The representation theory of generalized hyperoctahedral groups

We give an explicit decomposition of $\text{Ind}(1{)}_{B_n}^{S_{2n}}$, following Barbasch and Vogan [1]. We define two natural generalizations of $B_n$, and extend the proof in [1] to recursively compute these decompositions. Although the decompositions do not appear to follow a simple pattern, we prove enough of their structure to show that they are almost never multiplicity-free.