1,722,312 research outputs found
A remark on rational Cherednik algebras and differential operators on the cyclic quiver
Get PDFWe show that the spherical subalgebra Uk,c of the rational Cherednik algebra associated to Sn 2 Cl, the wreath product of the symmetric group and the cyclic group of order l, is isomorphic to a quotient of the ring of invariant differential operators on a space of representations of the cyclic quiver of size l. This confirms a version of [5, Conjecture 11.22] in the case of cyclic groups. The proof is a straightforward application of work of Oblomkov [12] on the deformed Harish–Chandra homomorphism, and of Crawley–Boevey, [3] and [4], and Gan and Ginzburg [7] on preprojective algebras
Macdonald difference operators and Harish-Chandra series
No full textWe analyse the centralizer of the Macdonald difference operator in an appropriate algebra of Weyl group invariant difference operators. We show that it coincides with Cherednik's commuting algebra of difference operators via an analog of the Harish-Chandra isomorphism. Analogs of Harish-Chandra series are defined and realized as solutions to the system of basic hypergeometric difference equations associated to the centralizer algebra. These Harish-Chandra series are then related to both Macdonald polynomials and Chalykh's Baker-Akhiezer functions
On the categories of Harish-Chandra modules
Get PDFGiven an algebra A, the study of the category of A-modules is often restricted to the study of
a more tractable subcategory of A-modules which are well-behaved with respect to a subalgebra Γ
of A. In 1973, Lepowsky and McCollum studied the category of A-modules which are weight
modules with respect to Γ. Later, in 1994, Drozd, Futorny, and Ovsienko studied the category of
A-modules which are generalized weight modules with respect to Γ, called Harish-Chandra
modules. They showed that the structure of Harish-Chandra modules can be described using
information about the relationship between A and the cofinite maximal ideals of Γ.
In Chapter 2 of this thesis, we generalize the framework of Drozd, Futorny, and Ovsienko by
introducing an equivalence relation ∼ on the set cfs(Γ) of cofinite maximal ideals of Γ. We define
Harish-Chandra block modules with respect to ∼ to be A-modules that are the direct sum of block
spaces corresponding to the equivalence classes cfs(Γ)/∼. We describe the structure of
Harish-Chandra block modules based on the relationship between A and cfs(Γ)/∼. In particular,
we give decompositions of the category of Harish-Chandra block modules and the collection of
isomorphism classes of simple Harish-Chandra block modules. We also define a topologically
enriched category A with objects cfs(Γ)/∼, and show the category of Harish-Chandra block
modules is equivalent to a category of continuous A-modules. Furthermore, we provide a
sufficient condition for when there are a finite number of isomorphism classes of simple
Harish-Chandra block modules with a given support.
In Chapter 3, we continue the study of (generalized) Harish-Chandra modules. We observe
that the equivalence of categories obtained in Chapter 2 is closely related to a Yoneda embedding.
We begin to define a Harish-Chandra pair as a pair (A, B) where A is an algebra, and B is a full
subcategory of AMod. We review some potential examples, and identify the presence of a left adjoint to the Yoneda embedding Y : AMod → [Bop, Vectk] as a key property. In many cases, we
can describe the subcategories on which this adjunction is an equivalence
Harish-Chandra bimodules over rational Cherednik algebras
Get PDFWe study Harish-Chandra bimodules over the rational Cherednik algebra associated to a complex reflection group with parameter . Our results allow us to partially reduce the study of these bimodules to smaller algebras. We classify those pairs of parameters for which there exist fully supported Harish-Chandra bimodules, and give a description of the category of all Harish-Chandra bimodules modulo those without full support. When is a symmetric group we are able to classify all irreducible Harish-Chandra bimodules. Our proofs are based on localization techniques, the action of the Namikawa-Weyl group on the set of parameters, and the study of partial KZ functors.15 pages, preliminary version; v2 28 pages, major changes; v3 27 pages, significant changes in Section 5; v4 38 pages, more changes in Section 5. Final versio
On the Category of Harish-Chandra Block Modules
Get PDFIf is a subalgebra of , then an -module is called a
Harish-Chandra module if it is the direct sum of its generalized weight spaces
with respect to . In 1994, Drozd, Futorny, and Ovsienko defined a
generalization of a central subalgebra called a Harish-Chandra subalgebra and
showed that when is a Harish-Chandra subalgebra of the structure
of Harish-Chandra -modules can be described using information about the
relationship between and the cofinite maximal ideals of .
We extend these results by dropping the assumption that is
quasicommutative. We facilitate this by introducing an equivalence relation
on the set of cofinite maximal ideals of
. We define Harish-Chandra block modules with respect to to be
-modules that are the direct sum of so called block spaces corresponding to
the equivalence classes . If is a
Harish-Chandra block subalgebra of with respect to , then the
structure of Harish-Chandra block modules can be described based on the
relationship between and . In particular, we
give a decomposition of the category of Harish-Chandra block modules and the
collection of isomorphism classes of irreducible Harish-Chandra block modules.
Furthermore, we define a category on
, and show the category of profinite
-modules is equivalent to the category of Harish-Chandra block
modules. Taking to be noetherian and quasicommutative, and to
be the equality relation, we recover (in fact, a slight refinement of) results
from Drozd, Futorny, and Ovsienko. Lastly, we provide a sufficient condition
for when there are a finite number of isoclasses of simple Harish-Chandra block
modules with a given support
Inductive local-global conditions and generalised Harish-Chandra theory
No full textWe study new properties of generalised Harish-Chandra theory aiming at explaining the inductive local-global conditions for finite groups of Lie type in nondefining characteristic. In particular, we consider a parametrisation of generalised Harish-Chandra series that is compatible with Clifford theory and with the action of automorphisms on irreducible characters and we reduce it to the verification of certain requirements on stabilisers and extendibility of characters. This parametrisation is used by the author in a separate paper to obtain new conjectures for finite reductive groups that can be seen as geometric realisations of the local-global counting conjectures and their inductive conditions. As a by-product, we extend the parametrisation of generalised Harish-Chandra series given by Broué–Malle–Michel to the nonunipotent case by assuming maximal extendibility
Harish-Chandra cuspidal pairs
Get PDFThe irreducible characters of a finite reductive group are partitioned into Harish-Chandra series that are labelled by cuspidal pairs. In this note, we describe how one can algorithmically calculate those cuspidal pairs using results of Lusztig.<br/
Inverse Harish-Chandra Transform and Difference Operators
Get PDFIn the paper we calculate the images of the operators of multiplication by Laurent polynomials with respect to the Harish-Chandra transform and its non-symmetric generalization due to Opdam. It readily leads to a new simple proof of the Harish-Chandra inversion theorem in the zonal case (see [HC,He1]) and the corresponding theorem from [O1]. We assume that k > 0 and restrict ourselves to compactly supported functions, borrowing the growth estimates from [O1]
Unitary Harish-Chandra representations of real supergroups
Get PDFWe give conditions for unitarizability of Harish-Chandra super
modules for Lie supergroups and superalgebra
Affine Jacquet functors and Harish-Chandra categories
No full textAbstractWe define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan on the structure of Verma modules in the Bernstein–Gelfand–Gelfand categories O for Kac–Moody algebras. This is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine Harish-Chandra modules of Lian and Zuckerman. The latter provides an extension of the works of Rocha-Caridi and Wallach [A. Rocha-Caridi, N.R. Wallach, Projective modules over infinite dimensional graded Lie algebras, Math. Z. 180 (1982) 151–177] and Deodhar, Gabber and Kac [V. Deodhar, O. Gabber, V. Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. Math. 45 (1982) 92–116] on block decompositions of BGG categories for Kac–Moody algebras. We also derive a compatibility relation between the affine Jacquet functor and the Kazhdan–Lusztig tensor product and apply it to prove that the affine Harish-Chandra category is stable under fusion tensoring with the Kazhdan–Lusztig category. This compatibility will be further applied in studying translation functors for the affine Harish-Chandra category, based on the fusion tensor product
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