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By Palais, Richard

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Example 2. (U (1), U (n)). Here U (1) is the center of the maximal compact subgroup K = U (n). This dual pair describes the restriction of the oscillator representation to K, which is quite important. 30 Jeffrey Adams Fix ψ. 6) the correspondence from U (1) to U (n) is (k + n2 ) → (k + 12 , 21 , . . , 12 ), k = 0, 1, 2, . . The K–types with k even (resp. odd) constitute the irreducible summand ωψ+ (resp. ωψ− ). The lowest K–types of these two summands (both in the sense of Vogan, and of lowest degree) are ( 21 , .

Morimoto and K. Yamaguchi (see, for instance, [5, 7, 10]). Rigidity is a blessing and a curse. It makes life simpler, but it makes it harder to choose coordinates. For smooth maps, much is known about rigidity. If N is the nilradical of a parabolic subgroup of a semisimple Lie group G, then, as shown by Yamaguchi, N is rigid (and Contact(U, V ) is essentially a subset of G) except in a limited number of cases. , [9]). The Heisenberg Group, SL(3, R), and Rigidity 43 In a fundamental paper, P. Pansu [6] studied quasiconformal and weakly contact maps on Carnot groups (stratified groups with a natural distance function).

The image 1 of 1 in the quotient F + /Nλ is non–zero, and X · 1 = z 2 = λ1. The case of F − is similar. The embedding of the second copy G2 of GL(1) is ι2 : x = diag(x−1 , x). The natural choice of ι2 is (x, ) → (ι2 (x), )). With this convention ι−1 2 ◦ι1 : G1 → G2 takes (x, ) to (x−1 , ) = (x, sgn(x))−1 . With these choices, the duality correspondence is χ → χ−1 sgn for any genuine (g1 , K1 ) character χ. The sgn term comes from the twist by sgn(x) in ι−1 2 ◦ ι1 . Of course ι2 can be modified to eliminate the twist by sgn.