= Solution
Let $U\leq\operatorname{SL}_2(p)$ be the Sylow p-subgroup of upper unitriangular matrices. It is cyclic of order $p$, generated by
$$
u=\begin{pmatrix}1&1\\0&1\end{pmatrix}.
$$
Realize $S^r(V_2)$ as the homogeneous polynomials of degree $r$ in $X,Y$, with $u$ acting by $X\mapsto X$ and $Y\mapsto X+Y$. For $0\leq r<p$, the only vectors fixed by $u$ are the multiples of $X^r$: successive comparison of the coefficients of $Y^r,Y^{r-1},\ldots$ proves this. Thus the nilpotent operator $u-1$ has one-dimensional kernel. Its <Jordan normal form> therefore has a single block, so
$$
\boxed{\operatorname{Res}_U^G S^r(V_2)\text{ is indecomposable of dimension }r+1.}
$$
This also follows from the <indecomposable modules of a cyclic p-group in characteristic p>.
For $r=p-1$, the restriction has dimension $p$ and is the regular $kU$-module, hence is projective. Since $[G:U]$ is prime to $p$, part (b)(ii) makes $S^{p-1}(V_2)$ a simple projective $kG$-module. It is therefore a <defect-zero representation> and lifts to an ordinary irreducible representation of the same dimension. Consequently
$$
\boxed{\operatorname{SL}_2(p)\text{ has an ordinary irreducible character of degree }p.}
$$
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