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We study the quasi-admissibility and raisablility of some nilpotent orbits of a covering group. In particular, we determine the degree of the cover such that a given split nilpotent orbit is quasi-admissible and non-raisable. The speculated wavefront sets of theta representations are also computed explicitly, and are shown to be quasi-admissible and non-raisable. Lastly, we determine the leading coefficients in the Harish-Chandra character expansion of theta representations of covers of the general linear groups.

We study a conjectural formula for the maximal elements in the wavefront set associated with a theta representation of a covering group over $p$-adic fields. In particular, it is shown that the formula agrees with the existing work in the literature for various families of groups. We also recapitulate the results of an analogous formula in the archimedean case, which motivated the conjectural formula in the $p$-adic setting., Comment: 34 pages

Let $\widetilde G$ be the nonlinear double cover of the real points of a connected, simply connected, semisimple complex group. In [Ts], we introduce a set of genuine small representations of $\widetilde G$ with infinitesimal character $\lambda$, denoted $\prod _\lambda ^s (\widetilde G)$. In this paper, we show that $\prod _{\rho/2} ^s (\widetilde G)$ is precisely the set of genuine irreducible representations arising from the Kazhdan-Patterson lifting of the trivial representation, when $\widetilde G$ is simply laced and split., Comment: 29 pages

This paper concerns the numerical solution of a fully nonlinear parabolic double obstacle problem arising from a finite portfolio selection with proportional transaction costs. We consider the optimal allocation of wealth among multiple stocks and a bank account in order to maximize the finite horizon discounted utility of consumption. The problem is mainly governed by a time-dependent Hamilton-Jacobi-Bellman equation with gradient constraints. We propose a numerical method which is composed of Monte Carlo simulation to take advantage of the high-dimensional properties and finite difference method to approximate the gradients of the value function. Numerical results illustrate behaviors of the optimal trading strategies and also satisfy all qualitative properties proved in Dai et al. (2009) and Chen and Dai (2013).

This paper provides a comparison between the $K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits for complex groups of type $D$. Precisely, let $ G_ 0 =Spin(2n,\mathbb C)$ be the Spin complex group viewed as a real group, and $K\cong G_0$ be the complexification of the maximal compact subgroup of $G_0$. We compute $K$-spectra of the regular functions on some small nilpotent orbits $\mathcal O$ transforming according to characters $\psi$ of $C_{ K}(\mathcal O)$ trivial on the connected component of the identity $C_{ K}(\mathcal O)^0$. We then match them with the ${K}$-types of the genuine (i.e. representations which do not factor to $SO(2n,\mathbb C)$) unipotent representations attached to $\mathcal O$., Comment: arXiv admin note: text overlap with arXiv:1702.04841

The results in this paper provide a comparison between the $K$-structure of unipotent representations and regular sections of bundles on nilpotent orbits. Precisely, let $\widetilde{G_0} =\widetilde{Spin}(a,b)$ with $a+b=2n$, the nonlinear double cover of $Spin(a,b)$, and let $\widetilde{K}=Spin(a, \mathbb C)\times Spin(b, \mathbb C)$ be the complexification of the maximal compact subgroup of $\widetilde{G_0}$. We consider the nilpotent orbit $\mathcal O_c$ parametrized by $[3 \ 2^{2k} \ 1^{2n-4k-3}]$ with $k>0$. We provide a list of unipotent representations that are genuine, and prove that the list is complete using the coherent continuation representation. Separately we compute $\widetilde{K}$-spectra of the regular functions on certain real forms $\mathcal O$ of $\mathcal O_c$ transforming according to appropriate characters $\psi$ under $C_{\widetilde{K}}(\mathcal O)$, and then match them with the $\widetilde{K}$-types of the genuine unipotent representations. The results provide instances for the orbit philosophy.

Let G be the real points of a simply connected, semisimple, simply laced complex Lie group, and let \tilde{G} be the nonlinear double cover of G. We discuss a set of small genuine irreducible representations of \tilde{G} which can be characterized by the following properties: (a) the infinitesimal character is \rho/2; (b) they have maximal tau-invariant; (c) they have a particular associated variety O. When G is split, we construct them explicitly. Furthermore, in many cases, there is a one-to-one correspondence between these small representations and the pairs (genuine central characters of \tilde{G}, real forms of O) via the map \pi mapped to (central character of \pi, real associated variety of \pi)., Comment: To appear in Transactions of the AMS