LIFTING TO GL(2) OVER A DIVISION QUATERNION ALGEBRA, AND AN EXPLICIT CONSTRUCTION OF CAP REPRESENTATIONS

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<jats:p>The aim of this paper is to carry out an explicit construction of CAP representations of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0027763016000155_inline1" /><jats:tex-math>$\text{GL}(2)$</jats:tex-math></jats:alternatives></jats:inline-formula> over a division quaternion algebra with discriminant two. We first construct cusp forms on such a group explicitly by lifting from Maass cusp forms for the congruence subgroup <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0027763016000155_inline2" /><jats:tex-math>${\rm\Gamma}_{0}(2)$</jats:tex-math></jats:alternatives></jats:inline-formula>. We show that this lifting is nonzero and Hecke-equivariant. This allows us to determine each local component of a cuspidal representation generated by such a lifting. We then show that our cuspidal representations provide examples of CAP (cuspidal representation associated to a parabolic subgroup) representations, and, in fact, counterexamples to the Ramanujan conjecture.</jats:p>

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