32 research outputs found
On the formal affine Hecke algebra
We study the action of the formal affine Hecke algebra on the formal group
algebra, and show that the the formal affine Hecke algebra has a basis indexed
by the Weyl group as a module over the formal group algebra. We also define a
concept called the normal formal group law, which we use to simplify the
relations of the generators of the formal affine Demazure algebra and the
formal affine Hecke algebra.Comment: v2: This is an essentially extended version from the previous one.
Note the title and abstract are changed. v3: The proof of the main theorem is
clarified and section 5,6 are simplifie
Elliptic affine Hecke algebras and their representations
We apply equivariant elliptic cohomology to the Steinberg variety in Springer
theory, and prove that the corresponding convolution algebra is isomorphic to
the elliptic affine Hecke algebra constructed by Ginzburg-Kapranov-Vasserot.
Under this isomorphism, we describe explicitly the cohomology classes that
correspond to the elliptic Demazure-Lusztig operators. As an application, we
study the Deligne-Langlands theory in the elliptic setting, and classify
irreducible representations of the elliptic affine Hecke algebra. The
irreducible representations are in one to one correspondence with certain
nilpotent Higgs bundles on the elliptic curve. We also study representations at
torsion points in type-.Comment: V2: 40 pages; Section 6.5 added; minor revisions throughou
Geometric representations of the formal affine Hecke algebra
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 Hecke algebra with a convolution algebra coming from the oriented
cohomology theory applied to the Steinberg variety. As a consequence, this
algebra acts on the corresponding cohomology of the Springer fibers. This
generalizes the action of classical affine Hecke algebra on the -theory of
the Springer fibers constructed by Lusztig. We also give a residue
interpretation of the formal affine Hecke algebra, which coincides with the
residue construction of Ginzburg, Kapranov, and Vasserot when the formal group
law comes from a 1-dimensional algebraic group.Comment: v2 32 pages, significant modification
Stable bases of the Springer resolution and representation theory
In this note, we collect basic facts about Maulik and Okounkov's stable bases
for the Springer resolution, focusing on their relations to representations of
Lie algebras over complex numbers and algebraically closed positive
characteristic fields, and of the Langlands dual group over non-Archimedean
local fields.Comment: Expository article for the `International Festival in Schubert
Calculus', Guangzhou, China, 201
On the \gamma-filtration of oriented cohomology of complete spin-flags
We study the characteristic map of algebraic oriented cohomology of complete
spin-flags and the ideal of invariants of formal group algebra. As an
application, we provide an annihilator of the torsion part of the
-filtration. Moreover, if the formal group law determined by the
oriented cohomology is congruent to the additive formal group law modulo 2,
then at degree 2 and 3, the -filtration of complete spin-flags is
torsion free.Comment: 15 pages, some minor change
Wall-crossings and a categorification of -theory stable bases of the Springer resolution
We compare the -theory stable bases of the Springer resolution associated
to different affine Weyl alcoves. We prove that (up to relabelling) the change
of alcoves operators are given by the Demazure-Lusztig operators in the affine
Hecke algebra. We then show that these bases are categorified by the Verma
modules of the Lie algebra, under the localization of Lie algebras in positive
characteristic of Bezrukavnikov, Mirkovi\'c, and Rumynin. As an application, we
prove that the wall-crossing matrices of the -theory stable bases coincide
with the monodromy matrices of the quantum cohomology of the Springer
resolution.Comment: V2: 31 pages; exposition changes throughou
Milnor-Witt -groups of local rings
We introduce Milnor-Witt -groups of local rings and show that the th
Milnor-Witt -group of a local ring which contains an infinite field of
characteristic not is the pull-back of the th power of the fundamental
ideal in the Witt ring of and the th Milnor -group of over the
th Milnor -group of modulo . This generalizes the work of
Morel-Hopkins on Milnor-Witt -groups of fields
On the K-theory stable bases of the Springer resolution
Cohomological and K-theoretic stable bases originated from the study of
quantum cohomology and quantum K-theory. Restriction formula for cohomological
stable bases played an important role in computing the quantum connection of
cotangent bundle of partial flag varieties. In this paper we study the
K-theoretic stable bases of cotangent bundles of flag varieties. We describe
these bases in terms of the action of the affine Hecke algebra and the twisted
group algebra of Kostant-Kumar. Using this algebraic description and the method
of root polynomials, we give a restriction formula of the stable bases. We
apply it to obtain the restriction formula for partial flag varieties. We also
build a relation between the stable basis and the Casselman basis in the
principal series representations of the Langlands dual group. As an
application, we give a closed formula for the transition matrix between
Casselman basis and the characteristic functions.Comment: V2: 43 pages; Section 7 in V1 removed; Section 8 significantly
revised; minor revisions throughou
Formal affine Demazure and Hecke algebras of Kac-Moody root systems
We define the formal affine Demazure algebra and formal affine Hecke algebra
associated to a Kac-Moody root system. We prove the structure theorems of these
algebras, hence, extending several result and construction (presentation in
terms of generators and relations, coproduct and product structures, filtration
by codimension of Bott-Samelson classes, root polynomials and multiplication
formulas) that were previously known for finite root system.Comment: 19 page
Invariants, exponents and formal group laws
Let W be the Weyl group of a crystallographic root system acting on the
associated weight lattice by reflections. In the present notes we extend the
notion of an exponent of the W-action introduced in [Baek-Neher-Zainoulline,
arXiv:1106.4332] to the context of an arbitrary algebraic oriented cohomology
theory of Levine-Morel, Panin-Smirnov and the associated formal group law. From
this point of view the classical Dynkin index of the associated Lie algebra
will be the second exponent of the deformation map from the multiplicative to
the additive formal group law. We apply this generalized exponent to study the
torsion part of an arbitrary oriented cohomology theory of a twisted flag
variety.Comment: 15 pages, the last section is revise