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

This quantity incorporates an identical identify as that of a global convention held on the nationwide collage of Singapore, Sep 11 January 2006 at the celebration of Roger E. Howe's sixtieth birthday. Authored by means of prime contributors of the Lie concept group, those contributions, extended from invited lectures given on the convention, are a becoming tribute to the originality, intensity and impression of Howe's mathematical paintings. the variety and variety of the subjects will entice a extensive viewers of study mathematicians and graduate scholars attracted to symmetry and its profound functions.

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**Extra resources for Harmonic analysis, group representations, automorphic forms, and invariant theory: in honor of Roger E. Howe**

**Example text**

Cm , 0, . . , 0, d1 , . . , dn )+ 1 (r − s, . . , r − s; s − r, . . , s − r) → 2 σ = (a1 , . . , ak , 0, . . , 0, d1 , . . , dn ; c1 , . . , cm , 0, . . , 0, b1 , . . , b )+ 1 (p − q, . . , p − q; q − p, . . , q − p) 2 where the obvious inequalities hold: k + ≤ p, m + n ≤ q, k + n ≤ r, m + ≤ s III. (G, G ) = (Sp(p, q), O ∗ (2n)), (K, K ) = (Sp(p) × Sp(q), U (n)). (a1 , . . , ar , 0, . . , 0; b1 , . . , bs , 0, . . , 0) → (a1 , . . , ar , 0, . . , 0, −b1 , . . , −bs ) + (p − q, .

Finally (O(p, q), Sp(2nR)) witih p + q = 2n + 1 is in [4], this is similar to [33] except that the covering groups are unavoidable. We first consider the case p, q even. In this case the covering of Sp(2n, R) splits and the correspondence can be written in terms of the linear groups. Roughly speaking the correspondence in these cases is “functorial”, and a number of nice properties hold which fail in general. In particular the minimal K–type in the sense of Vogan is always of minimal degree in this situation.

Ak , 0, . . , 0; ) 1− 2 Sp(2n, R) : (p−2k) p p p p p p τ = (a1 + , . . , ak + , + 1, . . , + 1, , . . , ) . 2 2 2 2 2 2 All such highest weights occur, subject to the constraints k ≤ [ p2 ] and k + 1− 2 (p − 2k) ≤ n. 20 Jeffrey Adams This means that the weight σ for O(p) is the highest weight of the irreducible representation σ, and the weight for Sp(2n, R) is the highest weight of the K –type of τ of lowest degree in π . II. (U (p), U (m, n)) The inverse image K of U (p) in Sp(2p(m + n), R) is isomorphic to the p m+n cover defined by the character det 2 .