The Schur polynomials $$s_\lambda(x_1, \ldots, x_n) = \frac{|x_i^{\lambda_j+n-j}|_{1\le i,j\le n}}{|x_i^{n-j}|_{1\le i,j\le n}}$$ naturally appear as polynomial representatives for Schubert classes in the cohomology ring of Grassmanians, and also have a representation-theoretic interpretation as characters of irreducible representations of $GL_n$.
The analogs of Schur polynomials for $T$-equivariant cohomology of Grassmannians are the double Schur polynomials, which are defined as $$s_\lambda(x_1, \ldots, x_n|t) = \frac{|(x_i|t)^{\lambda_j+n-j}|_{1\le i,j\le n}}{|(x_i|t)^{n-j}|_{1\le i,j\le n}}$$ where $$(x_i|t)^k=(x_i-t_1)\ldots(x_i-t_k).$$
Question: Is there any representation theoretic interpretation of double Schur polynomials analogous to the interpretation of Schur polynomials as characters of irreps of $GL_n$?
