Geodesic
Regular representation of the group $\mathrm{GL}(N,\mathbb{R})$: factorization, Casimir operators and Toda chain
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 27, Tome 494 (2020), pp. 23-47

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The note is devoted to a factorization formula for the matrix constructed from the generators of the group $\mathrm{GL}(N,\mathbb{R})$ in its regular representation. The factorization formula makes it possible to calculate these generators together with Casimir operators in the case of an arbitrary $N$, and it also clarifies a link between the group $\mathrm{GL}(N,\mathbb{R})$ and the quantum Toda chain. 
@article{ZNSL_2020_494_a1,
     author = {N. M. Belousov and S. E. Derkachev},
     title = {Regular representation of the group $\mathrm{GL}(N,\mathbb{R})$: factorization, {Casimir} operators and {Toda} chain},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {23--47},
     publisher = {mathdoc},
     volume = {494},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_494_a1/}
}
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N. M. Belousov; S. E. Derkachev. Regular representation of the group $\mathrm{GL}(N,\mathbb{R})$: factorization, Casimir operators and Toda chain. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 27, Tome 494 (2020), pp. 23-47. http://geodesic.mathdoc.fr/item/ZNSL_2020_494_a1/