A Formula for the Geometric Jacquet Functor and its Character Sheaf Analogue

Let (G,K) be a symmetric pair over the complex numbers, and let X=K\G be the corresponding symmetric space. In this paper we study a nearby cycles functor associated to a degeneration of X to MN\G, which we call the "wonderful degeneration". We show that on the category of character sheaves on X, this functor is isomorphic to a composition of two averaging functors (a parallel result, on the level of functions in the p-adic setting, was obtained in [BK, SV]). As an application, we obtain a formula for the geometric Jacquet functor of [ENV] and use this formula to give a geometric proof of the celebrated Casselman's submodule theorem and establish a second adjointness theorem for Harish-Chandra modules.Comment: Revised version. Equivariancy replaces stratification arguments, so that the results are applicable to all sheaf setting

On the Casselman-Jacquet functor

We study the Casselman-Jacquet functor $J$, viewed as a functor from the (derived) category of $(\mathfrak{g},K)$-modules to the (derived) category of $(\mathfrak{g},N^-)$-modules, $N^-$ is the negative maximal unipotent. We give a functorial definition of $J$ as a certain right adjoint functor, and identify it as a composition of two averaging functors $\text{Av}^{N^-}_!\circ \text{Av}^N_*$. We show that it is also isomorphic to the composition $\text{Av}^{N^-}_*\circ \text{Av}^N_!$. Our key tool is the pseudo-identity functor that acts on the (derived) category of (twisted) $D$-modules on an algebraic stack.Comment: Very minor modifications, compared to previous version (to appear in Proceedings of Symposia in Pure Mathematics volume, titled "Representations of Reductive Groups"

Analyzing the Decision to Get Flu Shot: An Empirical Study

Influenza vaccination has been shown to be cost effective in reducing morbidity and mortality and in decreasing work absenteeism and use of health-care resources. The purpose of this study was to identify predictors and beliefs regarding people's vaccination decision against the influenza. It was hypothesized that Health Belief Model (HBM) categories, such as severity of illness, vaccine effectiveness and side effects of the vaccine, affect the decision to get flu shot. In addition, we examined psychological effects, such as time preference, subjective probability of flu, and attitude toward risk. A questionnaire surveys was conducted in the USA, in 2004. The questions included HBM categories and the psychological effects. The results indicate that the main predictors of past immunization against influenza are: the estimated effectiveness of the vaccination, periodic blood test, perceived severity of flu illness, side effects of vaccine (negative effect), having health anxieties, and subjective probability of being infected. Based upon these results, it is recommended to enlarging people's knowledge regarding the influenza illness, its potential risks, and the potential benefits of the vaccine.