TY - JOUR
T1 - Dunkl kernel associated with dihedral groups
AU - Deleaval, L.
AU - Demni, N.
AU - Youssfi, H.
N1 - Publisher Copyright:
© 2015 Elsevier Inc..
PY - 2015/12/15
Y1 - 2015/12/15
N2 - In this paper, we pursue the investigations started in [18] where the authors provide a construction of the Dunkl intertwining operator for a large subset of the set of regular multiplicity values. More precisely, we make concrete the action of this operator on homogeneous polynomials when the root system is of dihedral type and under a mild assumption on the multiplicity function. In particular, we obtain a formula for the corresponding Dunkl kernel and another representation of the generalized Bessel function already derived in [7]. When the multiplicity function is everywhere constant, our computations give a solution to the problem of counting the number of factorizations of an element from a dihedral group into a fixed number of (non-necessarily simple) reflections. In the remainder of the paper, we supply another method to derive the Dunkl kernel associated with dihedral systems from the corresponding generalized Bessel function. This time, we use the shift principle together with multiple combinations of Dunkl operators in the directions of the vectors of the canonical basis of R2. When the dihedral system is of order six and only in this case, a single combination suffices to get the Dunkl kernel and agrees up to an isomorphism with the formula recently obtained by Amri [2, Lemma 1] in the case of a root system of type A2. We finally derive an integral representation for the Dunkl kernel associated with the dihedral system of order eight.
AB - In this paper, we pursue the investigations started in [18] where the authors provide a construction of the Dunkl intertwining operator for a large subset of the set of regular multiplicity values. More precisely, we make concrete the action of this operator on homogeneous polynomials when the root system is of dihedral type and under a mild assumption on the multiplicity function. In particular, we obtain a formula for the corresponding Dunkl kernel and another representation of the generalized Bessel function already derived in [7]. When the multiplicity function is everywhere constant, our computations give a solution to the problem of counting the number of factorizations of an element from a dihedral group into a fixed number of (non-necessarily simple) reflections. In the remainder of the paper, we supply another method to derive the Dunkl kernel associated with dihedral systems from the corresponding generalized Bessel function. This time, we use the shift principle together with multiple combinations of Dunkl operators in the directions of the vectors of the canonical basis of R2. When the dihedral system is of order six and only in this case, a single combination suffices to get the Dunkl kernel and agrees up to an isomorphism with the formula recently obtained by Amri [2, Lemma 1] in the case of a root system of type A2. We finally derive an integral representation for the Dunkl kernel associated with the dihedral system of order eight.
KW - Dihedral root systems
KW - Dunkl kernel
KW - Dunkl operators
KW - Generalized Bessel function
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U2 - 10.1016/j.jmaa.2015.07.029
DO - 10.1016/j.jmaa.2015.07.029
M3 - Article
AN - SCOPUS:84939267681
SN - 0022-247X
VL - 432
SP - 928
EP - 944
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 2
ER -