Paley-Wiener theorem for Whittaker functions on a \(p\)-adic reductive group (Q2894443)

Let \(P_0\) be a minimal parabolic subgroup of \(G\) and \(U_0\) be its unipotent radical. Consider the space of Whittaker functions \(C^\infty(U_0\backslash G,\psi)\) consisting of smooth functions \(f(g)\) such that \(f(ug)=\psi(u)f(g)\) with \(u\in U_0\) and \(\psi\) a non-degenerate character of \(U_0\). This paper gives a Paley-Wiener theorem for this space, namely it classifies the image of the Fourier transform of \(C_c^\infty(U_0\backslash G,\psi)\).NEWLINENEWLINEIn more detail, for a representation of finite length \(\pi\) of \(G\), let \(Wh(\pi)\) be the space of Whittaker functionals, namely linear forms \(\xi\) such that \(\xi(\pi(u)v)=\psi(u)\xi(v)\) (\(u\in U_0\) and \(v\in \pi\)). When \(\pi\) is an irreducible unitary supercuspidal representation, Schur's Lemma gives NEWLINE\[NEWLINE\int_{A_GU_0\backslash G}\xi(\pi(g)v)\overline{\xi'(\pi(g)v')}\;dg=\langle \xi,\xi'\rangle_{Wh} (v,v'),\quad v,v'\in\pi,\quad \xi,\xi'\in Wh(\pi).NEWLINE\]NEWLINE Here, \(\langle\cdot,\cdot\rangle_{Wh}\) is a paring on \(Wh(\pi)\). Now let \(P=MU\) be a semi-standard parabolic subgroup of \(G\) (thus \(M_0\subset M\), where \(M_0\) is the Levi subgroup of \(P_0\)). For \(\sigma\) a representation of \(M\), the Jacquet integral gives an isomorphism between \(Wh(\sigma)\) and \(Wh(i_P^G \sigma)\), where \(i_P^G \sigma\) is an induced representation. Thus, any \(\eta\in Wh(\sigma)\) defines a unique \(i(\eta)\in Wh(i_P^G \sigma)\).NEWLINENEWLINETo any \(f\in C_c^\infty(U_0\backslash G,\psi)\), a semi-standard parabolic subgroup \(P\) and an irreducible unitary supercuspidal representation \(\sigma\) of \(M\subset P\), we can associate a unique vector \(F_f(P,\sigma)=(\xi,v)\in Wh(\sigma)\otimes i_P^G\sigma\) such that NEWLINE\[NEWLINE\langle \xi,\xi'\rangle_{Wh}(v,v')=\int_{U_0\backslash G} f(g)\overline{W_{\xi',v'}(g)},\quad \xi'\in Wh(\sigma),\;v'\in i_P^G\sigma,NEWLINE\]NEWLINE where \(W_{\xi',v'}(g)=i(\xi')(i_P^G\sigma(g)v')\). The Paley-Wiener theorem describes the image of the map \(f\mapsto F_f(P,\sigma)\) (inside the space of maps \(F: (P,\sigma)\mapsto Wh(\sigma)\otimes i_P^G\sigma\)). The description is in terms of the behavior of \(F\) under Weyl elements and intertwining operators. The key to the description is a highly nontrivial adjointness relation of a certain \(B\)-matrix.NEWLINENEWLINEWe note that the paper does not assume \(G\) is quasi-split, thus the space \(Wh(\sigma)\) is not necessarily one-dimensional.NEWLINENEWLINEThe proof proceeds by an explicit construction of the function \(f\) from a given map \(F\) satisfying the right conditions. Roughly, given \(F\), we can associate wave packets to it which are functions in \(C_c^\infty(U_0\backslash G,\psi)\), then a linear combination of these wave packets gives the right function \(f\). For analytic considerations, it is necessary to use shifted wave packets (where the integration converges) in place of wave packets. A nice trick using Bernstein center allows one to establish the identity between the shifted wave packets and the wave packets. Another key ingredient is a result of Heiermann.NEWLINENEWLINEThe fact that \(f\mapsto F_f\) is injective is also established, using a result of Bernstein. The paper also establishes a conjecture of Lapid and Mao, which states that, if there is \(\xi\in Wh(\pi)\) such that \(\xi(\pi(g)v)\) are always square integrable, then \(\pi\) is square integrable. An analogue for supercuspidal representations is also proved, namely, if there is \(\xi\in Wh(\pi)\) such that \(\xi(\pi(g)v)\) are always compactly supported modulo \(A_GU_0\), then \(\pi\) is supercuspidal.