Chen-Bo Zhu

Conservation relations for local theta correspondence

We prove Kudla-Rallis conjecture on first occurrences of local theta correspondence, for all type I irreducible dual pairs and all local fields of characteristic zero.

Degenerate principal series and local theta correspondence

Following our previous paper [LZ] which deals with the groupU(n, n), we study the structure of certain Howe quotients Ω p,q and Ω p,q (1) which are natural Sp(2n,R) modules arising from the Oscillator representation associated with the dual pair (O(p, q), Sp(2n,R)), by embedding them into the degenerate principal series representations of Sp(2n,R) studied in [L2].

Degenerate principal series and local theta correspondence II

Following our previous paper [LZ] which deals with the groupU(n, n), we study the structure of certain Howe quotients Ωp,q and Ωp,q(1) which are natural Sp(2n,R) modules arising from the Oscillator representation associated with the dual pair (O(p, q), Sp(2n,R)), by embedding them into the degenerate principal series representations of Sp(2n,R) studied in [L2].

The explicit duality correspondence of (Sp(p, q), O{*}(2n))

Abstract We investigate the type I dual pairs over the quaternion algebra H , namely the family of dual pairs (Sp(p,q),O ∗ (2n)) . We give a complete and explicit description of duality correspondence for p + q ⩽ n as well as some of the cases for p + q > n , in terms of the Langlands parameters.

Uniqueness of Ginzburg-Rallis models: The Archimedean case

In this paper, we prove the uniqueness of Ginzburg-Rallis models in the archimedean case. As a key ingredient, we introduce a new descent argument based on two geometric notions attached to submanifolds, which we call metrical properness and unipotent

Theta lifting of unitary lowest weight modules and their associated cycles

\chi

Weyl’s construction and tensor power decomposition for ₂

-incompatibility.

Representations with scalarK-types and applications

We consider a reductive dual pair (G,G′) in the stable range with G′ the smaller member and of Hermitian symmetric type. We study theta lifting of (holomorphic) nilpotent K ′ C -orbits in relation to theta lifting of unitary lowest weight modules of G′. We determine the associated cycles of all such representations. In particular, we prove that the multiplicity in the associated cycle is preserved under theta lifting.

Theta lifting of holomorphic discrete series: The case of (,)×(,)

Let V be the 7-dimensional irreducible representations of G2. We decompose the tensor power V ⊗n into irreducible representations of G2 and obtain all irreducible representations of G2 in the decomposition. This generalizes Weyl’s work on the construction of irreducible representations and decomposition of tensor products for classical groups to the exceptional group G2.

Representations of real and p-adic groups

We discuss some results of Shimura on invariant differential operators and extend a folklore theorem about spherical representationas to representations with scalarK-types. We then apply the result to obtain non-trivial isomorphisms of certain representations arising from local theta correspondence, many of which are unipotent in the sense of Vogan.