Pseudovarieties defining classes of sofic subshifts closed under taking shift equivalent subshifts

For a pseudovariety of ordered semigroups, let be the class of sofic subshifts whose syntactic semigroup lies in . It is proved that if contains then is closed under taking shift equivalent subshifts, and conversely, if is closed under taking conjugate subshifts then contains and . Almost finite type subshifts are characterized as the irreducible elements of , which gives a new proof that the class of almost finite type subshifts is closed under taking shift equivalent subshifts.     

Moduli of metaplectic bundles on curves and Theta-sheaves

We give a geometric interpretation of the Weil representation of the metaplectic group, placing it in the framework of the geometric Langlands program.For a smooth projective curve X we introduce an algebraic stack of metaplectic bundles on X. It also has a local version , which is a gerbe over the affine Grassmanian of G. We define a categorical version of the (nonramified) Hecke algebra of the metaplectic group. This is a category of certain perverse sheaves on , which act on by Hecke operators. A version of the Satake equivalence is proved describing as a tensor category. Further, we construct a perverse sheaf on corresponding to the Weil representation and show that it is a Hecke eigen-sheaf with respect to .     

Estimates of weighted Hardy-Littlewood averages on the p-adic vector space

In the p-adic vector space , we characterize those non-negative functions ψ defined on for which the weighted Hardy-Littlewood average is bounded on (1?r?∞), and on . Also, in each case, we find the corresponding operator norm ‖Uψ‖.