We consider some mathematical properties of Weyl-like quantizations based on two families of orderings of e^ι(aP+bQ): the first family, W_{(λ, 0)}, interpolates between Wick (λ=1) and antiWick (λ=−1) ordering, while the second family, W_{(0, μ)}, interpolates between the Q−(μ=1) and P−(μ=−1) orderings. The ordering W_{(0, 0)} common to both families is the unordered Weyl system.
The most important property is that of the existence of quantizations. For all orderings W_{(0, μ)} and for W_{(λ, 0)} with −1≤λ≤1 quantization is a well-defined map from the tempered distributions on phase space into the continuous linear operators from \mathcal{S}(\mathbb{R}) into \mathcal{S}(\mathbb{R})'. For the orderings W_{(λ, 0)} with 0<λ≤0 we have to restrict the class of wave functions from \mathcal{S}(\mathbb{R}) to a certain dense subset of it, and the resulting quantization procedure sends tempered distributions on phase space into sesquilinear forms on this subspace. For Wick ordering itself we have not been able to find any useable quantization scheme, and we doubt whether any one exists that is based on tempered distributions.
We also consider questions of boundedness, and determine the matrix coefficients for the quantizations of phase space functions of radius or of angle. In particular, we consider various quantizations of the angle function in phase space.

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Part I of this paper develops various general concepts in generic representation and cohomology theories. Roughly speaking, we provide a general theory of orders in non-semisimple algebras applicable to problems in the representation theory of finite and algebraic groups, and we formalize the notion of a “generic” property in representation theory. Part II makes new contributions to the non-describing representation theory of finite general linear groups. First, we present an explicipt Morita equivalence connecting GL_n(q) with the theory of q-Schur algebras, extending a unipotent block equivalence of Takeuchi [T]. Second, we apply this Morita equivalence to study the cohomology groups H^•(GL_n(q), L), when L is an irreducible module in non-describing characteristic. The generic theory of Part I then yields stability results for various groups H^ι(GL_n(q), L), reminscent of our general theory [CPSK] with van der Kallen of generic cohomology in the describing characteristic case, (In turn, the stable value of such a cohomology group can be expressed in terms of the cohomology of the affine Lie algebra \widehat{\mathfrak{gl}}_n(\mathbb{C}).) The arguments entail new applications of the theory of tilting modules for q-Schur algebras. In particular, we obtain new complexes involving tilting modules associated to endomorphism algebras obtained from general finite Coxeter groups.

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