Nilpotent orbits of symplectic p-adic Lie algebras and quadratic forms

If V is an sl2Qp representation with an invariant symplectic bilinear form, then we describe a connection between nilpotent orbits of sp (V) and anisotropic isometry classes of quadratic forms over Qp . This connection relies heavily on lattices L of V that are stable under sl2Zp and good, meaning that p L* ⊆ L ⊆ L*. We begin by determining the anisotropic isometry classes of quadratic forms as well as the structure of the Witt ring whose elements correspond to the anisotropic isometry classes. Next we show that when V is an sl2Qp representation with invariant bilinear form, then V has a decomposition V = ⊕i( Wi ⊗ U i). The third chapter classifies the good stable lattices of V and shows that such a lattice has a decomposition into good lattices in the components of the decomposition of V from Chapter 2. In Chapter 4 we combine all of this to connect nilpotent orbits of sp (V) to quadratic forms via the lattices studied in Chapter 3.