We will discuss how to realize categories of highest weight representations of affine W-algebras as categories of perverse sheaves on affine flag manifolds, and the modules for affine Hecke algebras which they categorify. Applications, e.g. to character formulas and the Kac-Roan-Wakimoto conjecture, will be discussed. For principal nilpotent elements, this was worked out jointly with Raskin, and the general case is work in progress with Arakawa.
We realize the quantum loop groups and shifted quantum loop groups of arbitrary types, possibly non-symmetric, using critical K-theory. This gives a generalization of Nakajima’s construction of symmetric quantum loop groups via quiver varieties to non-symmetric types. This also yields a geometric realization of some simple modules, in particular the Kirillov-Reshethikin modules, and the tensor product of prefundamental modules.
The Steinberg variety and the equivariant coherent sheaves on it play a very important role in Geometric Representation Theory. In this talk we will discuss various t-structures on the equivariant derived category of the Steinberg of importance for Representation Theory in zero and positive characteristics.
In 2015, Brosnan and Chow, and independently Guay-Paquet, proved the Shareshian-Wachs conjecture, which links the combinatorics of chromatic symmetric functions to the geometry of Hessenberg varieties via a permutation group action on the cohomology ring of regular semisimple Hessenberg varieties. This talk will give a brief overview of that story and discuss how the dot action can be computed in all Lie types using the Betti numbers of certain nilpotent Hessenberg varieties. As an application, we obtain new geometric insight into certain linear relations satisfied by chromatic symmetric functions, known as the modular law.
Symplectic duality predicts that symplectic singularities should come in pairs. For example, Nakajima quiver varieties are conjecturally dual to BFN Coulomb branches (of the corresponding quiver theories). Another family of potentially symplectically dual pairs was described recently in the works of Losev, Mason-Brown, and Matvieievskyi: they describe symplectically duals to Słodowy slices to nilpotent orbits.
In this talk, we will discuss the Hikita-Nakajima conjecture that relates the geometry of symplectically dual varieties. It turns out that the conjecture is very likely to hold for quiver varieties (as was predicted by Nakajima) but does not quite hold for Słodowy slices and arbitrary Higgs branches. We will explain certain simplification of this conjecture that may work in general. We will discuss a possible approach toward the proof of this conjecture. The approach is highly based on the ideas of Bellamy, Braverman, Kamnitzer, Losev, Tingley, Webster, Weekes, Yacobi, and their co-authors.
We will illustrate the approach on the examples of ADHM space (for which Hikita-Nakajima conjecture is true as stated) and for certain Słodowy varieties.
The geometric construction of DAHA by the equivariant K-theory of the Steinberg-type variety for an affine flag variety by Vasserot, Varagnolo-Vasserot is a precursor of the Coulomb branch construction. It has been generalized to versions of DAHA, such as cyclotomic DAHA naturally in view of Coulomb branches. I would like to recall these results, and then add one new example which seems not be known before.
We introduce a generalization of K-k-Schur functions and k-Schur functions via the Pieri rule. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule is described explicitly in the cases of K-k-Schur functions and k-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for k-Schur functions, and explains it as a degeneration of the rule for K-k-Schur functions. In particular, many other special cases promise to be detailed in the future.
In this talk, I will discuss a virtual variant of the quantized Coulomb branch constructed by Braverman-Finkelberg-Nakajima, where the convolution product is modified by a virtual intersection. The resulting virtual Coulomb branch acts on the moduli space of
quasimaps into the holomorphic symplectic quotient T *N///G. When G is abelian, over the torus fixed points, this representation is a Verma module. The vertex function, a K-theoretic enumerative invariant introduced by A. Okounkov, can be expressed as a Whittaker function of the algebra. The construction also provides a description of the quantum q-difference module. As an application, this gives a proof of the invariance of the quantum q-difference module under the variation of GIT.
Motivated by recent works joint with Tomoyuki Arakawa and Jethro Van Ekeren on collapsing levels, we conjectured that if W is a finite extension of a vertex subalgebra V, then the natural morphism between the corresponding associated varieties is dominant. In the case where W is a simple W-algebra and V is its simple affine vertex algebra, the conjecture is deeply related with the singularities of nilpotent Słodowy slices. In this talk, I will explain some results toward the conjecture and interesting examples.
I will present some recent works with Gunningham, Safronov, Vazi-rani, and Yang (in various combinations) and which compute GLN-, SLN- and PGLN-skein modules for the 3-torus T 3, and related work of Kinnear which generalizes this to mapping tori T 2 ×γ S1, for γ ∈ SL2(ℤ).
The proofs for GLN and SLN start with a description of the skein category of T 2 via the representation theory of double affine Hecke algebras, while for PGLN they rely on an instance of electric-magnetic duality.
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