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For each nontrivial unipotent orbit O, there is a subset TO of the set of regular elements in T such that the following asymptotic expansion holds. FfT (t) = |D(t)|1/2 If (t) ∼ −AT |D(t)|1/2 f (1) + BT CO (t)ΛO (f ) dim O>0 where the Shalika germ CO (t) is the characteristic function of TO . The constants AT and BT depend on normalization of measures and whether T is ramified or unramified. By summing products of characters, we are led to the following expression. 1 μ(T )If (t) = χΠ (t)fˆ(Π) + χΠ (t)fˆ(Π) 2 Π∈D Π∈RPSV − q+1 μ(A1 ) 2q + q μ(A1 )κT |D(t)|−1/2 2 ξ ∈ F× ξ|Ah0 +1 = 1 |Γ(ξ)|−2 fˆ(ξ)dξ ξ ∈ F× ξ|Ah0 +1 = 1 fˆ(ξ)dξ This is the Fourier transform of the elliptic orbital integral corresponding to the regular element t.

Herb. Fourier inversion and the Plancherel theorem for semisimple real Lie groups. Amer. J. , 104(1):9–58, 1982. [49] Rebecca A. Herb. Discrete series characters and Fourier inversion on semisimple real Lie groups. Trans. Amer. Math. , 277(1):241–262, 1983. [50] Rebecca A. Herb. Elliptic representations for Sp(2n) and SO(n). Pacific J. , 161(2):347– 358, 1993. [51] Rebecca A. Herb. Supertempered virtual characters. , 93(2):139–154, 1994. [52] Rebecca A. Herb. Discrete series characters and two-structures.

Then apply the Mackey theory. 6. 1) The regular representation on L2 (G/G ) = L2 (GL(n, R)/O(n)) is unitarily equivalent to the restriction of a holomorphic discrete series representation of G := Sp(n, R) to G. 2) The regular representation on L2 (GL(n, R)/GL(p, R) × GL(q, R)) with (p + q = n) is unitarily equivalent to the restriction of a degenerate principal representation of G := GL(n, R) × GL(n, R) to G (namely, to the tensor product representation). 6. Wild aspects of branching laws. Retain our assumption that (G, G ) is a reductive symmetric pair.

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Complex Numbers and Complex Arithmetic [article] by Doe J.

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