin introduced the notion of quasiinvariants of Coxeter groups, which is a generalization of invariants. In [2], Bandlow and Musiker showed that for the symmetric group $S_{n}$ of order $n$, the space of quasiinvariants has a decomposition indexed by standard tableaux. They gave a description of a basis for the components indexed by standard tableaux of shape $(n-1,1)$. In this paper, we generalize their results to a description of a basis for the components indexed by standard tableaux of arbitrary hook shape.

We determine the location of all of the zeros of certain Poincaré series associated with the Fricke groups $\Gamma_{0}^{*}(2)$ and $\Gamma_{0}^{*}(3)$ in their fundamental domains by applying and extending the method of F.K.C. Rankin and H.P.F. Swinnerton-Dyer ([9]).

An $n$-dimensional Hartogs domain $D_{F}$ can be equipped with a natural Kähler metric $g_{F}$. This paper contains two results. In the first one we prove that if $g_{F}$ is an extremal Kähler metric then $(D_{F}, g_{F})$ is holomorphically isometric to an open subset of the $n$-dimensional complex hyperbolic space. In the second one we prove the same assertion under the assumption that there exists a real holomorphic vector field $X$ on $D_{F}$ such that $(g_{F}, X)$ is a Kähler--Ricci soliton.

We prove that the duality operator preserves the Frobenius--Schur indicators of characters of connected reductive groups of Lie type with connected center. This allows us to extend a result of D. Prasad which relates the Frobenius--Schur indicator of a regular real-valued character to its central character. We apply these results to compute the Frobenius--Schur indicators of certain real-valued, irreducible, Frobenius-invariant Deligne--Lusztig characters, and the Frobenius--Schur indicators of real-valued regular and semisimple characters of finite unitary groups.

In this article we construct all the primitive idempotents of the restricted quantum group $\bar{U}_{q} (\mathit{sl}_{2})$ and also describe $\bar{U}_{q} (\mathit{sl}_{2})$ as the subalgebra of the direct sum of matrix algebras. By using this result we construct a basis of the space of symmetric linear functions of $\bar{U}_{q}(\mathit{sl}_{2})$ and determine the decomposition of the integral of the dual of $\bar{U}_{q} (\mathit{sl}_{2})$ twisted by the balancing element to the basis of the space of symmetric linear functions.

In [7] the $q$ tetrahedron algebra $\boxtimes_{q}$ was introduced as a $q$ analogue of the universal enveloping algebra of the three point loop algebra $\mathit{sl}_{2} \otimes \mathbf{C}[t,t^{-1},(t-1)^{-1}]$. In this paper the relation between finite dimensional $\boxtimes_{q}$ modules and finite dimensional modules for $U_{q}(L(\mathit{sl}_{2}))$, a $q$ analogue of the loop algebra $L(\mathit{sl}_{2})$, is studied. A connection between the $\boxtimes_{q}$ module structure and $L$-operators for $U_{q}(L(\mathit{sl}_{2}))$ is also discussed.

In the previous paper, the authors gave criteria for $A_{k+1}$-type singularities on wave fronts. Using them, we show in this paper that there is a duality between singular points and inflection points on wave fronts in the projective space. As an application, we show that the algebraic sum of $2$-inflection points (i.e. godron points) on an immersed surface in the real projective space is equal to the Euler number of $M_{-}$. Here $M^{2}$ is a compact orientable 2-manifold, and $M_{-}$ is the open subset of $M^{2}$ where the Hessian of $f$ takes negative values. This is a generalization of Bleecker and Wilson's formula [3] for immersed surfaces in the affine $3$-space.