Let $V = {\mathbb R}^{p,q}$ be the pseudo-Euclidean
vector space of signature $(p,q)$, $p\ge 3$ and $W$
a module over the even Clifford algebra $C\! \ell^0 (V)$. A
homogeneous quaternionic manifold $(M,Q)$ is constructed for any
$\mathfrak{spin}(V)$-equivariant linear map $\Pi : \wedge^2 W
\rightarrow V$. If the skew symmetric vector valued bilinear form
$\Pi$ is nondegenerate then $(M,Q)$ is endowed with a
canonical pseudo-Riemannian metric $g$ such that $(M,Q,g)$ is
a homogeneous quaternionic pseudo-Kähler manifold. If the metric
$g$ is positive definite, i.e. a Riemannian metric, then the
quaternionic Kähler manifold $(M,Q,g)$ is shown to admit a simply
transitive solvable group of automorphisms. In this special case
($p=3$) we recover all the known homogeneous quaternionic
Kähler manifolds of negative scalar curvature (Alekseevsky spaces) in a
unified and direct way. If $p>3$ then $M$ does not admit any
transitive action of a solvable Lie group and we obtain new families of
quaternionic pseudo-Kähler manifolds. Then it is shown that for $q =
0$ the noncompact quaternionic manifold $(M,Q)$ can be endowed
with a Riemannian metric $h$ such that $(M,Q,h)$ is a
homogeneous quaternionic Hermitian manifold, which does not admit any
transitive solvable group of isometries if $p>3$.
The twistor bundle $Z \rightarrow M$ and the
canonical ${\mathrm SO}(3)$-principal bundle $S \rightarrow
M$ associated to the quaternionic manifold $(M,Q)$ are shown to
be homogeneous under the automorphism group of the base. More specifically,
the twistor space is a homogeneous complex manifold carrying an invariant
holomorphic distribution $\mathcal D$ of complex codimension one,
which is a complex contact structure if and only if $\Pi$ is
nondegenerate. Moreover, an equivariant open holomorphic immersion $Z
\rightarrow \bar{Z}$ into a homogeneous complex manifold
$\bar{Z}$ of complex algebraic group is constructed.
Finally, the construction is shown to have a natural mirror in
the category of supermanifolds. In fact, for any
$\mathfrak{spin}(V)$-equivariant linear map $\Pi : \vee^2 W
\rightarrow V$ a homogeneous quaternionic supermanifold $(M,Q)$
is constructed and, moreover, a homogeneous quaternionic pseudo-Kähler
supermanifold $(M,Q,g)$ if the symmetric vector valued bilinear form
$\Pi$ is nondegenerate.
Readership
Graduate students and research mathematicians interested in
differential geometry.