October 7, 2022
The aim of this paper is to give another proof of a theorem of D.Prasad, which calculates the character of an irreducible representation of $\text{GL}(mn,\mathbb{C})$ at the diagonal elements of the form $\underline{t} \cdot c_n$, where $\underline{t}=(t_1,t_2,\cdots,t_m)$ $\in$ $(\mathbb{C}^*)^{m}$ and $c_n=(1,\omega_n,\omega_n^{2},\cdots,\omega_n^{n-1})$, where $\omega_n=e^{\frac{2\pi \imath}{n}}$, and expresses it as a product of certain characters for $\text{GL}(m,\mathbb{C})$ at $\underline{t}^n$.
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For a fixed integer $t \geq 2$, we consider the irreducible characters of representations of the classical groups of types A, B, C and D, namely $\text{GL}_{tn}, \text{SO}_{2tn+1}, \text{Sp}_{2tn}$ and $\text{O}_{2tn}$, evaluated at elements $\omega^k x_i$ for $0 \leq k \leq t-1$ and $1 \leq i \leq n$, where $\omega$ is a primitive $t$'th root of unity. The case of $\text{GL}_{tn}$ was considered by D. J. Littlewood (AMS press, 1950) and independently by D. Prasad (Israel J. ...
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