Standard cyclic module of sl2

Question

Let $L=\mathfrak{sl}(2, \mathbb{F})$, $B$ a standard Borel subalgebra.
I am trying to solve exercise 20.4 from J.E. Humphreys "Introduction to Lie Algebras and Representation Theory", but I am stuck.
First of all $Z(\lambda)$ is constructed as $\mathfrak{U}(L)\otimes_{\mathfrak{U}(B)}D_{\lambda}$ by induced module construction. $1\otimes v^{+}$ is the maximal vector.
The exercise is to show that this is isomorphic to another definiton of $Z(\Lambda)$ given before. That means I have to finde a basis $(v_{0}, v_{1}, \dots )$ fulfilling:
a)$h.v_{i}=(\lambda-2i)v_{i}$
b)$y.v_{i}=(i+1)v_{i+1}$
c)$x.v_{i}=(\lambda-i+1)v_{i-1}$ with $v_{-1}=0$
for (x, h, y) the usual basis of $L$.
I don´t know exactly how $L$ acts on $Z(\lambda)$, but as $Z(\lambda)$ is an $\mathfrak{U}(L)$-module via the left multiplication, I thought it might be the same with $L$.
I tried the following:
$v_{0}=1\otimes v^{+}$
$v_{1}=y\otimes v^{+}$
$\dots$
$v_{i}=\frac{1}{i!} y^{i} \otimes v^{+}$
$\dots$
With this b) is obvious, but I don´t know how to continue with a) and c) or if this is even the correct idea.
Thank you for helping me.