We study the relationship between non-trivial values of generalized cardinal invariants at an inaccessible cardinal \(\kappa \) and compactness principles at \(\kappa ^+\) and \(\kappa ^{++}\) . Let \(\textsf {TP}(\kappa ^{++})\) , \(\textsf {SR}(\kappa ^{++})\) and \(\lnot \textsf {wKH}(\kappa ^+)\) denote the tree property and stationary reflection on \(\kappa ^{++}\) and the negation of the weak Kurepa Hypothesis on \(\kappa ^+\) , respectively. 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 \(\textsf {SR}(\kappa ^{++})\) hold, and (ii) There is an inaccessible cardinal \(\kappa \) such that \(\kappa ^+ = \mathfrak {t}(\kappa )< \mathfrak {u}(\kappa )< 2^\kappa \) and \(\textsf {SR}(\kappa ^{++}), \textsf {TP}(\kappa ^{++})\) and \(\lnot \textsf {wKH}(\kappa ^+)\) hold. The cardinals \(\mathfrak {u}(\kappa )\) and \(2^\kappa \) can have any reasonable values in these models. We obtain these results by combining the forcing construction from [4] due to Brooke-Taylor, Fischer, Friedman and Montoya with the Mitchell forcing and with (new and old) indestructibility results related to \(\textsf {TP}(\kappa ^{++})\) , \(\textsf {SR}(\kappa ^{++})\) and \(\lnot \textsf {wKH}(\kappa ^+)\) . Apart from \(\mathfrak {u}(\kappa )\) and \(\mathfrak {t}(\kappa )\) we also compute the values of \(\mathfrak {b}(\kappa )\) , \(\mathfrak {d}(\kappa )\) , \(\mathfrak {s}(\kappa )\) , \(\mathfrak {r}(\kappa )\) , \(\mathfrak {a}(\kappa )\) , \(\textrm{cov}({\mathcal {M}}_\kappa )\) , \(\textrm{add}({\mathcal {M}}_\kappa )\) , \(\textrm{non}({\mathcal {M}}_\kappa )\) , \(\textrm{cof}({\mathcal {M}}_\kappa )\) which will all be 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 \(\kappa ^+\) . Finally, as a corollary of the construction, we observe that items (i) and (ii) hold also for the traditional invariants on \(\kappa = \omega \) , using Mitchell forcing up to a weakly compact cardinal; in this case we also obtain the disjoint stationary sequence property \(\textsf {DSS}(\omega _2)\) , which implies the negation of the approachability property \(\lnot \textsf {AP}(\omega _2)\) .
