Representations of quantum groups arising from the Stokes phenomenon and applications

Abstract

In this paper, we show that the quantum Stokes matrices, of certain meromorphic linear system of ordinary differential equations, give rise to a family of Drinfeld isomorphisms from quantum groups to the undeformed universal enveloping algebra of gl(n). In particular, we compute explicitly the Drinfeld isomorphisms corresponding to caterpillar points on the parameter space. Our computation unveils a relation between the asymptotics of confluent hypergeometric functions and the Gelfand-Zeitlin subalgebras.As by products, we show that the Drinfeld isomorphisms at caterpillar points coincide with the Appel-Gautam isomorphisms. Then by going to the semiclassical limit, we place the results in this paper into the context of Poisson geometry. As an application, we prove the conjecture of Appel and Gautam, i.e., their isomorphisms are canonical quantization of the Alekseev-Meinrenken diffeomorphisms arising from the linearization problem in Poisson geometry.

Crystals arising from Stokes phenomenon

“…Here a gl n -crystal is a finite set which models a weight basis for a representation of gl n , and crystal operators indicate the leading order behaviour of the simple root vectors on the basis under the crystal limit q → 0 in quantum group U q (gl n ). From this perspective, the above theorem has the following interpretation: in [18] we proved that the entries s…”

“…• the set B(λ, u) of vectors ξ d for all possible d is a base of L(λ) with a natural parameterization; In this paper, we first prove the statement for u being a special/limit point, as a technical preliminary of the general u cases. In [17,18], using the theory of isomonodromy deformation [7], we studied the renormalized limit of S ± (u) at an "infinite" point u = diag(u 1 , ..., u n ) with u 1 ≪ • • • ≪ u n . The infinite point, called a caterpillar point u cat , belongs to the 0-dimensional stratum of the de Concini-Procesi wonderful compactification h reg (R) (see e.g., [9]) of h reg (R).…”

“…The Stokes matrices S ± (u, h), which measure the difference of the asymptotics of these solutions as z → ∞ within different Stokes sectors, can be expressed under the eigenbasis Ei(u). In particular, in [18] we computed the Stokes matrices at a caterpillar point under the eigenbasis Ei(u cat ) of A(u cat ). In this case, Ei(u cat ) is just Gelfand-Tsetlin basis, and the integral representations for the solutions give rise to confluent hypergeometric functions, see [18].…”

“…In particular, in [18] we computed the Stokes matrices at a caterpillar point under the eigenbasis Ei(u cat ) of A(u cat ). In this case, Ei(u cat ) is just Gelfand-Tsetlin basis, and the integral representations for the solutions give rise to confluent hypergeometric functions, see [18]. The rest is to use the integral expression to prove that the WKB datum (E(λ, u), ẽk (u), fk (u)) continuously depend on u ∈ h reg (R).…”

“…Thus without losing much information, we can simply study the corresponding problem "at a caterpillar point". See [18] for more details and examples.…”