Resumen de Tempered representations of p-adic groups: special idempotents and topology
Peter Schneider, Ernst Wilhelm Zink
Let G = G(k) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra H = H(G) and the Schwartz algebra S = S(G) forming abelian categories M(G) and Mt (G), respectively. Idempotents e ∈ H or S define full subcategories Me(G) = {V : HeV = V} and Mt e(G) = {V : SeV = V}. Such an e is said to be special (in H or S) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for e ∈ H we will prove that, for special e ∈ S, Mt e(G) = ∈θe Mt () is a finite direct product of component categoriesMt (), now referring to connected components of the center of S. A special e ∈ H will be also special in S, but idempotents e ∈ H not being special can become special in S. To obtain conditions we consider the sets Irrt (G) ⊂ Irr(G) of (tempered) smooth irreducible representations of G, and we view Irr(G) as a topological space for the Jacobson topology defined by the algebra H. We use this topology to introduce a preorder on the connected components of Irrt (G). Then we prove that, for an idempotent e ∈ H which becomes special in S, its support θe must be saturated with respect to that preorder. We further analyze the above decomposition of Mt e(G) in the case where G is k-split with connected center and where e = eJ ∈ H is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support θeJ to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case G = G Ln, where θeJ identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.
