Regularized limits of Stokes matrices, isomonodromy deformation and crystal basis

Abstract

In the first part of the paper, we solve the boundary and monodromy problems for the isomonodromy equation of the $n\times n$ meromorphic linear system of ordinary differential equations with Poncar\'{e} rank $1$. In particular, we derive an explicit expression of the Stokes matrices of the linear system, via the boundary value of the solutions of the isomonodromy equation at a critical point. Motivated by this result, we then describe the regularized limits of Stokes matrices as the irregular data $u={\rm diag}(u_1,...,u_n)$ in the linear system degenerates, i.e., as some $u_i, u_j,...,u_k$ collapse. The prescription of the regularized limit is controlled by the geometry of the De Concini-Procesi wonderful compactification space. As applications, many analysis problems about higher rank Painlev\'e transcendents can be solved. In the second part of the paper, we show some important applications of the above analysis results in representation theory and Poisson geometry: we obtain the first transcendental realization of crystals in representations of $\frak{gl}_n$ via the Stokes phenomenon in the WKB approximation; we develop a wall-crossing formula that characterizes the discontinuous jump of the regularized limits of Stokes matrices as crossing walls in the compactification space, and interpret the known cactus group actions on crystals arising from representation theory as a wall-crossing phenomenon; and we find the first explicit linearization of the standard dual Poisson Lie group for $U(n)$.