Weil Representation, Howe Duality, and the Theta correspondence

5 Howe Conjecture in the Archimedean caseThe Howe conjecture makes sense in the archimedean case too. In this case oneworks with the Harish-Chandra module of the unitary metaplectic representationdefined on square integrable functions on a maximal totally isotropic subspace byformulae exactly similar to the one in the non-archimidean case. We give below theHarish-Chandra module in the Fock model which is more convenient to work with.We will work with the two sheeted covering Sp(n, ˆ R) of Sp(n, R) and the inverseimage Û(n) of U(n) ⊂ Sp(n, R) as the maximal compact subgroup of Sp(n, ˆ R).Let sp be the Lie algebra of Sp(n, ˆ R) and u, the Lie algebra of U(n). One has theCartan decompositionsp = u ⊕ p = sp 1,1 ⊕ sp 2,0 ⊕ sp 0,2 ,with u = sp 1,1 . The Fock model of the Weil representation of (sp, Û(n)) is realisedon the space S of polynomial functions on C n , with representations of u, p 2,0 , p 0,2given as follows.(1) u operates via the differential operatorsz i∂+ 1 ∂z j 2 δ ij, 1 ≤ i, j ≤ n,(2) p 2,0 operates by multiplication by z i z j , 1 ≤ i, j ≤ n,(3) p 0,2 operates via the differential operators∂ 2∂z i ∂z j, 1 ≤ i, j ≤ n.The action of Û(n) is the standard representation of U(n) on polynomials onC n twisted by a character of Û(n) which does not factor through U(n).Let now (G, G ′ ) be a dual reductive pair in Sp(n) with maximal compact subgroupK in Ĝ and K′ in Ĝ′ such that K · K ′ is contained inÛ(n), and let g bethe Lie algebra of G, and g ′ of G ′ . For any irreducible, admissible (g, K)-module(π, V π ), let S(π) and S[π] be defined as in the non-archimedean case but now takinghomomorphisms in the sense of (g, K)-modules. Then Howe proves in [H2] thatjust as in Lemma 4.2,S[π] ∼ = π ⊗ σ 0 (π)17