In this article, we give a geometric interpretation of the Hitchin component $T^4(\Sigma )\subset \mathrm{Rep}({\pi }_1(\Sigma ),{\mathrm{PSL}}_4(R))$ of a closed oriented surface of genus $g\ge 2$. We show that representations in $T^4(\Sigma )$ are precisely the holonomy representations of properly convex foliated projective structures on the unit tangent bundle of $\Sigma$. From this, we also deduce a geometric description of the Hitchin component $T(\Sigma ,{\mathrm{Sp}}_4(R))$ of representations into the symplectic group

We study closures of ${\mathrm{GL}}_2^{+}(\mathbb{R})$-orbits in the total space $\Omega M_g$ of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that in the generic stratum, such manifolds are the whole stratum, the hyperelliptic locus, or parameterize curves whose Jacobian has additional endomorphisms. This follows from a cohomological description of the tangent bundle to $\Omega M_g$. For nongeneric strata, similar results can be shown by a case-by-case inspection. We also propose to study a notion of linear manifold that comprises Teichmüller curves, Hilbert modular surfaces, and the ball quotients of Deligne and Mostow [DM]. Moreover, we give an explanation for the difference between Hilbert modular surfaces and Hilbert modular threefolds with respect to this notion of linearity

Let $G$ be a complex, linear algebraic group acting on an algebraic space $X$. The purpose of this article is to prove a Riemann-Roch theorem (Theorem 6.5) that gives a description of the completion of the equivariant Grothendieck group $G_0(G,X)\otimes \mathbb{C}$ at any maximal ideal of the representation ring $R(G)\otimes \mathbb{C}$ in terms of equivariant cycles. The main new technique for proving this theorem is our nonabelian completion theorem (Theorem 5.3) for equivariant $K$-theory. Theorem 5.3 generalizes the classical localization theorems for diagonalizable group actions to arbitrary groups

We construct a microlocalization of the rational Cherednik algebras $H$ of type $S_n$. This is achieved by a quantization of the Hilbert scheme ${\mathrm{Hilb}}^n{\mathbb{C}}^2$ of $n$ points in ${\mathbb{C}}^2$. We then prove the equivalence of the category of $H$-modules and that of modules over its microlocalization under certain conditions on the parameter