Calculus of principal series Whittaker functions on \(\text{GL}(3,\mathbb C)\)
Let \(G=\text{GL}(3,\mathbb C)\) with Lie algebra \(\mathfrak g\) and \(K\) a maximal compact subgroup of \(G\). Consider the Iwasawa decomposition \(G=NAK\) of \(G\) and the corresponding minimal parabolic subgroup \(P=MAN\) of \(G\). A nondegenerate character \(\eta\) of \(N\) defines a bundle over \(N\backslash G\). Denote by \(C^{\infty}_{\eta}(N\backslash G)\) the space of its smooth sections on which the group \(G\) acts by right translation \(C^{\infty}\text{Ind}_{N}^{G}\eta\). Let \(\pi\) be an irreducible admissible representation of \(G\) and choose a \(K\)-type \((\tau^{\star},V_{\tau}^{\star})\) which occurs with multiplicity one in \(\pi\). Here \((\tau^{\star},V_{\tau}^{\star})\) stands for the contragredient representation of a finite dimensional representation \((\tau,V_{\tau})\) of \(K\). Fix an injective \(K\)-homomorphism \(i\in \Hom_{K}(\tau^{\star},\pi\mid_{K})\). Associated with an element \(T\in \Hom_{(\mathfrak g,K)}(\pi,C^{\infty}\text{Ind}_{N}^{G}\eta)\) is an element \(T_{i}\in \Hom_{K}(\tau^{\star},C^{\infty}\text{Ind}_{N}^{G}\eta)\) defined by: \(T(i(v^{\star}))(h)=\langle v^{\star},T_{i}(h)\rangle\) for all \(v^{\star}\in V_{\tau}^{\star}\), \(h\in G\). The elements of the space \[ \text{Wh}(\pi,\eta,\tau)=\cup_{i\in \Hom_{K}(\tau^{\star},\pi\mid_{K})}\{T_{i}\in \Hom_{K}(\tau^{\star},C^{\infty}\text{Ind}_N^G\eta)\mid T\in \Hom_{(\mathfrak g,K)}(\pi,C^{\infty}\text{Ind}_N^G\eta)\} \] are called Whittaker functions on \(G\) with respect to \((\pi,\eta,\tau)\). It is difficult to give explicit formulas for these functions for a general \(G\), even in low dimensional cases. In the paper under review, the authors give explicit formulas for Whittaker functions on \(\text{GL}(3,\mathbb C)\). They use the so-called Gelfand-Zelevinsky basis of simple \(K\)-modules and the Dirac-Schmid operators. They also relate Whittaker functions on \(\text{GL}(3,\mathbb C)\) with Whittaker functions on \(\text{GL}(2,\mathbb C)\), using some propagation formula.
scientific article; zbMATH DE number 5553029
scientific article