In quantum geometric Langlands, the Satake equivalence plays a less prominent role than in classical theory. Gaitsgory--Lurie proposed conjectural substitute, later termed fundamental local equivalence. With few exceptions, we prove this conjecture and its extension to affine flag variety by using what amount Soergel module techniques.

We use a result of van Geemen [vG4] to determine the endomorphism algebra of the Kuga–Satake variety of a K3 surface with real multiplication. This is applied to prove the Hodge conjecture for self-products of double covers of P which are ramified along six lines.

We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalization of the Satake isomorphism for these algebras, relating it to integrable representations of the Langlands dual affine Kac-Moody group. In the next publication we shall use these results to define and study the notion of Hecke eigenfunction for the group Ga...

We determine several necessary and sufficient conditions for a closed almost-complex orbifold Q with abelian local groups to admit a nonvanishing vector field. These conditions are stated separately in terms of the orbifold Euler-Satake characteristics of Q and the connected components of its twisted sectors, the Euler characteristics of the underlying topological spaces of Q and the components...

We determine several necessary and sufficient conditions for a closed almost-complex orbifold Q to admit a nonvanishing vector field. These conditions are stated separately in terms of the orbifold Euler-Satake characteristics of Q and the connected components of its twisted sectors, the Euler characteristics of the underlying topological spaces of Q and the components of its twisted sectors, a...

A bstract We provide, through the framework of extended geometry, a geometrisation duality symmetries appearing in magical supergravities. new ingredient is general formulation geometry with structure group non-split real form. simple diagrammatic rule for solving section constraint by inspection Satake diagram derived.