Geometry and representations of Cherednik algebras and categories O
French National Research Agency (ANR)
Funder: French National Research Agency (ANR)Project code: ANR-10-BLAN-0110
Funder Contribution: 180,000 EUR
_x000D_ _x000D_ _x000D_ Considerable breakthroughs have occurred recently in representation theory. The general purpose of the project is to study the application of new tools to _x000D_ some classical subjects in this field : the category O of semi-simple complex Lie algebras (and their affine version) and the representations of (affine) Hecke algebras._x000D_ _x000D_ _x000D_ First, we'll be interested in double affine Hecke algebras (DAHA's) and the rational degenerations (called Cherednik algebras). The DAHA's are supposed to play a similar role, for higher representation theory, to that of affine Hecke algebras for p-adic groups. Little is known about their representations. Their simple_x000D_ modules can be described via the equivariant K-theory of the affine Steinberg variety. An important question is to understand better the affine Springer fibers and the DAHA's representations in their cohomology. Cherednik algebras have a nice category of representations, called again the category O. It is equipped with a remarkable functor to Hecke algebra representations, the Hecke functor. From that point of view they are important tools to study the latter._x000D_ Our goal is to understand better this category O. We'll focus in particular on the following points. Understand the finite dimensional modules and their relations_x000D_ with the radial parts of invariant differential operators. Understand the category O for cyclotomic Cherednik algebras and its relation with representations of affine Lie algebras and cyclotomic q-Schur algebras._x000D_ _x000D_ _x000D_ Next, we'll be interested in W-algebras. The affine W-algebras are indeed vertex algebra introduced some time ago in mathematical physic. The finite W-algebras have been introduced recently by Premet to study modular representations of Lie algebras. They are also efficient tools to understand the primitive ideals of the enveloping algebra of a semi-simple Lie algebra. Finally they yield a new way to understand the singular blocks of the parabolic category O. We would like to understand better the representation theory of W-algebras. We'll focus in particular on the possible relations of affine W-algebras with the parabolic category O of affine Lie algebras. This should enter in order to prove an equivalence of category between affine Lie algebra and_x000D_ cyclotomic q-Schur algebras, see above. We'll also be particularly interested in the geometry behind affine W-algebras, in particular in the possible relation with affine Springer fibers._x000D_ _x000D_ _x000D_ The third object consists of the KLR-algebras. They have been introduced one year ago to categorify some standard objects in representation theory_x000D_ (integrable modules of Kac-Moody algebras, canonical bases, modified quantized enveloping algebra). They are indeed very basic objects which can serve to understand the representations of all affine Hecke algebras of classical type. One goal of the project is to study the representation theory of KLR algebras to, hopefully, obtain new deep results on affine Hecke algebras and canonical bases.
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