A note on the Kirillov model for representations of GL(n)(C)

作者：Kemarsky Alexander^*

来源：Comptes Rendus Mathematique, 2015, 353(7): 579-582.

DOI：10.1016/j.crma.2015.04.002

Let G = GL(n)(C) and 1 not equal psi : C -> C-x be an additive character. Let U be the subgroup of upper triangular unipotent matrices in G. Denote by theta the character
theta : U -> C given by theta(u) : = psi (u(1.2) + u(2.3) + ... + u(n-1).n).
Let P be the mirabolic subgroup of G consisting of all matrices in G with the last row equal to (0, 0,..., 0, 1). We prove that if pi is an irreducible generic representation of GL(n)(C) and W(pi, psi) is its Whittaker model, then the space {f vertical bar(P) : P -> C: f is an element of W(pi, psi)) contains the space of infinitely differentiable functions f : P -> C that satisfy f(up) = psi(u) f (p) for all u is an element of U and p is an element of P and that have a compact support modulo U. A similar result was proven for GL(n)(F), where F is a p-adic field by Gelfand and Kazhdan (1975) [1] and for GL(n)(R) by Jacquet (2010) [2].

• 出版日期2015-7