A theorem of density for Kloosterman integrals
This article proves density theorems for Kloosterman orbital integrals for the group \(GL_n\) and for the relative orbital integrals attached to a quadratic extension. The integrals in question arise in the study of the relative trace formula. The theorems here show in particular that the big cell matching of certain orbital integrals (the fundamental lemma) implies the matching for the remaining cells. Let \(F\) be a nonarchimedean local field of characteristic zero, and \(\psi\) be a non-trivial additive character of \(F\). Let \(N(F)\) be the upper triangular unipotent subgroup of \(GL_n(F)\), and let \(\theta\) be the character of \(N(F)\) defined by \(\theta(n)=\psi(\sum n_{i,i+1})\). Let \(w\) be a permutation matrix, \(a\) be a diagonal matrix, and \(\Phi\) be a smooth function of compact support on \(GL(n,F)\). Then the Kloosterman orbital integral is \(I(wa,\Phi)= \int \Phi(^tn_2wan_1)\theta(n_1n_2) dn_1 dn_2\). Here \(wa\) is assumed to be relevant, and the integral is over a suitable quotient of \(N(F)\times N(F)\). The author's first main theorem states that if \(I(a,\Phi)=0\) for all diagonal \(a\), then in fact all integrals \(I(wa,\Phi)\) vanish for all \(w,a\) with \(wa\) relevant. Similarly, let \(E/F\) be a quadratic extension, \(\Psi\) a smooth function of compact support on the Hermitian invertible matrices in \(GL_n(E)\), and \(J(wa,\Psi)=\int \Psi(^t{\overline n} wa n) \theta(\overline n n) dn\) be the relative orbital integral. Here bar denotes the Galois conjugate. The author's second main theorem states that if \(I(a,\Phi)=\mu_n(a) J(a,\Psi)\) for all diagonal \(a\), then for all relevant \(wa\), \(I(wa,\Phi)=\mu_n(a)J(wa,\Psi)\). Here \(\mu_n(a)\) is a specific character of the group \(A(F)\) of diagonal matrices given in terms of the character of \(E/F\). The proof involves the study of the germ expansions of the orbital integrals and an explicit computation of asymptotics.
scientific article; zbMATH DE number 1531005