Back to the Weyl Calculus

Abstract

We come back to the constructions of Section 3: our standing assumption in this chapter is that the number N is 4 times a squarefree odd integer, but it does not have to coincide with 4 or 12 any more. Then, not every element of (ℤ/Nℤ)^× is a square, and we have to consider the full set of distributions ϖ_ρ as defined in Lemma 3.2: we introduce the linear combination

$$ W_N \left( {x,\xi } \right) = \sum\limits_{\rho \in R_N } {W\left( {\varpi _\rho ,\varpi _\rho } \right)\left( {x,\xi } \right)} , $$

again a Γ-invariant distribution. Recall that ? is the set of squares in (ℤ/Nℤ)^× and that R _N is a set of representatives of (ℤ/Nℤ)^× mod Λ.