Geometric quantization and the generalized Segal-Bargmann transform for Lie groups of compact type

Let K be a connected Lie group of compact type and let T*(K) be its cotangent bundle. This paper considers geometric quantization of T*(K), first using the vertical polarization and then using a natural Kahler polarization obtained by identifying T*(K) with the complexified group K_C. The first main result is that the Hilbert space obtained by using the Kahler polarization is naturally identifiable with the generalized Segal-Bargmann space introduced by the author from a different point of view, namely that of heat kernels. The second main result is that the pairing map of geometric quantization coincides with the generalized Segal-Bargmann transform introduced by the author. This means that in this case the pairing map is a constant multiple of a unitary map. For both results it is essential that the half-form correction be included when using the Kahler polarization. Together with results of the author with B. Driver, these results may be seen as an instance of "quantization commuting with reduction."Comment: Final version. To appear in Communications in Mathematical Physic

Six Challenges in Designing Equity-Based Pay

This paper analyzes why the primary goal of the equity-pay explosion--creating long-run ownership incentives for top executives--has often been difficult to achieve in practice. More generally, I describe six challenges in the design of equity-based pay plans and discuss potential solutions. The six challenges involve: 1. mismatched time horizons; 2. gaming; 3. the value-cost wedge'; 4. the leverage-fragility tradeoff; 5. aligning risk-taking incentives; and 6. avoiding excessive compensation. The paper also discussed the merits of stock versus options and concludes that restricted stock is often a superior form of compensation.