By Jian-Shu Li, Eng-Chye Tan, Nolan R Wallach, Chen-Bo Zhu

This quantity contains an identical name as that of a world convention held on the nationwide college of Singapore, nine eleven January 2006 at the celebration of Roger E. Howe s sixtieth birthday. Authored via major contributors of the Lie idea neighborhood, those contributions, increased from invited lectures given on the convention, are a becoming tribute to the originality, intensity and effect of Howe s mathematical paintings. the variety and variety of the subjects will entice a large viewers of analysis mathematicians and graduate scholars attracted to symmetry and its profound functions.

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22] Roger Howe. Perspectives on invariant theory: Schur duality, multiplicityfree actions and beyond. In The Schur lectures (1992) (Tel Aviv), volume 8 of Israel Math. Conf. , pages 1–182. , Ramat Gan, 1995. [23] Roger Howe and Eng-Chye Tan. Nonabelian harmonic analysis. Universitext. Springer-Verlag, New York, 1992. Applications of SL(2, R). [24] James E. Humphreys. Introduction to Lie algebras and representation theory, volume 9 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1978.

In the p–adic case this is covered in the lectures of Brooks Roberts [43], see also [33], [3]. For unexplained notation see [3]. 1 Let π1 = L(µ1 , ν1 ) be an irreducible representation of G1 . The Theta Correspondence over R 33 Define the integer k = k[µ1 ] by writing µ1 = (a1 , . . , ak , 0, . . , 0; ) with a1 ≥ a2 ≥ · · · ≥ ak > 0. Write ν1 = (b1 , . . , bm ), and define the integer 0 ≤ q = q[µ1 , ν1 ] ≤ m − k to be the largest integer such that 2q − 2 + τ, 2q − 4 + τ, . . , τ all occur (in any order) in {±bk+1 , ±bk+2 , .

2 2 2 2 All such weights occur, subject to the obvious constraints k + ≤ p, k ≤ m, ≤ n. U (m, n) : III. [32] (Sp(p), O ∗ (2n)). Both Sp(p) and O ∗ (2n) are connected, and the coverings split, so we consider only the linear groups. The maximal compact subgroup of O ∗ (2n) is isomorphic to U (n). Sp(p) : O∗ (2n)) : σ = (a1 , . . , ak , 0, . . , 0) → τ = (a1 + p, . . , ak + p, p, . . , p) . All such weights occur, subject to the obvious constraints k ≤ p, n. 7. Joint Harmonics The proof of the results in Section 5 depends heavily on information about K–types.