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The answer is $$ u(x) = (-\Delta)^{-\alpha/2} \chi_B(x) = 2^{-\alpha} G_{2,2}^{1,1}\biggl(\begin{matrix}1-(N-\alpha)/2 & 1+\alpha/2 \\ 0 & 1 - N/2\end{matrix} \, \bigg\vert \, |x|^2\biggr) , $$ where $G$ is the Meijer G-function; see Corollary 3(i) in this paper. This reduces to the hypergeometric function $_2F_1$; if I copied the expressions correctly from Mathematica, one has: $$ u(x) = \frac{\Gamma((N-\alpha)/2)}{2^\alpha \Gamma(1+\alpha/2) \Gamma(N/2)} {_2F_1}(-\alpha/2, (N-\alpha)/2; N/2; |x|^2) $$ for $x \in B$ and $$ u(x) = \frac{\Gamma((N-\alpha)/2)}{2^\alpha \Gamma(\alpha/2) \Gamma(N/2-\alpha)} |x|^{\alpha-N} {_2F_1}(1-\alpha/2, (N-\alpha)/2; 1+N/2; 1/|x|^2) $$ for $x \in \mathbb{R}^N \setminus B$. The calculation dates back to Blumenthal–Getoor–Ray (or a subset), I believe.