A geometric construction of the Iwahori-Hecke algebra for unramified groups
Let \(G\) be a reductive group over a nonarchimedean local field \(F\). The Langlands correspondence conjecture states that the irreducible admissible representations of \(G\) should be parametrized by representations of the Galois group of \(F\) into the complex dual group of \(G\). P. Deligne and G. Lusztig refined this conjecture by inserting more specific data on the Galois representation side. In [Invent. Math. 87, 153-215 (1987; Zbl 0613.22004)] \textit{D. Kazhdan} and \textit{G. Lusztig} were able to prove the conjecture for \(G\) being split over \(F\) and representations \(V\) of \(G\) which have nontrivial fixed vectors under the Iwahori subgroup \(I\). These representations are in bijection with the finite dimensional simple modules of the Iwahori-Hecke algebra \({\mathcal H}_I =C_c (I\setminus G/I)\). The key step in the proof is the construction of an isomorphism of \({\mathcal H}_I\) as a module over itself to the equivariant \(K\)-homology of a generalized flag variety. In the paper under consideration this step is extended to unramified groups, i.e. reductive groups \(G\) which are quasisplit and are split over an unramified extension \(F'\) over \(F\). The author fixes a generator \(\sigma\) of the Galois group of \(F'| F\) and replaces equivariant \(K\)-theory by a quotient \(^\sigma K\) of it which depends on \(\sigma\). The author then goes through the proof which partly can be taken over and partly has to be built anew using different ideas. It does however not seem possible to complete the proof of the Deligne-Langlands conjecture along the lines of [loc. cit.] since there are some properties of equivariant \(K\)-homology, like the existence of a Künneth formula spectral sequence, which are not clear for the modified functor \(^\sigma K\). Fortunately, the author was able to get around these difficulties by other methods. This final step appeared in a different paper by the author [Math. Nachr. 191, 19-58 (1998; see the following review)].
scientific article; zbMATH DE number 1066028