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$\DeclareMathOperator\SL{SL}$Let $\lambda = (\lambda_1,\ldots,\lambda_n)$ be a sequence of positive integers and let $$ R_\lambda(x;t) = \sum_{w\in S_n} w\cdot \left( x_1^{\lambda_1}\ldots x_n^{\...

I need some help about the problem below. Let $d\geq 4$ and $f$ a symmetric polynomial, homogeneous of degree $d$, in $n$ variables $x_1,\dots,x_n$, with real coefficients. We set $$ E(f):=\sum_{j=1}^{...

I am working with polynomial representations of spherical subalgebra of double affine Hecke algebra (DAHA) for $\mathfrak{gl}_n$. Let's call this algebra $\mathfrak{A}_n$ for short. Typically we think ...