Parity preservation of $K$-types under theta correspondence
  This note shows a property of degree-parity preservation for $K$-types under Howe's theta correspondence. As its application, we deduce the preservation of parity of all $K$-types occurring in an arbitrary irreducible $(\mathfrak{g},K)$-module of any Lie group in reductive dual pairs. 
Xiang Fan
math.RT