On the Mui\'c conjecture--the irreducibility of the big theta lift

Abstract

Let $F$ be a non-archimedean local field of characteristic zero. We study theta correspondence for (complex) representations of symplectic--even orthogonal dual reductive pairs over $F;$ more specifically, the big theta lifts. We prove that, starting from a discrete series representation $\pi$ of a symplectic (even orthogonal group) over $F,$ its big theta lift $\Theta (\pi)$ (as a representation of an even orthogonal (symplectic) group) if non-zero, is an irreducible representation, thus proving a conjecture of Mui\'c. Building upon this result, we completely describe the situations in which the theta lifts of tempered representations are irreducible and when they are not.