  We study the relationship between non-trivial values of generalized cardinal
invariants at an inaccessible cardinal $\kappa$ and compactness principles at
$\kappa^{++}$. We show that if the existence of a supercompact cardinal
$\kappa$ with a weakly compact cardinal $\lambda$ above $\kappa$ is consistent,
then the following are consistent as well (where $\mathfrak{t}(\kappa)$ and
$\mathfrak{u}(\kappa)$ are the tower number and the ultrafilter number,
respectively):
  (i) There is an inaccessible cardinal $\kappa$ such that $\kappa^+ <
\mathfrak{t}(\kappa)= \mathfrak{u}(\kappa)< 2^\kappa$ and stationary reflection
and the disjoint stationary sequence property at $\kappa^{++}$ hold.
  (ii) There is an inaccessible cardinal $\kappa$ such that $\kappa^+ =
\mathfrak{t}(\kappa) < \mathfrak{u}(\kappa)< 2^\kappa$ and the principles in
(i) hold plus the tree property at $\kappa^{++}$ and the negation of the weak
Kurepa Hypothesis at $\kappa^+$.
  The cardinals $\mathfrak{u}(\kappa)$ and $2^\kappa$ can have any reasonable
values in these models. We compute several other values of cardinal invariants,
such $\mathfrak{a}(\kappa)$ and $\mathfrak{b}(\kappa)$, and the invariants of
the meager ideal (they are all equal to $\mathfrak{u}(\kappa)$). In (ii), we
compute $\mathfrak{p}(\kappa) = \mathfrak{t}(\kappa) = \kappa^+$ by observing
that the $\kappa^+$-distributive quotient of the Mitchell forcing adds a tower
of size $\kappa^+$.
  As a corollary of the construction, we obtain that (i) and (ii) are also true
for $\kappa = \omega$ (starting with a weakly compact cardinal in the ground
model).
