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Let me parse your question on three levels. First, there's an obvious counterpart to W-algebras in the p-adic world that you allude to. Namely, "W-algebras = endomorphisms of Whittaker models". More p …

Ryan Reich has a paper called Notes on Beilinson's "How to glue perverse sheaves" -- it's also available off of Dennis Gaitsgory's Geometric Representation Theory page which is an amazing resource i …

The videos from the LMS lectures and all of the GRASP videos are now available again from the links you gave (for download, not streaming). Many apologies for their long hiatus offline and many thanks …

I'm not close to familiar enough with the references you cite or examples you ask about to address them, but here's a picture coming out of Sakellaridis and Venkatesh [SV]. Let us label a period not b …

For Question 1 I agree with dhy that Namikawa's work is the relevant place to start, and from there the booming field of symplectic representation theory, which from one perspective is all about gener …

I don't know of a DAG mechanism that implies these statements, any more formally than the standard proofs -- i.e., we reduce to a maximal torus, where the statement follows from the shape of the cohom …

[This is an elaboration of parts of Mark Penney's answer] A natural source of categorical actions of the mapping class group is the category assigned by any 4d TFT to a surface. Such categories are o …

Local spaces:finite subschemes::Factorization spaces:finite subsets. A local space over X is a compatible collection of spaces over the Hilbert schemes of arbitrary numbers of points in X satisfying …

To elaborate on Kevin's excellent answer, one can account for the current absence of "higher loop" representation theory using physics. Namely, all of the representation theoretic structures you menti …

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 …

It appears this question is resolved in a definitive fashion in today's preprint Algebraic Groups and compact generation of their derived categories of representations by Hall and Rydh. Their first th …

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 …

OK, this is a very broad question so I'll be telegraphic. There is a sequence of increasingly detailed conjectures going by the name GL -- it's really a "program" (harmonic analysis of $\mathcal{D}$-m …

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 …

One natural generalization is the center of the group algebra (i.e., algebra of class functions - so in the abelian case you consider, just the group algebra itself). In the continuous case there are …