Theta representations appear globally as the residues of Eisenstein series on covers of groups; their unramified local constituents may be characterized as subquotients of certain principal series. A cuspidal theta representation is one which is equal to the local twisted theta representation at almost all places. Cuspidal theta representations are known to exist but only for covers of GLj, j≤ 3. In this paper we establish necessary conditions for the existence of cuspidal theta representations on the r-fold metaplectic cover of the general linear group of arbitrary rank.
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