Explicit evaluation of the Stokes matrices for certain quantum confluent hypergeometric equations (Q6668499)

Denoting by \(\mathfrak{gl}_n\) the Lie algebra over the field of complex numbers, \(U(\mathfrak{gl}_n)\) its universal enveloping algebra generated by \((e_{ij})_{1\leq i,j\leq n}\) subject to the relation \([e_{ij},e_{k\ell}]=\delta_{jk}e_{i\ell}-\delta_{\ell i}e_{kj}\), and by \(T=(T_{ij})\) the \(n\times n\)-matrix with entries valued un \(U(\mathfrak{gl}_n)\): \(T_{ij}=e_{ij}\) for all \(i,j=1,...,n\), the authors are interested, for any finite-dimensional irreducible representation \(L(\lambda)\) of \(\mathfrak{gl}_n\) with a highest weight \(\lambda\), in the quantum confluent hypergeometric system \N\[\N\dfrac{dF}{dz}=h\left(iu+\dfrac{1}{2\pi i}\dfrac{T}{z}\right)F,\N\]\Nwhere\N\begin{itemize} \N\item\(F(z)\in\text{End}(L(\lambda))\otimes\text{End}(\mathbb C^n)\) is a \(n\times n\)-matrix function with entries in \(\text{End}(L(\lambda))\); \item\(h\) is a non-zero real parameter; \N\item\(u=\sum_{i=1}^{n}u_i\otimes E_{ii}\in U(\mathfrak{gl}_n)\otimes\text{End}(\mathbb C^n)\) with \(u_1,...,u_n\) complex parameters and \(E_{ii}\) the \(n\times n\)-matrix whose the \((i,i)\)-entry is \(1\) and other entries are \(0\).\N\end{itemize}\NSuch equation has a unique formal solution \(\widetilde{F}(z)\) around \(z=\infty\), and the resummation theory tells that there exist certain sectorial regions around \(z=\infty\), such that on each of these sectorial regions there is a unique (therefore canonical) holomorphic solution with the prescribed asymptotics \(\widetilde{F}(z)\). These solutions are in general different (that reflects the Stokes phenomenon), and the transition between them can be measured by a pair of Stokes matrices.\N\NThe aim of this paper is to give an explicit form of these Stokes matrices in the special case where \(u_1=...=u_{n-1}=0\) and \(u_n=1\), case which plays an important role from the perspective of isomonodromy deformation.