Symmetry classification for Jackson integrals associated with irreducible reduced root systems (Q2769563)

The author defines the Jackson integral associated with an irreducible reduced root system \(R\) as certain sums which are invariant under the Weyl group \(W\) of \(R\). The summation is over full-rank \(W\)-invariant sublattices \(L\) of the coweight lattice \(P\) of \(R\). When \(L=P\) or when \(L\) is the coroot lattice \(Q\) of \(R\), the Jackson integrals admit a product formula in terms of Jacobi elliptic theta functions (Theorem 4.5). The product formula allows their classification, which shows that they include as special instances the sums investigated by \textit{K. Aomoto} in [J. Algebr. Comb. 8, No. 2, 115--126 (1998; Zbl 0918.33013)] and the \(B_n\)- and \(G_2\)-type sums of \textit{R. A. Gustafson} [Trans. Am. Math. Soc. 341, No. 1, 69--119 (1994; Zbl 0796.33012)]. NEWLINENEWLINENEWLINEThe author announces a sequel to the present paper, in which Jackson integrals associated with nonreduced root systems will also include Gustafson's \(C_n\)- and \(D_n\)-type sums as well as the \(BC_n\)-type sums of \textit{J. F. van Diejen} [Publ. Res. Inst. Math. Sci. 33, No. 3, 483--508 (1997; Zbl 0894.33007)].