Geodesic
Note on Weight Spaces of Irreducible Linear Representations
Canadian mathematical bulletin, Tome 11 (1968) no. 3, pp. 399-403

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Let L denote a finite dimensional, simple Lie algebra over an algebraically closed field F of characteristic zero. It is well known that every weight space of an irreducible representation (ρ, V) admitting a highest weight function is finite dimensional. In a previous paper [2], we have established the existence of a wide class of irreducible representations which admit a one-dimensional weight space but no highest weight function. In this paper we show that the weight spaces of all such representations are finite dimensional. 
Lemire, F. W. Note on Weight Spaces of Irreducible Linear Representations. Canadian mathematical bulletin, Tome 11 (1968) no. 3, pp. 399-403. doi: 10.4153/CMB-1968-045-2
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     title = {Note on {Weight} {Spaces} of {Irreducible} {Linear} {Representations}},
     journal = {Canadian mathematical bulletin},
     pages = {399--403},
     year = {1968},
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     number = {3},
     doi = {10.4153/CMB-1968-045-2},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1968-045-2/}
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