Suppose $G=GL_n(\mathbb{F}_q)$, with $T$ and $U$ the standard maximal torus and unipotent radical. Assume that the characteristic is such that $|G|$ and $|T|$, $|U|$ are invertible in the field to avoid dividing by $0$.
If $\chi$ is an irreducible character of $T$, afforded by some representation $E$, is it true that $$ k[G/U]\otimes_{k[T]}E \simeq kGe_U\otimes_{k[T]}E\simeq kGe_Ue_\chi $$ as $k[G]$-modules, where $e_\chi$ is the isotypic projection associated to $\chi$? The first isomorphism is okay I believe since $k[G/U]\simeq kGe_U$, where $e_U=|U|^{-1}\sum_{u\in U}u$ is the idempotent corresponding to the trivial representation, but I am unsure of the second. Here $k[G/U]$ is the $k$-vector space with basis the quotient set $G/U$, viewed as a $(k[G],k[T])$-bimodule via left and right multiplication.
You can choose the representation affording the character $\chi$ just to be $k[T]e_\chi$. Note that multiplication gives an isomorphism $k[G] e_U \otimes_{k[T]} k[T] \cong k[G] e_U$ as left $k[G]$-modules which maps the submodule $k[G] e_U \otimes_{k[T]} k[T] e_\chi$ onto $k[G]e_U e_\chi$.