scholarly journals Elliptic Double Affine Hecke Algebras

2020 ◽  
Author(s):  
Eric M. Rains ◽  
Keyword(s):  
Hilbert Scheme ◽  
Hecke Algebra ◽  
Rational Surface ◽  
Symmetric Power ◽  
Affine Hecke Algebra ◽  
Double Affine Hecke Algebra ◽  
Affine Weyl Group ◽  
Double Affine Hecke Algebras ◽  
Hilbert Scheme Of Points ◽  
Affine Hecke Algebras

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the C<sub>n</sub> version of the construction to construct a flat noncommutative deformation of the nth symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.

scholarly journals HECKE ALGEBRAS FOR INNER FORMS OF -ADIC SPECIAL LINEAR GROUPS

2015 ◽  
Vol 16 (2) ◽  
pp. 351-419 ◽  
Author(s):  
Anne-Marie Aubert ◽  
Paul Baum ◽  
Roger Plymen ◽  
Maarten Solleveld
Keyword(s):  
Finite Group ◽  
Local Field ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Linear Groups ◽  
Special Linear Groups ◽  
Affine Hecke Algebra ◽  
Morita Equivalent ◽  
Affine Hecke Algebras ◽  
A Finite Group

Let$F$be a non-Archimedean local field, and let$G^{\sharp }$be the group of$F$-rational points of an inner form of$\text{SL}_{n}$. We study Hecke algebras for all Bernstein components of$G^{\sharp }$, via restriction from an inner form$G$of$\text{GL}_{n}(F)$.For any packet of L-indistinguishable Bernstein components, we exhibit an explicit algebra whose module category is equivalent to the associated category of complex smooth$G^{\sharp }$-representations. This algebra comes from an idempotent in the full Hecke algebra of$G^{\sharp }$, and the idempotent is derived from a type for$G$. We show that the Hecke algebras for Bernstein components of$G^{\sharp }$are similar to affine Hecke algebras of type$A$, yet in many cases are not Morita equivalent to any crossed product of an affine Hecke algebra with a finite group.

scholarly journals Double affine Hecke algebras and generalized Jones polynomials

2016 ◽  
Vol 152 (7) ◽  
pp. 1333-1384 ◽  
Author(s):  
Yuri Berest ◽  
Peter Samuelson
Keyword(s):  
Torus Knots ◽  
Knot Invariants ◽  
Skein Module ◽  
Double Affine Hecke Algebra ◽  
Double Affine Hecke Algebras ◽  
Jones Polynomials ◽  
Kauffman Bracket Skein Module ◽  
Affine Hecke Algebras ◽  
Figure Eight Knot ◽  
Colored Jones Polynomials

In this paper we propose and discuss implications of a general conjecture that there is a natural action of a rank 1 double affine Hecke algebra on the Kauffman bracket skein module of the complement of a knot $K\subset S^{3}$. We prove this in a number of nontrivial cases, including all $(2,2p+1)$ torus knots, the figure eight knot, and all 2-bridge knots (when $q=\pm 1$). As the main application of the conjecture, we construct three-variable polynomial knot invariants that specialize to the classical colored Jones polynomials introduced by Reshetikhin and Turaev. We also deduce some new properties of the classical Jones polynomials and prove that these hold for all knots (independently of the conjecture). We furthermore conjecture that the skein module of the unknot is a submodule of the skein module of an arbitrary knot. We confirm this for the same example knots, and we show that this implies that the colored Jones polynomials of $K$ satisfy an inhomogeneous recursion relation.

scholarly journals ON THE SPECTRAL DECOMPOSITION OF AFFINE HECKE ALGEBRAS

2004 ◽  
Vol 3 (4) ◽  
pp. 531-648 ◽  
Author(s):  
Eric M. Opdam
Keyword(s):  
Probability Measure ◽  
Spectral Decomposition ◽  
Hecke Algebra ◽  
Plancherel Formula ◽  
Root Lattice ◽  
Abelian Subalgebra ◽  
Group Invariant ◽  
Affine Hecke Algebra ◽  
Algebraic Torus ◽  
Affine Hecke Algebras

An affine Hecke algebra $\mathcal{H}$ contains a large abelian subalgebra $\mathcal{A}$ spanned by the Bernstein–Zelevinski–Lusztig basis elements $\theta_x$, where $x$ runs over (an extension of) the root lattice. The centre $\mathcal{Z}$ of $\mathcal{H}$ is the subalgebra of Weyl group invariant elements in $\mathcal{A}$. The natural trace (‘evaluation at the identity’) of the affine Hecke algebra can be written as integral of a certain rational $n$-form (with values in the linear dual of $\mathcal{H}$) over a cycle in the algebraic torus $T=\textrm{Spec}(\mathcal{A})$. This cycle is homologous to a union of ‘local cycles’. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum $W_0\setminus T$ of $\mathcal{Z}$. From this result we derive the Plancherel formula of the affine Hecke algebra.AMS 2000 Mathematics subject classification: Primary 20C08; 22D25; 22E35; 43A32

scholarly journals Algebraic and analytic Dirac induction for graded affine Hecke algebras

2013 ◽  
Vol 13 (3) ◽  
pp. 447-486 ◽  
Author(s):  
Dan Ciubotaru ◽  
Eric M. Opdam ◽  
Peter E. Trapa
Keyword(s):  
Weyl Group ◽  
Discrete Series ◽  
Hecke Algebras ◽  
Reductive Group ◽  
Dirac Operators ◽  
Hecke Algebra ◽  
Semisimple Lie Groups ◽  
Affine Hecke Algebra ◽  
Affine Hecke Algebras ◽  
Dirac Induction

AbstractWe define the algebraic Dirac induction map ${\mathrm{Ind} }_{D} $ for graded affine Hecke algebras. The map ${\mathrm{Ind} }_{D} $ is a Hecke algebra analog of the explicit realization of the Baum–Connes assembly map in the $K$-theory of the reduced ${C}^{\ast } $-algebra of a real reductive group using Dirac operators. The definition of ${\mathrm{Ind} }_{D} $ is uniform over the parameter space of the graded affine Hecke algebra. We show that the map ${\mathrm{Ind} }_{D} $ defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded affine Hecke algebra analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

Sign in / Sign up

Export Citation Format

Share Document