Tempered representations of p-adic groups: special idempotents and topology

Abstract

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 ${\mathcal {H}}={\mathcal {H}}(G)$ and the Schwartz algebra ${\mathcal {S}}={\mathcal {S}}(G)$ forming abelian categories ${\mathcal {M}}(G)$ and ${\mathcal {M}}^t(G)$, respectively. Idempotents $e\in {\mathcal {H}}$ or ${\mathcal {S}}$ define full subcategories ${\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}$ and ${\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}$. Such an e is said to be special (in ${\mathcal {H}}$ or ${\mathcal {S}}$) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for $e\in {\mathcal {H}}$ we will prove that, for special $e \in {\mathcal {S}}$, ${\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )$ is a finite direct product of component categories ${\mathcal {M}}^t(\Theta )$, now referring to connected components of the center of ${\mathcal {S}}$. A special $e\in {\mathcal {H}}$ will be also special in ${\mathcal {S}}$, but idempotents $e\in {\mathcal {H}}$ not being special can become special in ${\mathcal {S}}$. To obtain conditions we consider the sets $\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)$ of (tempered) smooth irreducible representations of G, and we view $\mathrm{Irr}(G)$ as a topological space for the Jacobson topology defined by the algebra ${\mathcal {H}}$. We use this topology to introduce a preorder on the connected components of $\mathrm{Irr}^t(G)$. Then we prove that, for an idempotent $e \in {\mathcal {H}}$ which becomes special in ${\mathcal {S}}$, its support $\theta _e$ must be saturated with respect to that preorder. We further analyze the above decomposition of ${\mathcal {M}}_e^t(G)$ in the case where G is k-split with connected center and where $e = e_J \in {\mathcal {H}}$ is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support $\theta _{e_J}$ to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case $G=GL_n$, where $\theta _{e_J}$ 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.