On a Casselman-Shalika type formula for unramified Speh representations

We give a Casselman-Shalika type formula for unramified Speh representations. Our formula computes values of the normalized spherical element of the $(k,c)$ model of a Speh representation at elements of the form $\mathrm{diag}(g,I_{(k-1)c})$, where $g\in {\mathrm{G}\mathrm{L}}_c\left(F\right)$ for a non-archimedean local field $F$. The formula expresses these values in terms of modified Hall–Littlewood polynomials evaluated at the Satake parameter attached to the representation. Our proof is combinatorial and very simple. It utilizes Macdonald’s formula and the unramified computation of the Ginzburg–Kaplan integral. This addresses a question of Lapid-Mao.