Ph.D. Recipients and their Thesis Abstracts

Lie Groups

Algebra, Analysis, Combinatorics, Differential Equations / Dynamical Systems, Differential Geometry, Geometry, Interdisciplinary, Lie Groups, Logic, Mathematical Physics, PDE / Numerical Analysis, Probability, Statistics, Topology

Warped Cohomology

Abstract: This work concerns the computation of various L_2 cohomology theories, where L_2 cohomology is an analogue of de Rham cohomology on complete Riemannian manifolds which demands the forms under consideration be square integrable. These computations take place on a class of manifolds defined herein, which generalize the arithmetic quotients of rank one symmetric spaces. In particular, they are all with the sole exception of the Euclidean line finite volume. Nevertheless, infinite dimensionality problems can still arise for this L_2 cohomology, so the square integrability condition is tightened or loosened by multiplying various exponential weighting functions into the Riemannian measure. This often produces finite dimensional weighted L_2 cohomology groups. Finally, the manifolds under study are sufficiently close to arithmetic quotients of rank one symmetric spaces that the cohomology of the links of the cusp points at infinity admits a weight space decomposition, allowing the definition of an analogue of the weighted cohomology of [GHM94] on them. This analogue, called warped cohomology, will in fact compute the weighted L_2 cohomologies defined above and is defined on many nonarithmetic spaces. Warped cohomology allows results on such spaces which are similar to the weighted L_2 construction of weighted cohomology (theorem A) derived on arithmetic quotients of symmetric spaces (of any rank) in [Nai99].

Twisted Torsion on Compact Hyperbolic Spaces: A Representation-Theoretic Approach

Abstract: We consider a twisted version \tau_\theta of the Ray-Singer analytic torsion on compact locally symmetric spaces X = K \ G / \Gamma (with G a noncompact connected semisimple Lie group, K its maximal compact subgroup, and \Gamma a discrete torsion-free cocompact subgroup), where \theta is an automorphism of X with the property that \theta^2 = 1. We obtain a representation-theoretic interpretation of the twisted torsion. By considering \theta = Cartan involution, we show that |\tau_\theta| = 1 for the compact locally symmetric spaces associated to the families of Lie groups locally isomorphic to SO_0(2n+1, 1).

Abelian Sesquisymplectic Convexity for Orbifolds

Abstract: We show how the quotient of the symplectic normal bundle to a suborbifold of a symplectic orbifold can be viewed as the symplectic normal bundle to the image of the suborbifold in the reduction of the symplectic orbifold. We use this to prove an orbifold sesquisymplectic version of the isotropic embedding theorem. We apply this to prove a symplectic slice theorem in the same setting and in turn use this to prove a convexity theorem in this setting. Finally we apply these results to toric orbifolds.

Representations and Characters of an Extension of SL(3,R) by an Outer Automorphism

Abstract: Not available.

Holomorphic Functions and Heat Kernel Measure on an Infinite Dimensional Complex Orthogonal Group

Abstract: The heat kernel measure \mu_t is constructed on an infinite dimensional complex group using a diffusion in a Hilbert space. Then it is proved that holomorphic polynomials on the group are square integrable with respect to the heat kernel measure. The closure of these polynomials, {\Cal H} L^2(SO_{HS},\mu_t), is one of two spaces of holomorphic functions we consider. The second space, {\Cal H} L^2(SO(\infty)), consists of functions which are holomorphic on an analog of the Cameron-Martin subspace for the group. It is proved that there is an isometry from the first space to the second one.

The main theorem is that an infinite dimensional nonlinear analog of the Taylor expansion defines an isometry from {\Cal H} L^2(SO(\infty)) into the Hilbert space associated with a Lie algebra of the infinite dimensional group. This is an extension to infinite dimensions of an isometry of B. Driver and L. Gross for complex Lie groups.

All the results of this paper are formulated for one concrete group, the Hilbert-Schmidt complex orthogonal group, though our methods can be applied in more general situations.

Short Time Behavior of Hermite Functions on Compact Lie Groups

Abstract: Let p_t(x) be the (Gaussian) heat kernel on R^n at time t. The classical Hermite polynomials at time t may be defined by a Rodriguez formula, given by H_\alpha(–x, t) p_t(x) = \alpha p_t(x), where \alpha is a constant coefficient differential operator on R^n. Recent work of Gross (1993) and Hijab (1994) has led to the study of a new class of functions on a general compact Lie group, G. In analogy with the R^n case, these "Hermite functions" on G are obtained by the same formula, wherein p_t(x) is now the heat kernel on the group, –x is replaced by x^{–1}, and \alpha is a right invariant differential operator. Let {\frak g} be the Lie algebra of G. Composing a Hermite function on G with the exponential map produces a family of functions on {\frak g}. We prove that these functions, scaled appropriately in t, approach the classical Hermite polynomials at time 1 as t tends to 0, both uniformly on compact subsets of {\frak g} and in L^p({\frak g}, d\mu), where 1 <= p< \infty, and d\mu is a Gaussian measure on {\frak g}. Similar theorems are established when G is replaced by G/K, where K is some closed, connected subgroup of G.

Local Theta Correspondence and Unrefined Minimal K-Types

Abstract: The local theta correspondence is a one-to-one correspondence between the irreducible admissible representations of two p-adic classical groups which form a reductive dual pair in a symplectic group. In this doctoral dissertation, we study the local theta correspondence by applying A. Moy and G. Prasad's theory of unrefined minimal K-types for p-adic reductive groups. After strengthening an important theorem of J.-L. Waldspurger, we prove that the depths of the two irreducible admissible representations paired by the local theta correspondence for any (type I) reductive dual pairs are equal. Moreover, we prove that the unrefined minimal K-types of the paired representations are related by the theta correspondence for finite reductive dual pairs when the depth is zero, and are related by the orbit correspondence when the depth is positive. In particular, an irreducible admissible representation has nontrivial vectors fixed by an Iwahori subgroup if and only if the irreducible admissible representation paired with it also has nontrivial vectors fixed by an Iwahori subgroup. As an application of our main results, we can describe the first occurrences of depth zero irreducible supercuspidal representations in the theta correspondence for the p-adic reductive dual pairs completely in terms of the first occurrences of irreducible cuspidal representations in the theta correspondence for finite reductive dual pairs. It is expected that this result is useful for further studies of the first occurrences of irreducible supercuspidal representations in the local theta correspondence.

The Bargmann-Segal "Coherent State" Transform for Compact Lie Groups

Abstract: Let K be an arbitrary compact, connected Lie group. We describe on K an analog of the Bargmann-Segal "coherent state" transform, and we prove that this generalized coherent state transform maps L^2(K) isometrically onto the space of holomorphic functions in L^2(G, \mu), where G is the complexification of K and where \mu is an appropriate heat kernel measure on G. The generalized coherent state transform is defined in terms of the heat kernel on the compact group K, and its analytic continuation to the complex group G. We also define a "K-averaged" version of the coherent state transform, and we prove a similar result for it.

Last modified: April 7, 2003