This the first in a series of papers on special Lagrangian submanifolds in ℂ^m. We study special Lagrangian submanifolds in ℂ^m with large symmetry groups, and we give a number of explicit constructions. Our main results concern special Lagrangian cones in ℂ^m invariant under a subgroup G in SU(m) isomorphic to U(1)^m−2. By writing the special Lagrangian equation as an ordinary differential equation (ODE) in G-orbits and solving the ODE, we find a large family of distinct, G-invariant special Lagrangian cones on T^m−2 in ℂ^m. These examples are interesting as local models for singularities of special Lagrangian submanifolds of Calabi-Yau manifolds. Such models are needed to understand mirror symmetry and the Strominger-Yau-Zaslow (SYZ) conjecture.

We give a dimension formula for the space of logarithm-free series solutions to an A-hypergeometric (or a Gel’fand-Kapranov-Zelevinskiĭ (GKZ) hypergeometric) system. In the case where the convex hull spanned by A is a simplex, we give a rank formula for the system, characterize the exceptional set, and prove the equivalence of the Cohen-Macaulayness of the toric variety defined by A with the emptiness of the exceptional set. Furthermore, we classify A-hypergeometric systems as analytic $\mathcal{D}$-modules.

We prove a formula for the structure sheaf of a quiver variety in the Grothendieck ring of its embedding variety. This formula generalizes and gives new expressions for Grothendieck polynomials. Our formula is stated in terms of coefficients that are uniquely determined by the geometry and can be computed by an explicit combinatorial algorithm. We conjecture that these coefficients have signs that alternate with degree. The proof of our formula involves K-theoretic generalizations of several useful cohomological tools, including the Thom-Porteous formula, the Jacobi-Trudi formula, and a Gysin formula of P. Pragacz.

We develop a theory of Fourier coefficients for modular forms on the split exceptional group G₂ over ℚ.