In our earlier paper ([AKT1]), by interpreting the formal transformation to the Airy equation near a simple turning point as the symbol of a microdifferential operator, we derived the Voros connection formula or, equivalently, the discontinuity function of a Borel transformed WKB solution at its movable singularities. In this paper we extend this approach to the two turning points problem; by constructing the formal transformation which brings a Schrödinger equation with two paired simple turning points that merge (i.e., a merging-turning-points equation or an MTP equation for short) to the Weber equation and by interpreting it as the symbol of a microdifferential operator, we reduce the analysis of an MTP equation to that of the Weber equation. Then, combining this transformation theory with the so–called "Sato's conjecture" for the Weber equation, we obtain the discontinuity function of a Borel transformed WKB solution of an MTP equation at its fixed singularities.

We study a class of representations of the Lie algebra $\mathfrak{n} \otimes \mathbb{C} [t, t^{-1}]$, where $\mathfrak{n}$ is a nilpotent subalgebra of $\mathfrak{sl}_3$. We derive Weyl-type (bosonic) character formulas for these representations. We establish a connection between the bosonic formulas and the Whittaker vector in the Verma module for the quantum group $U_v (\mathfrak{sl}_3)$. We also obtain a fermionic formula for an eigenfunction of the $\mathfrak{sl}_3$ quantum Toda Hamiltonian.

A module over an affine Kac–Moody algebra $\widehat{\mathfrak{g}}$ is called spherical if the action of the Lie subalgebra $\mathfrak{g} [[t]]$ on it integrates to an algebraic action of the corresponding group $G [[t]]$. Consider the category of spherical $\widehat{\mathfrak{g}}$-modules of critical level. In this paper we prove that this category is equivalent to the category of quasi-coherent sheaves on the ind-scheme of opers on the punctured disc which are unramified as local systems. This result is a categorical version of the well-known description of spherical vectors in representations of groups over local non-archimedian fields. It may be viewed as a special case of the local geometric Langlands correspondence proposed in [FG2].

Let $D$ be the classifying space of mixed Hodge structures with polarized graded quotients. We construct a real analytic manifold with corners $D_{\mathrm{BS}}$ which contains $D$ as a dense open subset. This is the mixed Hodge theoretic version of the Borel–Serre space in the case of pure weight constructed by Borel–Ji and Kato–Usui.

We prove the algebraic version of a conjecture of C. Sabbah on the existence of the good formal structure for meromorphic flat connections on surfaces after some blow up.

In the first part of this paper we prove the multiplicative property of the relative Quillen Chern character. Then we obtain a Riemann-Roch formula between the relative Chern character of the Bott morphism and the relative Thom form.

We propose a definition of (polarized) wild twistor $\mathscr{D}$-modules, generalizing to objects with irregular singularities that of (polarized) regular twistor $\mathscr{D}$-modules. We give a precise analysis in dimension one.

We survey some recent developments in the theory of $b$-function, spectrum, and multiplier ideals together with certain interesting relations among them including the case of arbitrary subvarieties.