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Register a new userFlat G-bundles and regular strata for reductive groups
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Let LG be an algebraic loop group associated to a reductive group G. A fundamental stratum is a triple consisting of a point x in the Bruhat-Tits building of LG, a nonnegative real number r, and a character of the corresponding depth r Moy-Prasad subgroup that satisfies a non-degeneracy condition. The authors have shown in previous work how to associate a fundamental stratum to a formal flat G-bundle and used this theory to define its slope. In this paper, the authors study fundamental strata that satisfy an additional regular semisimplicity condition. Flat G-bundles that contain regular strata have a natural reduction of structure to a (not necessarily split) maximal torus in LG, and the authors use this property to compute the corresponding moduli spaces. This theory generalizes a natural condition on algebraic connections (the GL_n case), which plays an important role in the global analysis of meromorphic connections and isomonodromic deformations.
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Let G be a split reductive group. We introduce the moduli problem of bundle chains parametrizing framed principal G-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of G. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple G, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev-Manins spaces of weighted pointed curves and with Kauszs compactification of GL(n).
The theory of minimal K-types for p-adic reductive groups was developed in part to classify irreducible admissible representations with wild ramification. An important observation was that minimal K-types associated to such representations correspond to fundamental strata. These latter objects are triples (x, r, beta), where x is a point in the Bruhat-Tits building of the reductive group G, r is a nonnegative real number, and beta is a semistable functional on the degree r associated graded piece of the Moy-Prasad filtration corresponding to x. Recent work on the wild ramification case of the geometric Langlands conjectures suggests that fundamental strata also play a role in the geometric setting. In this paper, we develop a theory of minimal K-types for formal flat G-bundles. We show that any formal flat G-bundle contains a fundamental stratum; moreover, all such strata have the same rational depth. We thus obtain a new invariant of a flat G-bundle called the slope, generalizing the classical definition for flat connections. The slope can also be realized as the minimum depth of a stratum contained in the flat G-bundle, and in the case of positive slope, all such minimal depth strata are fundamental. Finally, we show that a flat G-bundle is irregular singular if and only if it has positive slope.
This paper generalizes the classical theory of Newton polygons from the case of general linear groups to the case of split reductive groups. It also gives a root-theoretic formula for dimensions of Newton strata in the adjoint quotients of reductive groups.
We construct the Frobenius structure on a rigid connection $mathrm{Be}_{check{G}}$ on $mathbb{G}_m$ for a split reductive group $check{G}$ introduced by Frenkel-Gross. These data form a $check{G}$-valued overconvergent $F$-isocrystal $mathrm{Be}_{check{G}}^{dagger}$ on $mathbb{G}_{m,mathbb{F}_p}$, which is the $p$-adic companion of the Kloosterman $check{G}$-local system $mathrm{Kl}_{check{G}}$ constructed by Heinloth-Ng^o-Yun. By exploring the structure of the underlying differential equation, we calculate the monodromy group of $mathrm{Be}_{check{G}}^{dagger}$ when $check{G}$ is almost simple (which recovers the calculation of monodromy group of $mathrm{Kl}_{check{G}}$ due to Katz and Heinloth-Ng^o-Yun), and establish functoriality between different Kloosterman $check{G}$-local systems as conjectured by Heinloth-Ng^o-Yun. We show that the Frobenius Newton polygons of $mathrm{Kl}_{check{G}}$ are generically ordinary for every $check{G}$ and are everywhere ordinary on $|mathbb{G}_{m,mathbb{F}_p}|$ when $check{G}$ is classical or $G_2$.
In this paper, we study the analytic continuation to complex time of the Hamiltonian flow of certain $Gtimes T$-invariant functions on the cotangent bundle of a compact connected Lie group $G$ with maximal torus $T$. Namely, we will take the Hamiltonian flows of one $Gtimes G$-invariant function, $h$, and one $Gtimes T$-invariant function, $f$. Acting with these complex time Hamiltonian flows on $Gtimes G$-invariant Kahler structures gives new $Gtimes T$-invariant, but not $Gtimes G$-invariant, Kahler structures on $T^*G$. We study the Hilbert spaces ${mathcal H}_{tau,sigma}$ corresponding to the quantization of $T^*G$ with respect to these non-invariant Kahler structures. On the other hand, by taking the vertical Schrodinger polarization as a starting point, the above $Gtimes T$-invariant Hamiltonian flows also generate families of mixed polarizations $mathcal{P}_{0,sigma}, sigma in {mathbb C}, {rm Im}(sigma) >0$. Each of these mixed polarizations is globally given by a direct sum of an integrable real distribution and of a complex distribution that defines a Kahler structure on the leaves of a foliation of $T^*G$. The geometric quantization of $T^*G$ with respect to these mixed polarizations gives rise to unitary partial coherent state transforms, corresponding to KSH maps as defined in [KMN1,KMN2].
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