Duality between \(\mathrm{GL}(n,\mathbb{R})\), \(\mathrm{GL}(n,\mathbb{Q}_{p})\), and the degenerate affine Hecke algebra for \(\mathfrak {gl}(n)\) (Q2879627)

The authors prove an instance of the Lefschetz principle by proving a beautiful relationship between representations of \(GL(n, \mathbb{R})\) and of \(GL(n, \mathbb{Q}_p)\). The latter comes in the guise of the degenerate affine Hecke algebra \({\mathcal{H}}_n\) corresponding to \(\mathfrak{gl}(n)\) which controls the category of Iwahori-spherical representations of \(GL(n, \mathbb{Q}_p)\). More precisely, the authors define a functor \(F_{n,k}\) from the category \(HC_n\) of Harish-Chandra modules for \(GL(n, \mathbb{R})\) to the category of finite-dimensional representations of \({\mathcal{H}}_k\). This functor is defined rather simply as follows.NEWLINENEWLINELet \(V\) denote the standard representation of \(GL(n, \mathbb{R})\) and let `sgn' denote the determinant representation of \(O(n)\). Given a Harish-Chandra module \(X\) for \(GL(n, \mathbb{R})\), the authors define NEWLINE\[NEWLINEF_{n,k}(X) = \Hom_{O(n)} (1, (X \otimes \text{sgn}) \otimes V^{\otimes^k}).NEWLINE\]NEWLINE It is known that \({\mathcal{H}}_k\) acts on \(Y \otimes V^{\otimes^k}\) for any \(U(\mathfrak{gl}(n, \mathbb{C}))\)-module \(Y\). This action commutes with the \(O(n)\)-action which makes \(F_{n,k}(X)\) a module over \({\mathcal{H}}_k\). The functor \(F_{n,k}\) is exact and covariant. But, unfortunately the functor is not well-behaved on the whole category \(HC_n\). One of the accomplishments of the authors is to identify a subcategory which is the real analogue of subquotients of spherical principal series and show that the functor is nice on the objects there. They introduce a notion of a level of modules in \(HC_n\) and define a subcategory of modules of level at least \(k\). The main theorem asserts the following.NEWLINENEWLINE Suppose \(X\) is an irreducible Harish-Chandra module for \(GL(n, \mathbb{R})\) whose level is at least \(n\). Then, \(F_{n,n}(X)\) is irreducible or zero. Moreover, \(F_{n,n}\) implements a bijection between irreducible Harish-Chandra modules of level equal to \(n\) and irreducible \({\mathcal{H}}_n\)-modules with the same infinitesimal character.NEWLINENEWLINEIn a later paper [``Functors for unitary representations of classical real groups and affine Hecke algebras'', Adv. Math. 227, No. 4, 1585--1611 (2011; Zbl 1241.22018)], the authors generalize the functors of this paper to other classical groups. Those results are not as neat the ones in this paper for \(GL(n)\).