Howe Correspondence for Real Unitary Groups

Arithmetic theta lifting andL-derivatives for unitary groups, I

“…By the theta dichotomy proved in [Paul 1998;Gong and Grenié 2011], we get a factor (π, χ) (see Section 2D for a precise definition) which is the product of the local factors (π v , χ v ) for each place v of F, such that (π v , χ v ) ∈ {±1} and (π v , χ v ) = 1 for almost all v. Although it is conjectured that this (π v , χ v ) is related to the local -factor in representation theory (see [Harris et al 1996]), it is not by our definition. From these local factors, we can construct a hermitian space ‫(ޖ‬π, χ ) over ‫ށ‬ E of rank 2n which is coherent (resp.…”

“…Moreover, Bruinier and Yang [2009] used regularized theta lifting and related the inner product to L-derivatives to give another proof of the original Gross-Zagier formula. A certain p-adic (or rigid analytic) version of the Gross-Zagier formula has been studied in [Bertolini and Darmon 1997;1998]. There is another approach to studying L-derivatives via doubling integrals and in general derivatives of Eisenstein series, discovered by Kudla [1997;2002;2003;Kudla et al 2006].…”

Howe's correspondence for a generic harmonic analyst

“…Anyone interested in the efforts to describe the correspondence in terms of Langlands parameters might consult [Pau98], [Pau00] and [AB95]. There are connections with particle physics [How85] and the theory of automorphic forms, [How79].…”

On Doubling Construction for Real Unitary Dual Pairs

“…The Howe duality correspondence asserts that the relation Hom G V × G W (ω W , π V ⊗ π W ) = 0 determines a one-to-one correspondence between R( G V , ω W ) and R( G W , ω W ) (see [How89]). In this unitary dual pair case, an explicit description of the correspondence was obtained by Paul [Pau98,Pau00].…”