Macdonald ’ s Evaluation Conjectures and Difference Fourier Transform
نویسنده
چکیده
Generalizing the characters of compact simple Lie groups, Ian Macdonald introduced in [M1,M2] and other works remarkable symmetric trigonometric polynomials dependent on the parameters q, t. He came up with four main conjectures formulated for arbitrary root systems. A new approach to the Macdonald theory was suggested in [C1] on the basis of double affine Hecke algebras (new objects in mathematics). In [C2] the norm conjecture (including the famous constant term conjecture [M3]) and the conjecture about the denominators of the coefficients of the Macdonald polynomials were proved. This paper contains the proof of the remaining two (the duality and evaluation conjectures). The evaluation conjecture (now a theorem) is in fact a q, t-generalization of the classic Weyl dimension formula. One can expect interesting applications of this theorem since the so-called q-dimensions are undoubtedly important. It is likely that we can incorporate the Kac-Moody case as well. The necessary technique was developed in [C4]. As to the duality theorem (in its complete form), it states that the generalized trigonometric-difference zonal Fourier transform is self-dual (at least formally). We define this q, t-transform in terms of double affine Hecke algebras. The most natural way to check the self-duality is to use the connection of these algebras with the so-called elliptic braid groups (the Fourier involution will turn into the transposition of the periods of an elliptic curve).
منابع مشابه
Macdonald ’ s Evaluation Conjectures , Difference Fourier Transform , and applications
Generalizing the characters of compact simple Lie groups, Ian Macdonald introduced in [M1,M2] and other works remarkable orthogonal symmetric polynomials dependent on the parameters q, t. He came up with three main conjectures formulated for arbitrary root systems. A new approach to the Macdonald theory was suggested in [C1] on the basis of (double) affine Hecke algebras and related difference ...
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