Elliptic representations for \(\text{Sp}(2n)\) and \(\text{SO}(n)\)
Let \(G\) be a connected, reductive \(p\)-adic group and \(G^ e\) the set of regular elliptic elements. This is an open subset of \(G\). Let \(\pi\) be an irreducible tempered representation of \(G\) with character distribution \(\Theta_ \pi\) and write \(\Theta^ e_ \pi\) for the restriction to \(G^ e\). The representation \(\pi\) is said to be elliptic if \(\Theta^ e_ \pi\) is nonzero. In the line of the philosophy of cusp forms lies a result by \textit{D. Kazhdan} [J. Anal. Math. 47, 1-36 (1986; Zbl 0634.22009)], who showed that an irreducible tempered representation is elliptic just in case it is not an integer linear combination of induced representations. In the paper under consideration this is sharpened for \(G=\text{Sp}(2n)\) and \(G=\text{SO}(2n+1)\) to the assertion that every irreducible tempered representation is either elliptic or is induced from an elliptic representation. It is further shown that this fails for \(G=\text{SO}(2n)\). In this case there are irreducible tempered representations which are not induced representations and are not elliptic.
scientific article; zbMATH DE number 540674