Representations of the associated Lie conformal algebra of the [formula omitted] algebra and beyond

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We introduce the so-called infinite contact Lie conformal superalgebras KN(p) from finite Lie conformal superalgebras of Cartan type K. The annihilation algebra of the first member L in degenerate cases is exactly the associated graded Lie algebra of the filtered W-infinity algebra W1+∞. Using the conformal version of Lie's theorem, we first classify finite conformal modules over KN(p) and its derived subalgebra. Then we classify finite irreducible conformal modules over positive subalgebras of L and its derived subalgebra, which are shown to be weight modules, parameterized by polynomials, in the sense of D'Andrea and Kac [13]. Finally, we conceptually classify extensions between these conformal modules.