Related papers: $Sp(n)$-orbits of isoclinic subspaces in the real …

We determine the invariants characterizing the $Sp(n)$-orbits in the real Grassmannian $Gr^\R(2k,4n)$ of the $2k$-dimensional complex and $\Sigma$-complex subspaces of a $4n$-dimensional Hermitian quaternionic vector space. A…

The odd symplectic Grassmannian $\mathrm{IG}:=\mathrm{IG}(k, 2n+1)$ parametrizes $k$ dimensional subspaces of $\mathbb{C}^{2n+1}$ which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic…

Let $V$ and $V'$ be $2n$-dimensional vector spaces over fields $F$ and $F'$. Let also $\Omega: V\times V\to F$ and $\Omega': V'\times V'\to F'$ be non-degenerate symplectic forms. Denote by $\Pi$ and $\Pi'$ the associated…

Let $\Pi$ be a polar space of type $\textsf{D}_{n}$. Denote by ${\mathcal G}_{\delta}(\Pi)$, $\delta\in \{+,-\}$ the associated half-spin Grassmannians and write $\Gamma_{\delta}(\Pi)$ for the corresponding half-spin Grassmann graphs. In…

Let G be the group preserving a nondegenerate sesquilinear form on a vector space V, and H a symmetric subgroup of G of the type G1 x G2. We explicitly parameterize the H-orbits in the Grassmannian of r-dimensional isotropic subspaces of V…

Let $\mathcal{F}_h^i(k,n)$ be the $i$th ordered configuration space of all distinct points $H_1,\ldots,H_h$ in the Grassmannian $Gr(k,n)$ of $k$-dimensional subspaces of $\mc^n$, whose sum is a subspace of dimension $i$. We prove that…

We prove that, for m greater than 3 and k greater than m-2, the Grassmannian of m-dimensional subspaces of the space of skew-symmetric forms over a vector space of dimension 2k is birational to the Hilbert scheme of Palatini scrolls in…

Let $n$ be an integer greater than or equal to $3$ and $(R,\Delta)$ a Hermitian form ring where $R$ is commutative. We prove that if $H$ is a subgroup of the odd-dimensional unitary group $U_{2n+1}(R,\Delta)$ normalised by a relative…

Let X be an orthogonal Grassmannian parametrizing isotropic subspaces in an even dimensional vector space equipped with a nondegenerate symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in the singular…

Let $Sp(2,1)$ be the isometry group of the quaternionic hyperbolic plane ${{\bf H}_{\mathbb H}}^2$. An element $g$ in $Sp(2,1)$ is `hyperbolic' if it fixes exactly two points on the boundary of ${{\bf H}_{\mathbb H}}^2$. We classify pairs…

A moving parallel frame method is applied to geometric non-stretching curve flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable systems with unitary invariance. These systems consist of a bi-Hamiltonian modified…

We establish a derived geometric Satake equivalence for the quaternionic general linear group GL_n(H). By applying the real-symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the…

We give a generalization to higher dimensions of Silverman's result on finiteness of integer points in orbits. Assuming Vojta's conjecture, we prove a sufficient condition for morphisms on P^N so that (S,D)-integral points in each orbit are…

For any rank-one Riemannian symmetric space S of non-compact type and any discrete, cofinite, non-cocompact, torsion-free group $\Gamma$ of orientation-preserving Riemannian isometries on S, we develop a cohomological interpretation for the…

Let X be a symplectic or odd orthogonal Grassmannian parametrizing isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove a Giambelli formula which expresses an arbitrary Schubert class in…

In this paper we give a characterization of real hypersurfaces in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, $m \geq 2$ with Reeb vector field $\xi$ belonging to the maximal quaternionic subbundle $\mathcal Q$. Then it…

Denote by $Sp(k,l)$ the quaternionic symplectic group of signature $(k,l)$. We study the deformation rigidity of the embedding $Sp(k,l) \times Sp(1) \hookrightarrow H$, where $H$ is either $Sp(k+1,l)$ or $Sp(k,l+1)$, this is done by…

Let $\Gamma$ be a nonelementary discrete subgroup of $\mathrm{Sp}(n,1)$. We show that if the trace skew-field of $\Gamma$ is commutative, then $\Gamma$ stabilizes a copy of complex hyperbolic subspace of quaternionic hyperbolic $n$-space.

Let V be a vector space with a nondegenerate symmetric form and OG be the orthogonal Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH^*(OG) and show that its…

Let $G$ be a linear connected non-compact real simple Lie group and let $K\subset G$ be a maximal compact subgroup of $G$. Suppose that the centre of $K$ isomorphic to $\mathbb{S}^1$ so that $G/K$ is a global Hermitian symmetric space. Let…