  Let ${\mathscr L}=(X,\preceq)$ be a lattice. For ${\cal P}\subseteq X$ we say
that ${\cal P}$ is $t$-{\it intersecting} if ${\sf rank}(x\wedge y)\ge t$ for
all $x,y\in{\cal P}$. The seminal theorem of Erd\H{o}s, Ko and Rado describes
the maximum intersecting ${\cal P}$ in the lattice of subsets of a finite set
with the additional condition that ${\cal P}$ is contained within a level of
the lattice. The Erd\H{o}s-Ko-Rado theorem has been extensively studied and
generalized to other objects and lattices.
  In this paper, we focus on intersecting families of permutations as defined
with respect to the weak Bruhat lattice. In this setting, we prove analogs of
certain extremal results on intersecting set systems. In particular we give a
characterization of the maximum intersecting families of permutations in the
Bruhat lattice. We also characterize the maximum intersecting families of
permutations within the $r^{\textrm{th}}$ level of the Bruhat lattice of
permutations of size $n$, provided that $n$ is large relative to $r$.
