Geodesic
The intermediate Lie algebra $\mathfrak d_{n-1/2}$, the weight scheme and finite-dimensional representations with highest weight
Izvestiya. Mathematics , Tome 68 (2004) no. 2, pp. 375-404

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Multiple points of the spectrum in the reduction $D_n\downarrow D_{n-1}$ are separated by introducing a non-semisimple intermediate subalgebra and a weight scheme different from the Gel'fand–Tsetlin scheme. We suggest a method of constructing a weight basis in the space of a finite-dimensional irreducible representation of $D_n$. The elements of this basis are labelled by such weight schemes. We also study the category of finite-dimensional highest-weight representations of this intermediate Lie algebra. 
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     author = {V. V. Shtepin},
     title = {The intermediate {Lie} algebra $\mathfrak d_{n-1/2}$, the weight scheme and finite-dimensional representations with highest weight},
     journal = {Izvestiya. Mathematics },
     pages = {375--404},
     publisher = {mathdoc},
     volume = {68},
     number = {2},
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     url = {http://geodesic.mathdoc.fr/item/IM2_2004_68_2_a7/}
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V. V. Shtepin. The intermediate Lie algebra $\mathfrak d_{n-1/2}$, the weight scheme and finite-dimensional representations with highest weight. Izvestiya. Mathematics , Tome 68 (2004) no. 2, pp. 375-404. http://geodesic.mathdoc.fr/item/IM2_2004_68_2_a7/