By the late Hans Reiter, Jan D. Stegeman

A revised and multiplied moment version of Reiter's vintage textual content Classical Harmonic research and in the neighborhood Compact teams (Clarendon Press 1968). It offers with numerous advancements in research centring round round the primary paintings of Wiener, Carleman, and particularly A. Weil. It starts off with the classical idea of Fourier transforms in euclidean house, maintains with a learn at sure normal functionality algebras, after which discusses capabilities outlined on in the neighborhood compact teams. the purpose is, first of all, to carry out basically the family members among classical research and staff conception , and secondly, to review uncomplicated houses of services on abelian and non-abelian teams. The booklet offers a scientific creation to those themes and endeavours to supply instruments for extra study. within the new version correct fabric is additional that was once no longer but on hand on the time of the 1st version.

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1 , 0, . . , 0; +) n−m (a , . . , a , 1, . . , 1, 0, . . , 0; −) 1 k where a1 ≥ · · · ≥ ak > 0, k ≤ [ m 2 ]. 7) 24 Jeffrey Adams V. [3] (G, G ) = (GL(m, C), GL(n, C)), (K, K ) = (U (m), U (n)). (a1 , . . , ak , 0, . . , 0, b1 , . . , b ) → (−b , . . , −b1 , 0, . . , 0, −ak , . . 8) where a1 ≥ · · · ≥ ak > 0 > b1 ≥ · · · ≥ b , k + ≤ m, n. VI. [3] (G, G ) = (O(m, C), Sp(2n, C)), (K, K ) = (O(m), Sp(n)). 1− 2 (m−2k) (a1 , . . , ak , 0, . . , 0; ) → (a1 , . . , ak , 1, . .

If RS is as shown, then the absolute values of the slopes of LS and RM are less than 1/(2n − 1), so if n is large enough, the images of the chords LS and RM lie in lines. Further, no matter what slopes are involved, if R and S lie on AC and CB, then R S is parallel to LM if and only if LS ∩ R M lies on the vertical line T U through the midpoint of LM . Since T U is also preserved, it follows that the image of a horizontal chord is horizontal. We can √ √ repeat this argument to deduce that the images of chords with slopes ± 3 that meet the interior of the triangle also have slopes of ± 3.

The lowest K–types of these two summands (both in the sense of Vogan, and of lowest degree) are ( 21 , . . , 21 ) and ( 32 , 21 , . . , 12 ) respectively. e. their highest weights are obtained from the highest weight of the lowest K–type by adding multiples of a single vector. This is a condition of [53]. In fact the four irreducible summands of the oscillator representations are the only non–trivial unitary representations of Sp(2n, R) (n ≥ 1) with K–types along a line. The oscillator representation is particularly “small”, and the duality correspondence is due in part to this.