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Let be a semisimple linear algebraic group defined over an algebraically closed field . Fix a smooth projective curve defined over , and also fix a closed point . Given any strongly semistable principal -bundle over , we construct an affine algebraic group scheme defined over , which we call the monodromy of . The monodromy group scheme is a subgroup scheme of the fiber over of the adjoint bundle for . We also construct a reduction of structure group of the principal -bundle to its monodromy group scheme. The construction of this reduction of structure group involves a choice of a closed point of over . An application of the monodromy group scheme is given. We prove the existence of strongly stable principal -bundles with monodromy
We consider a model describing premixed combustion in the presence of fluid flow: a reaction-diffusion equation with passive advection and ignition-type nonlinearity. What kinds of velocity profiles are capable of quenching (suppressing) any given flame, provided the velocity's amplitude is adequately large? Even for shear flows, the solution turns out to be surprisingly subtle. In this article we provide a sharp characterization of quenching for shear flows; the flow can quench any initial data if and only if the velocity profile does not have an interval larger than a certain critical size where it is identically constant. The efficiency of quenching depends strongly on the geometry and scaling of the flow. We discuss the cases of slowly and quickly varying flows, proving rigorously scaling laws that have been observed earlier in numerical experiments. The results require new estimates on the behavior of the solutions to the advection-enhanced diffusion equation (also known as passive scalar in physical literature), a classical model describing a wealth of phenomena in nature. The technique involves probabilistic and partial-differential-equation (PDE) estimates, in particular, applications of Malliavin calculus and the central limit theorem for martingales
Let be the rational Cherednik algebra of type with spherical subalgebra . Then is filtered by order of differential operators with associated graded ring , where is the nth symmetric group. Using the -algebra construction from [GS], it is also possible to associate to a filtered - or -module a coherent sheaf on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of and , and we relate it to Hilb(n) and to the resolution of singularities . For example, we prove the following.
• If so that is the unique one-dimensional simple -module, then , where is the punctual Hilbert scheme.
• If for , then under a canonical filtration on the finite-dimensional module , has a natural bigraded structure that coincides with that on , where ; this confirms conjectures of Berest, Etingof, and Ginzburg [BEG2, Conjectures 7.2, 7.3].
• Under mild restrictions on , the characteristic cycle of equals , where are Kostka numbers and the are (known) irreducible components of
We prove Langlands functoriality for the generic spectrum of general spin groups (both odd and even). Contrary to other recent instances of functoriality, our resulting automorphic representations on the general linear group are not self-dual. Together with cases of classical groups, this completes the list of cases of split reductive groups whose -groups have classical derived groups. The important transfer from to follows from our result as a special case
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