The Gangolli estimates for the coefficients of the Harish-Chandra expansions of the Eisenstein integrals on real reductive Lie groups
Let G be a real reductive Lie group and \(G=KAN\) an Iwasawa decomposition of G. Let M be the centralizer of A in K and W(A) the Weyl group of (G,A). We denote by \(F_ R\), \(R_ C\) and F the real dual space of the Lie algebra \({\mathfrak a}\) of A, its complexification and \((-1)^{1/2}F_ R\) respectively; \(F^+_ R\) and \(A^+\) the positive Weyl chambers of \(F_ R\) and A. Let \(\tau =(\tau_ 1,\tau_ 2)\) be a unitary double representation of K on V and let \(V_ M\) be the set of all \(v\in V\) satisfying \(\tau_ 1(m)v=v\tau_ 2(m)\) (m\(\in M)\). Then the Eisenstein integral E(v,\(\nu\),h) \((v\in V_ M\), \(\nu \in F_ c\), \(h\in A^+)\) is expanded into the series of the form: \(h^{\rho}E(v,\nu,h)=\sum_{w\in W(A)}\Phi (W\nu:h)C_{\tau}(w:\nu)\) and \(\Phi (\nu:h)=\sum_{\lambda \in L}\Gamma_{\lambda}(\nu -\rho)h^{\nu -\rho}\), where L is a lattice of simple roots for (G,A). When \(\tau\) is trivial, a remarkable estimate for \(\Gamma_{\lambda}(\nu)\) was obtained by G. Gangolli: there exist constants d, D such that \(| \Gamma_{\lambda}(\nu -\rho)| \leq Dm(\lambda)^ d\) for all \(\nu \in R=\{\nu =\xi +\eta\); \(\xi\in F\), \(- \eta \in CL(F^+_ R)\}\) and \(\lambda\in L\). However, for general \(\tau\) some singularities of \(\Gamma_{\lambda}\) arise, and thus we need a modification of the estimate. Actually, by multiplying a polynomial \(P_{\lambda}\) which vanishes the singularities, the authors deduce that there exist constants D, d, \(d_ 1\) such that \[ \| P_{\lambda}(\nu)\Gamma_{\lambda}(\nu -\rho)\| \leq D(1+\| \nu \| +m(\lambda))^{2d}m(\lambda)^{d_ 1}\text{ for all }\nu \in R,\quad \lambda \in L. \]
scientific article; zbMATH DE number 4069216