Unipotent representations of complex semisimple groups. (English) Zbl 0582.22007

Barbasch, Dan; Vogan, David A.

Ann. Math. (2) 121, 41-110 (1985).

The authors give a generalization of families of representations, which were introduced by 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 \(\mathrm{Ind}^G_B {\mathcal C}(\lambda,\mu)\), 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 \(\text{WF}(X)\) 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) \(\text{WF}(X)=\mathcal O\), (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 \[ R_x=| W_{\lambda_{\mathcal O}}|^{-1}\sum_{w\in W} \operatorname{tr}(\sigma_x(w)) X(\lambda_{\mathcal O},w\lambda_{\mathcal O}), \] 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: \[ X_{\pi}=| \bar A({\mathcal O})|^{-1}\sum_{[x]\in [\bar A(\mathcal O)]}\operatorname{tr}\pi(x) | [x] | R_x \] and \[ R_x=\sum_{\pi \in (\bar A(\mathcal O)){\hat{\;}}}\operatorname{tr} \pi (x) X_{\pi}. \] If \(\pi\) is trivial, \(X_{\pi}\) is one of Arthur’s representations.

Reviewer: Takeshi Kawazoe (Yokohama)

MSC:

22E46	Semisimple Lie groups and their representations
20G05	Representation theory for linear algebraic groups
17B08	Coadjoint orbits; nilpotent varieties
22E50	Representations of Lie and linear algebraic groups over local fields

Keywords:

complex semisimple Lie group; character formula; unipotent representation; Langlands subquotient; nilpotent orbit; wavefront set; Weyl group; Lusztig quotient

Citations:

Zbl 0541.22011