321 research outputs found

### Intertwining operator for $AG_2$ Calogero-Moser-Sutherland system

We consider generalised Calogero-Moser-Sutherland quantum Hamiltonian $H$
associated with a configuration of vectors $AG_2$ on the plane which is a union
of $A_2$ and $G_2$ root systems. The Hamiltonian $H$ depends on one parameter.
We find an intertwining operator between $H$ and the Calogero-Moser-Sutherland
Hamiltonian for the root system $G_2$. This gives a quantum integral for $H$ of
order 6 in an explicit form thus establishing integrability of $H$.Comment: 24 page

### On the evaluation formula for Jack polynomials with prescribed symmetry

The Jack polynomials with prescribed symmetry are obtained from the
nonsymmetric polynomials via the operations of symmetrization,
antisymmetrization and normalization. After dividing out the corresponding
antisymmetric polynomial of smallest degree, a symmetric polynomial results. Of
interest in applications is the value of the latter polynomial when all the
variables are set equal. Dunkl has obtained this evaluation, making use of a
certain skew symmetric operator. We introduce a simpler operator for this
purpose, thereby obtaining a new derivation of the evaluation formula. An
expansion formula of a certain product in terms of Jack polynomials with
prescribed symmetry implied by the evaluation formula is used to derive a
generalization of a constant term identity due to Macdonald, Kadell and Kaneko.
Although we don't give the details in this work, the operator introduced here
can be defined for any reduced crystallographic root system, and used to
provide an evaluation formula for the corresponding Heckman-Opdam polynomials
with prescribed symmetry.Comment: 18 page

### Algebraic and analytic Dirac induction for graded affine Hecke algebras

We define the algebraic Dirac induction map \Ind_D for graded affine Hecke
algebras. The map \Ind_D is a Hecke algebra analog of the explicit
realization of the Baum-Connes assembly map in the $K$-theory of the reduced
$C^*$-algebra of a real reductive group using Dirac operators. The definition
of \Ind_D is uniform over the parameter space of the graded affine Hecke
algebra. We show that the map \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 Hecke algebra analogue of the construction of discrete series
representations for semisimple Lie groups due to Parthasarathy and
Atiyah-Schmid.Comment: 37 pages, revised introduction, updated references, minor correction

### On the r-matrix structure of the hyperbolic BC(n) Sutherland model

Working in a symplectic reduction framework, we construct a dynamical
r-matrix for the classical hyperbolic BC(n) Sutherland model with three
independent coupling constants. We also examine the Lax representation of the
dynamics and its equivalence with the Hamiltonian equation of motion.Comment: 20 page

### Extended trigonometric Cherednik algebras and nonstationary Schr\"odinger equations with delta-potentials

We realize an extended version of the trigonometric Cherednik algebra as
affine Dunkl operators involving Heaviside functions. We use the quadratic
Casimir element of the extended trigonometric Cherednik algebra to define an
explicit nonstationary Schr\"odinger equation with delta-potential. We use
coordinate Bethe ansatz methods to construct solutions of the nonstationary
Schr\"odinger equation in terms of generalized Bethe wave functions. It is
shown that the generalized Bethe wave functions satisfy affine difference
Knizhnik-Zamolodchikov equations in their spectral parameter. The relation to
the vector valued root system analogs of the quantum Bose gas on the circle
with pairwise delta-function interactions is indicated.Comment: 23 pages; Version 2: expanded introduction and misprints correcte

### On the elliptic nonabelian Fourier transform for unipotent representations of p-adic groups

In this paper, we consider the relation between two nonabelian Fourier
transforms. The first one is defined in terms of the Langlands-Kazhdan-Lusztig
parameters for unipotent elliptic representations of a split p-adic group and
the second is defined in terms of the pseudocoefficients of these
representations and Lusztig's nonabelian Fourier transform for characters of
finite groups of Lie type. We exemplify this relation in the case of the p-adic
group of type G_2.Comment: 17 pages; v2: several minor corrections, references added; v3:
corrections in the table with unipotent discrete series of G

### Trigonometric Cherednik algebra at critical level and quantum many-body problems

For any module over the affine Weyl group we construct a representation of
the associated trigonometric Cherednik algebra $A(k)$ at critical level in
terms of Dunkl type operators. Under this representation the center of $A(k)$
produces quantum conserved integrals for root system generalizations of quantum
spin-particle systems on the circle with delta function interactions. This
enables us to translate the spectral problem of such a quantum spin-particle
system to questions in the representation theory of $A(k)$. We use this
approach to derive the associated Bethe ansatz equations. They are expressed in
terms of the normalized intertwiners of $A(k)$.Comment: 31 page

### Baker-Akhiezer functions and generalised Macdonald-Mehta integrals

For the rational Baker-Akhiezer functions associated with special
arrangements of hyperplanes with multiplicities we establish an integral
identity, which may be viewed as a generalisation of the self-duality property
of the usual Gaussian function with respect to the Fourier transformation. We
show that the value of properly normalised Baker-Akhiezer function at the
origin can be given by an integral of Macdonald-Mehta type and explicitly
compute these integrals for all known Baker-Akhiezer arrangements. We use the
Dotsenko-Fateev integrals to extend this calculation to all deformed root
systems, related to the non-exceptional basic classical Lie superalgebras.Comment: 26 pages; slightly revised version with minor correction

- â€¦