Unipotent representations of complex semisimple groups
The authors give a generalization of families of representations, which were introduced by \textit{J. Arthur} [Lect. Notes Math. 1041, 1-49 (1984; Zbl 0541.22011)], of a complex semisimple Lie group G. Especially, they give a character formula for a finite set of the resulting representations which is called ''special unipotent representation''. Let B and K be a Borel and a maximal compact subgroup of G, respectively, and put \(H=Z_ G(B\cap K)\). Suppose that \(\lambda,\mu\in {\mathfrak h}^*\) and \(\lambda-\mu\) is a weight of a finite dimensional holomorphic representation of G. Let \(\bar X(\lambda,\mu)\) be the Langlands subquotient of the K-finite part of \(Ind^ G_ B {\mathcal C}(\lambda,\mu)\), where \({\mathcal C}(\lambda,\mu)\) is the trivial extension on B of the character of H with differential \((\lambda,\mu)\). Then a nilpotent orbit \({\mathcal O}\in {\mathfrak g}^*\) is called ''special'' if there is \(X=\bar X(\lambda,\mu)\) such that (a) \(WF(X)={\mathcal O}\), (b) \(\lambda,\mu\) are integral, where WF(X) is the wavefront set of X. Let \(^ L{\mathfrak g}\) be the dual Lie algebra of \({\mathfrak g}\). Then there exists an order-reversing bijection \(\eta: {\mathcal O}\mapsto^ L{\mathcal O}\) between special nilpotent orbits in \({\mathfrak g}\) and \(^ L{\mathfrak g}\). Suppose that \({\mathcal O}\) is a special nilpotent orbit with \(^ L{\mathcal O}\) even. Then a ''special unipotent representation'' attached to \({\mathcal O}\) is an \(X=\bar X(\lambda,\mu)\) such that (a) \(WF(X)={\mathcal O}\), (b) \(\lambda,\mu\) are both conjugate to \(\lambda_{{\mathcal O}}\) under W, where W is the Weyl group of \((g,h)\) and \(2\lambda_{{\mathcal O}}\in {\mathfrak h}^*\) corresponding to the semisimple element \(^ Lh\) in \(^ L{\mathfrak h}\) attached to \(^ L{\mathcal O}.\) The main result can be stated as follows. First there exists a bijection \(\pi \mapsto X_{\pi}\) between the set of irreducible representations of the finite group \(\bar A({\mathcal O})\), the Lusztig quotient of the group of components of the centralizer of an element of \({\mathcal O}\), and the set of special unipotent representations of G attached to \({\mathcal O}\). We put \[ R_ x=| W_{\lambda_{{\mathcal O}}}|^{-1}\sum_{w\in W}tr(\sigma_ x(w)) X(\lambda_{{\mathcal O}},w\lambda_{{\mathcal O}}), \] where \(W_{\lambda_{{\mathcal O}}}=\{w\in W\); \(w\lambda_{{\mathcal O}}=\lambda_{{\mathcal O}}\}\) and \(\sigma_ x\in \hat W\) corresponds to [x]\(\in [\bar A({\mathcal O})]\), the set of conjugacy classes in \(\bar A({\mathcal O})\). Then the character formula is given as follows: \[ X_{\pi}=| \bar A({\mathcal O})|^{-1}\sum_{[x]\in [\bar A({\mathcal O})]}tr \pi(x) | [x] | R_ x \] and \[ R_ x=\sum_{\pi \in (\bar A({\mathcal O})){\hat{\;}}}tr \pi (x) X_{\pi}. \] If \(\pi\) is trivial, \(X_{\pi}\) is one of Arthur's representations.
scientific article; zbMATH DE number 3931302