On the correspondence of representations between 𝐺𝐿(𝑛) and division algebras

Abstract

For a division algebra D D over a p p -adic field F , F, we prove that depth is preserved under the correspondence of discrete series representations of G L n ( F ) GL_n(F) and irreducible representations of D ∗ D^* by proving that an explicit relation holds between depth and conductor for all such representations. We also show that this relation holds for many (possibly all) discrete series representations of G L 2 ( D ) . GL_2(D).