We study multivariate Chebyshev polynomials associated with root systems. Using properties of specialized singular elements corresponding to a root system , we construct explicitly the measure weight function γ. The latter ensures that these polynomials are orthonormal; it defines the scalar product in the function space where multivariate U-type Chebyshev polynomials constitute a basis. The obtained results are illustrated by constructing and studying 2-variate polynomials for root systems , and .
We consider the Knizhnik-Zamolodchikov and Dynamical operators, both differential and difference, in the context of the (gl k , gl n )-duality for the space of polynomials in kn anticommuting variables. We show that the Knizhnik-Zamolodchikov and Dynamical operators naturally exchange under the duality.•
We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907]. Starting with a space of quasi-exponentials W = ⟨α x i p ij (x), i = 1, . . . , n, j = 1, . . . , n i ⟩, where α i ∈ C * and p ij (x) are polynomials, we consider the formal conjugate Š † W of the quotient difference operator ŠW satisfying S = ŠW S W . Here, S W is a linear difference operator of order dim W annihilating W , and S is a linear difference operator with constant coefficients depending on α i and deg p ij (x) only. We construct a space of quasi-exponentials of dimension ord Š † W , which is annihilated by Š † W and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form x z q(x), where z ∈ C and q(x) is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the (gl k , gl n )-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in kn anticommuting variables.
We study the difference analog of the quotient differential operator from [13].Starting with a space of quasi-exponentials W = α x i p ij (x), i = 1, . . . , n, j = 1, . . . , n i , where α i ∈ C * and p ij (x) are polynomials, we consider the formal conjugate Š † W of the quotient difference operator ŠW satisfying S = ŠW S W . Here, S W is a linear difference operator of order dim W annihilating W , and S is a linear difference operator with constant coefficients depending on α i and deg p ij (x) only. We show that ker Š † W is also a space of quasi-exponentials, describe its basis and discrete exponents.We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form x z q(x), where z ∈ C and q(x) is a polynomial.Combining our results with the results on the bispectral duality obtained in [6], we relate the construction of the quotient difference operator to the (gl k , gl n )-duality of the trigonometric Gaudin Hamiltonians and trigonometric Dynamical Hamiltonians acting on the space of polynomials in kn anticommuting variables.
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