arXiv subjects
Publications and source records attributed to Hee Oh.
We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and, more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz.
Let K be a global field of characteristic not 2. Let Z be a symmetric variety defined over K and S a finite set of places of K. We obtain counting and equidistribution results for the S-integral points of Z. Our results are effective when K is a number field.
Let G be the group of k-points of a connected reductive k-group and H a symmetric subgroup associated to an involution s of G. We prove a polar decomposition G=KAH for the symmetric space G/H over any local field k of characteristic not 2. Here K is a compact subset of G and A is a finite union of the groups of k-points of maximal (k,s)-split tori, one for each H-conjugacy class. This decomposition is analogous to the well-known polar decomposition G=KAH for a real symmetric space G/H.
We study a refined version of the Linnik problem on the asymptotic behavior of the number of representations of integer $m$ by an integral polynomial as $m$ tends to infinity. We assume that the polynomial arises from invariant theory, and use methods from the theory of unipotent flows.
We use the theory of unipotent flows to prove, in a general setting, the equidistribution of Hecke points. We follow the approach of Burger-Sarnak, and use a theorem of Mozes-Shah.
Let X be a symmetric space of noncompact type and Γa lattice in the isometry group of X. We study the distribution of orbits of Γacting on the symmetric space X and its geometric boundary X(\infty). More precisely, for any y in X and b in X(\infty), we investigate the distribution of the set {(yγ,bγ^{-1}):γ\inΓ} in X\times X(\infty). It is proved, in particular, that the orbits of Γin the Furstenberg boundary are equidistributed, and that the orbits of Γin X are equidistributed in ``sectors'' defined with respect to a Cartan decomposition. We also discuss an application to the Patterson-Sullivan theory. Our main tools are the strong wavefront lemma and the equidistribution of solvable flows on homogeneous spaces.
We announce the folowing result: Any finitely generated non virtually solvable linear group over a field of characteristic zero has uniform exponential growth.
For G = SL(3,R) and G = SO(2,n), we give explicit, practical conditions that determine whether or not a closed, connected subgroup H of G has the property that there exists a compact subset C of G with CHC = G. To do this, we fix a Cartan decomposition G = KAK of G, and then carry out an approximate calculation of the intersection of KHK with A, for each closed, connected subgroup H of G.
A homogeneous space G/H is said to have a compact Clifford-Klein form if there exists a discrete subgroup D of G that acts properly discontinuously on G/H, such that the quotient space D\G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2,n), such that G/H has a compact Clifford-Klein form, but our classification is not quite complete when n is odd. The work reveals new examples of homogeneous spaces of SO(2,n) that have compact Clifford-Klein forms, if n is even. Furthermore, we show that if H is a closed, connected subgroup of G = SL(3,R), and neither H nor G/H is compact, then G/H does not have a compact Clifford-Klein form, and we also study noncompact Clifford-Klein forms of finite volume.