Howe ' s Rank and Dual Pair Correspondence in Semistable Range

ABSRACT Let G be a classical group of type I. For an irreducible unitary representation, Howe defined the notion of rank in analytic terms. On the algebraic side, there is the theory of primitive ideals and associated variety. In the first part of this thesis, we relate Howe's rank with the associated variety. In the second part, We study the Bargmann-Segal model of the oscillator representation. Based on this model, we construct an analytic compactification of the symplectic group. We also construct an analytic compactification of the orthogonal group. All the compactifications are compact symmetric spaces. In the third part, we define semistable range in the dual pair correspondence, and give an explicit construction of the dual pair correspondence in the semistable range. Finally, we prove the nonvanishing theorems of the dual pair correspondence in the semistable range for (Op,q, SP2n (R)). Our proof is based on some density theorems on some compact symmetric spaces. The LORD is my shepherd; I shall not want. He maketh me to lie down in green pastures: he leadeth me beside the still waters. He restoreth my soul: he leadeth me in the paths of righteousness for his name's sake. Yea, though I walk through the valley of the shadow of death, I will fear no evil: for thou art with me; thy rod and thy staff they comfort me. Thou preparest a table before me in the presence of mine enemies: thou anointest my head with oil; my cup runneth over. Surely goodness and mercy shall follow me all the days of my life: and I will dwell in the house of the LORD for ever.