Equivariant resolutions of singularities for orbits in generalized quiver varieties arising in the local Langlands program for p-adic groups

In this thesis, we investigate geometric aspects of the Langlands parameters arising in the local Langlands Program for p-adic groups. This work was inspired by David Vogan’s “The Local Langlands Conjecture”, and we built off of the work of Clifton Cunningham, Andrew Fiori, Ahmed Moussaoui, James Mracek, and Bin Xu in their book “Arthur Packets for p-Adic Groups by Way of Microlocal Vanishing Cycles of Perverse Sheaves, with Examples”. This thesis follows the construction detailed in Part II of their book, and we give more concrete examples to demonstrate their construction. Additionally, we provide an exposition on the local Langlands program for p-adic groups and give an algorithm for computing the resolutions of singularities arising from the study of the orbits of certain generalized quiver varieties for their respective Langlands parameters in the groups GLn, SOn, and Sp2n.