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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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