  In this paper, we study the Cauchy problem for the generalized Keller-Segel
system with the cell diffusion being ruled by fractional diffusion:
\begin{equation*} \begin{cases}
  \partial_{t}u+\Lambda^{\alpha}u-\nabla\cdot(u\nabla \psi)=0\quad &\mbox{in}\
\
  \mathbb{R}^n\times(0,\infty),
  -\Delta \psi=u\quad &\mbox{in}\ \
  \mathbb{R}^n\times(0,\infty),
  u(x,0)=u_0(x), \ \ &\mbox{in}\ \ \mathbb{R}^n. \end{cases} \end{equation*} In
the case that $1<\alpha\leq 2$, we prove local well-posedness for any initial
data and global well-posedness for small initial data in critical Besov spaces
$\dot{B}^{-\alpha+\frac{n}{p}}_{p,q}(\mathbb{R}^{n})$ with $1\leq p<\infty$,
$1\leq q\leq \infty$, and analyticity of solutions for initial data $u_{0}\in
\dot{B}^{-\alpha+\frac{n}{p}}_{p,q}(\mathbb{R}^{n})$ with $1< p<\infty$, $1\leq
q\leq \infty$. Moreover, the global existence and analyticity of solutions with
small initial data in critical Besov spaces
$\dot{B}^{-\alpha}_{\infty,1}(\mathbb{R}^{n})$ is also established. In the
limit case that $\alpha=1$, we prove global well-posedness for small initial
data in critical Besov spaces $\dot{B}^{-1+\frac{n}{p}}_{p,1}(\mathbb{R}^{n})$
with $1\leq p<\infty$ and $\dot{B}^{-1}_{\infty,1}(\mathbb{R}^{n})$, and show
analyticity of solutions for small initial data in
$\dot{B}^{-1+\frac{n}{p}}_{p,1}(\mathbb{R}^{n})$ with $1<p<\infty$ and
$\dot{B}^{-1}_{\infty,1}(\mathbb{R}^{n})$, respectively.
