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Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters $q,t_1,t_2,...,t_r$, where r (=1,2 or 3) is the number of W-orbits in R. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and p-adic symmetric spaces. Also when R=S is of type $A_n$, they conincide with the symmetric polynomials described in I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press (1995), Chapter VI.
Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all
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Let $R$ be a finite ring and define the hyperbola $H={(x,y) in R times R: xy=1 }$. Suppose that for a sequence of finite odd order rings of size tending to infinity, the following square root law bound holds with a constant $C>0$ for all non-trivial
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