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Intertwining operators and residues. II: Invariant distributions. (English) Zbl 0936.22012
Let \(G\) be a connected reductive group over a local field \(F\). The paper is a continuation of Part I [the author, J. Funct. Anal. 84, No. 1, 19–84 (1989; Zbl 0679.22011)], but also makes use of results on the families of invariant distributions \(I_M(\pi,X,f)\), \(I_M(\gamma,f)\) (where \(\pi\in\Pi(M(F))\), \(X\in a_M\), \(\gamma\in M(F))\) in another paper by the author [J. Am. Math. Soc. 1, No. 2, 323–383 (1988; Zbl 0682.10021)]. The main result of this paper, roughly speaking, is an expansion for \(I_M(\pi,X,f)\) in terms of (a) the residues of the \(J_L^M(\pi_{L,\gamma},f_L)\) studied in Part I, (b) the normalizing factors for the intertwining operators also studied in Part I (and in recent work of Shahidi), and (c) the functions \(I_L(\sigma,X_L,f_L)\), where \(L\) is a Levi “subgroup” of \(M\). As a prerequisite to accomplishing this, the author introduces the interesting notion of an admissible family of operators: for two fixed parabolic subgroups \(P,P'\) with Levi \(M\), a set \(\{A(\sigma)\colon\text{Ind}_P^G\sigma\to\text{Ind}_{P'}^G\sigma|\sigma\) a standard representation of \(M\)} is an admissible family if \(A(\sigma)\) is represented by a \(K\)-finite \(\sigma(\mathcal H(M(F))\))-valued kernel function. Attached to any admissible family is an “induced distribution” and to this distribution one may formally associate its Fourier transform. There is a simple reciprocity identity between the induced distribution and its Fourier transform which is used in proving the expansion for \(I_M(\pi,X,f)\) mentioned above. As a consequence of this expansion, the author shows that the residues of \(J_M(\pi,f)\) studied in Part I are supported on characters. In the case where \(f\) is cuspidal, the expansion for \(I_M(\pi,X,f)\) simplifies considerably. If, in addition, \(\pi\) has unitary central character then even further simplification occurs. The author then begins to study the \(I_M(\gamma,f)\) on real groups, for \(f\) cuspidal and discrete, and obtains some very interesting formulas. The first formula might be roughly regarded as a weighted Fourier expansion for \(I_M(\gamma,f)\), where \(\gamma\in T(\mathbf R)_{\text{reg}}\) and \(T\) a maximal torus over \(\mathbf R\) in \(M\) which is antisotropic \(\text{mod} A_M\). The author remarks that this formula is used in a paper on traces of Hecke operators. The second formula, Theorem 6.5, seems roughly speaking to say that for this torus \(T\subset M\), the Fourier expansion of the first formula actually may be rewritten as a Fourier expansion on \(M\). In concluding, the author shows how the residues of \(\{J_M(\pi_\lambda,f)\}\), the distributions \(\{I_M(\pi,X,f)\}\), and the asymptotic behavior of \(\{I_M(\gamma,f)\}\) can be computed from one another in the case \(G\) is rank 1.

MSC:
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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References:
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