Harmonic Analysis, Group Representations, Automorphic Forms by Jian-Shu Li, Eng-Chye Tan, Nolan R Wallach, Chen-Bo Zhu

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 get together of Roger E. Howe s sixtieth birthday. Authored by way of prime individuals of the Lie idea neighborhood, those contributions, elevated from invited lectures given on the convention, are a becoming tribute to the originality, intensity and effect of Howe s mathematical paintings. the diversity and variety of the themes will entice a vast viewers of analysis mathematicians and graduate scholars drawn to symmetry and its profound functions.

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7] P. Cartier. Quantum mechanical commutation relations and θ-functions, volume 9 of Proc. Symp. Pure Math. American Math. , Providence, Rhode Island, 1966. [8] Mark G. Davidson, Thomas J. Enright, and Ronald J. Stanke. Differential operators and highest weight representations. Mem. Amer. Math. , 94(455):iv+102, 1991. [9] Thomas Enright, Roger Howe, and Nolan Wallach. A classification of unitary highest weight modules. In Representation theory of reductive groups (Park City, Utah, 1982), volume 40 of Progr.

I. (O(p), Sp(2n, R)). The irreducible representations of SO(p) are parameterized by highest weights (a1 , . . , a[ p2 ] ) with ai ∈ Z and a1 ≥ a2 ≥ · · · ≥ a[ p2 ]−1 ≥ |a[ p2 ] |. If p is odd we may assume a[ p2 ] ≥ 0. Following Weyl [55] we parameterize representations of O(p) by restriction from U (p). Consider the irreducible representations of U (p) given by Young’s diagrams with p rows of length a1 , . . , ap , or equivalently with highest weight (a1 , . . , ap ) with a1 ≥ a2 ≥ · · · ≥ ap ≥ 0.

Given an irreducible representation π1 of G1 , there exists a non–negative integer n(π1 ) such that π occurs in the duality correspondence for (G1 , G2 (n)) if and only if n ≥ n(π1 ). 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.

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