Related papers: Motivic Springer Theory
A key tool for the study of an affine Hecke algebra $\mathcal{H}$ is provided by Springer theory of the Langlands dual group via the realization of $\mathcal{H}$ as equivariant $K$-theory of the Steinberg variety. We prove a similar…
We show that the motive of a Springer fiber is pure Tate. We then consider a category of equivariant Springer motives on the nilpotent cone and construct an equivalence to the derived category of graded modules over the graded affine Hecke…
We construct a geometric realization of categories of representations of affine Hecke algebras and split reductive $p$-adic groups via a $K$-motivic Springer theory. We suggest a connection to the coherent Springer theory of Ben-Zvi, Chen,…
This is a survey paper about affine Hecke algebras. We start from scratch and discuss some algebraic aspects of their representation theory, referring to the literature for proofs. We aim in particular at the classification of irreducible…
We give an explicit expression for the central elements of affine Hecke algebras of type A in the Coxeter presentation, in terms of (parabolic) affine Kazhdan-Lusztig polynomials. Our approach is based on a version of quantum affine…
The Hecke algebras and quantum group of affine type A admit geometric realizations in terms of complete flags and partial flags over a local field, respectively. Subsequently, it is demonstrated that the quantum group associated to partial…
We construct a bijection between admissible representations for an affine Lie algebra $\mathfrak{g}$ at boundary admissible levels and $\mathbb{C}^\times$ fixed points in homogeneous elliptic affine Springer fibres for the Langlands dual…
Let G be a split semisimple linear algebraic group over a field k0. Let E be a G-torsor over a field extension k of k0. Let h be an algebraic oriented cohomology theory in the sense of Levine-Morel. Consider a twisted form E/B of the…
We prove a conjecture by Lusztig, which describes the tensor categories of perverse sheaves on affine flag manifolds, with tensor structure provided by truncated convolution, in terms of the Langlands dual group. We also give a geometric…
For any formal group law, there is a formal affine Hecke algebra defined by Hoffnung, Malag\'on-L\'opez, Savage, and Zainoulline. Coming from this formal group law, there is also an oriented cohomology theory. We identify the formal affine…
This is not standard in the sense that we understand a Springer map to be a collapsing of homogeneous bundles. Apart from that we use mostly techniques from Chriss and Ginzbergs book but we work in the equivariant derived category of…
An affine Hecke algebras can be realized as an equivariant K-group of the corresponding Steinberg variety. This gives rise naturally to some two-sided ideals of the affine Hecke algebra by means of the closures of nilpotent orbits of the…
We use the Hecke algebras of affine symmetric groups and their associated Schur algebras to construct a new algebra through a basis, and a set of generators and explicit multiplication formulas of basis elements by generators. We prove that…
We present simple conditions which guarantee a geometric convolution algebra to behave like a variant of the quasi-hereditary algebra. In particular, standard modules of the affine Hecke algebras of type $\mathsf{BC}$, and the quiver Schur…
The flag variety of a complex reductive linear algebraic group G is by definition the quotient G/B by a Borel subgroup. It can be regarded as the set of Borel subalgebras of Lie(G). Given a nilpotent element e in Lie(G), one calls Springer…
After establishing a geometric Schur-Weyl duality in a general setting, we recall this duality in type A in the finite and affine case. We extend the duality in the affine case to positive parts of the affine algebras. The positive parts…
We study basic geometric properties of some group analogue of affine Springer fibers and compare with the classical Lie algebra affine Springer fibers. The main purpose is to formulate a conjecture that relates the number of irreducible…
We develop a notion of exponential motives on general prestacks equipped with a $\mathbf{G}_a$-action, and compare them with Whittaker motives via Gaitsgory's Kirillov model. We then establish foundational results for exponential motives on…
We relate two apparently different bases in the representations of affine Lie algebras of type A: one arising from statistical mechanics, the other from gauge theory. We show that the two are governed by the same combinatorics and therefore…
In this paper we provide a "combinatorial" description of the category of tilting perverse sheaves on the affine flag variety of a reductive algebraic group, and its free-monodromic variant, with coefficients in a field of positive…