Construction of central elements in the affine Hecke algebra via nearby cycles
Let \(G\) be a connected reductive group over a finite field \(\mathbb{F}_q\). The corresponding group \(G(K)\) over the local field \(K= \mathbb{F}_q((t))\) contains \(G({\mathcal O})\) as a maximal compact subgroup, where \({\mathcal O}:= \mathbb{F}_qt\), and the latter group gives rise to the Hecke algebra \(H\) of compactly supported bi-\(G({\mathcal O})\)-invariant functions \(f: G(K)\to\overline{\mathbb{Q}}_\ell\), equipped with the usual convolution product. It is known that the Hecke algebra \(H\) is commutative, whereas the Hecke algebra \(H_I\) associated with the Iwahori subgroup \(I\subset G({\mathcal O})\) is noncommutative. Moreover, if \(G\) is a split group, then \(H\) is isomorphic to the Grothendieck ring of the category of finite-dimensional representations of the Langlands dual group \(\check{G}\). The existence of such an isomorphism has been established by I. Satake many years ago. On the other hand, a result by \textit{I. N. Bernstein} [in: Représentations des groupes réductifs sur un corps local, 1--32 (1984; Zbl 0599.22016)] states that the representation ring of the Langlands dual group \(\check{G}\) is also isomorphic to the center of the Hecke algebra \(H_I\). Therefore there is a well-defined isomorphism \(\pi: Z(H_I)\widetilde\to H\), the so-called Satake-Bernstein isomorphism, and the main goal of the paper under review is to describe explicitely the inverse of this isomorphism. This is done by giving geometric interpretations of both \(H\) and \(H_I\) in terms of certain perverse sheaves. More precisely, the author considers the quotients \(Gr:= G(K)/G({\mathcal O})\) and \(Fl:= G(K)/I\) as group ind-schemes over \(\mathbb{F}_q\) and studies their associated categories \(P_{G({\mathcal O})}(Gr)\) and \(P_I(Fl)\) of equivariant perverse sheaves (on \(Gr\)) or \(I\)-equivariant perverse sheaves (on \(Fl\)), respectively. The better part of the paper is then devoted to the construction of a functor \(Z: P_{G({\mathcal O})}(Gr)\to P_I(Fl)\) which, the level of Grothendieck groups, induces the inverse Satake-Bernstein map \(\pi^{-1}\). This crucial functor, whose subtle and rather involved construction uses the operation of taking ``nearby cycles'' of perverse sheaves on the affine Grassmannian \(Gr= G(K)/G({\mathcal O})\), is shown to have extremely favorable properties, and supposedly it encodes a deeper representation-theoretic meaning to be explored in the future. As for a different approach toward a geometric interpretation of the Satake-Bernstein isomorphism, the reader is referred to the just as recent paper by \textit{I. Mirković} and \textit{K. Vilonen} [Math. Res. Lett. 7, No. 1, 13--24 (2000; Zbl 0987.14015)].
scientific article; zbMATH DE number 1655624