Invariants of theta correspondences

2 Jan 2015. NUS mathematicians describe one powerful method called the local theta correspondence or Howe correspondence.

There are many methods to produce representations. A team led by LOKE Hung Yean from the Department of Mathematics in NUS describes one powerful method called the local theta correspondence or Howe correspondence. They found a formula that computes the associated cycles of local theta lifts using geometric pictures called the moment maps. Let W be a symplectic real vector space. Let (G,G‘) be a pair of classical group in the metaplectic double cover Mp(W) of the symplectic group Sp(W) which are mutual centralizers of each other. They call this a dual pair. The dual pair correspondences or Howe correspondences is to use the oscillator representation of Mp(W) to produce a representation (V) of G‘ from an irreducible representation V of G. The representation (V) is called the full local theta lift of V. It has an irreducible quotient (V) called the local theta lift of V.

There is a technical term called the stable range condition if the group G is small compared to G’. It is a general consensus that under the stable range condition, the Howe correspondence is well behaved. For example, it is previously known that (V) is nonzero under stable range. In practice (V) is much easier to handle but (V) is more useful. The team shows that for a type I dual pair, if V is an irreducible unitary representation then (V) is equal to (V) with one exception.

They have also investigated the relationship between the associated cycles of V and that of (V). The main result is that under the stable range condition, the associated cycle of (V) could be computed from that of V via a purely geometric picture (see Figure).

Hereand are called the moment maps. They do not require that V is unitary.

If they do not assume the stable range condition, then there are counterexamples in which the result on associated cycles fail. On the other hand, these counterexamples occur under more acute situations. The team shows that the main result continues to hold for some specific representations if they relax the stable range condition moderately.

References

1. Hung YL, JJ Ma. Invariants and K-spectrums of local theta lifts, to appear in Compositio Math. (2013), available at ArXiv:1302.1031.
2. Hung YL, JJ Ma, UL Tang. Associated cycles of local theta lifts of unitary characters and unitary lowest weight modules, available at ArXiv:1207.6451 (2012).