Mean asymmetry of polynomials on compact homogeneous spaces

Let $M=G/H$ be a homogeneous space of a compact Lie group $G$, ${{\mathcal E}}$ be an $G$-invariant finite dimensional subspace of $L^2_{\mathord{\mathbb{R}}}(M)$, and ${\mathord{\mathcal{S}}}$ be the unit sphere in it. Set $\eta_a(u)=\int_M\left(u_+^a(x)-u_-^a(x)\right)\,dx$, where $u_+(x)=\max\{u(x),0\}$, $u_-(x)=-\min\{u(x),0\}$. We consider the asymptotic behavior of the variance of the random variable $\eta_a$ as $a\to\infty$ or $\dim{{\mathcal E}}\to\infty$ for the uniform distribution of $u$ in ${\mathord{\mathcal{S}}}$. For instance, if ${{\mathord{\mathcal{E}}}}$ is the space of trigonometrical polynomials of degree less or equal to $n$, then $\mathop{\mathrm{Var}}(\eta_a)\sim \frac{A}{n}$ as $n\to\infty$.