Lattice points and Lie groups. II

Cahn, Robert S.

doi:10.1090/S0002-9947-73-99952-2

Lattice points and Lie groups. II
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by Robert S. Cahn

Trans. Amer. Math. Soc. 183 (1973), 131-137

DOI: https://doi.org/10.1090/S0002-9947-73-99952-2

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Abstract:

Let $C$ be the Casimir operator on a compact, simple, simply connected Lie group $G$ of dimension $n$. The number of eigenvalues of $C$, counted with their multiplicities, of absolute value less than or equal to $t$ is asymptotic to $k t^{n/2}$, $k$ a constant. This paper shows the error of this estimate to be $O({t^{2b + a(a - 1)/(a + 1)}})$; where $a$ = rank of $G$ and $b = 1/2 (n - a)$.

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