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The phenomenon seems the same as for affine Kac-Moody algebras, where rational level is where everything special happens, or quantum groups, where roots of unity are the exceptional locus (and of cour …

Up to the central extension (which doesn't affect the classification) this is a special case of the classification of Cartan subgroups of a reductive group over a field. Since they all split after an …

I don't know the answer to your question, but maybe one can venture to guess that the Beilinson-Bernstein localization picture carries over to this setting? (for the enveloping algebra this is done by …

The equivalence (or something very close, I haven't checked carefully what is written) follows from Beilinson-Bernstein localization. The two categories can be realized roughly speaking as D-modules o …

Some of these questions are addressed (in the derived setting) in my paper Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry with John Francis and David Nadler --- for the underiv …

It's probably relevant to note that the translation functors are shadows of "obvious" functors: the categories of $\lambda$- and $\lambda+\nu$-twisted $D$-modules on the flag varieties are equivalent …

[Edited to reflect Reimundo's comment] The question addresses categorified versions of the Borel-Weil-Bott theorem (and more generally Beilinson-Bernstein localization), which states an equivalence b …

This is a wonderful and highly suggestive question. In addition to the fascinating hints provided by Stephen's answer, there are also reasons from geometric representation theory (as well as from phys …

I think it's very far from true, and misleading to state too informally, that every representation is highest weight. For example if we think of representations (with a fixed infinitesimal character) …

For context for Tom's answer, let me state the naive version of the conjecture, which has been around since around 1997 I think (due to Beilinson-Drinfeld). It calls for an equivalence of (dg) catego …

From the point of view of geometry, the crucial fact about $\rho$ is that the corresponding line bundle on the flag manifold is (upt to a sign) a (the) square-root of the canonical bundle (top exteri …

Well the first thing to say is to look at the very enthusiastic and world-encompassing papers of Cherednik himself on DAHA as the center of the mathematical world (say his 1998 ICM). I'll mention a c …

The de Rham space of a scheme is essentially never a scheme or algebraic space (unless I guess you're Spec of an Artin ring, in which case you'll get a discrete set of points). In particular this appl …

The $K_2$ functor has an important role in representation theory, starting from works of Bloch in which he identified the Kac-Moody central extension of loop algebras in terms of $K_2$ - a more updat …

I think Thiem's thesis Unipotent Hecke algebras of GL_n(F_q) discusses this in detail -- if I'm not mistaken the Hecke algebra you're asking about goes by the name Yokonuma Hecke algebra and there's a …