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**Characters of non-connected, reductive p-adic groups.**
*(English)*
Zbl 0629.22008

Let G be a reductive group over a p-adic field of characteristic 0. Harish-Chandra has shown that, for connected G, the distribution character of an irreducible admissible representation is given by a locally integrable function which is locally constant on the regular set. The author shows that this is also the case for non-connected reductive groups. One gets precise information concerning the singularities of the characters, just as in the connected case.

This extension is needed in work of Arthur and Clozel on base change. In particular, certain applications of the trace formula to the study of automorphic forms come from the consideration of outer automorphisms. For example, the “twisted orthogonality relations” arising in base change may be considered as ordinary orthogonality relations between elliptic characters of a non-connected group given as a semi-direct product of a connected group by a Galois automorphism.

The generalization to the non-connected case is an easy extension of Harish-Chandra’s methods. However, since the original article of Harish-Chandra [Lie Theor. Appl., Proc. Ann. Semin. Can. Math. Congr., Kingston 1977, 281-347 (1978; Zbl 0433.22012)] does not contain proofs of crucial lemmas (which do appear in unpublished notes), it is not clear from existing literature why the theorem extends.

Let Ad denote the action of G on its Lie algebra. Define the rank \(r_ i\) on each (Zariski) connected component \(G^ i\) of G to be the first non-zero power of T in the polynomial \(P_ i(T)=\det (T-Ad(g)+1),\) for g in \(G^ i\). Define the discriminant \(D_ G\) on each component \(G^ i\) to be the coefficient of \(T^{r_ i}\). An element \(\gamma\) of G is called regular if \(D_ G(\gamma)\neq 0\). First, one uses elementary representation theory of G, as in Harish-Chandra, to show that the distribution character of an irreducible admissible representation \(\pi\) with a non-zero \(K_ 0\)-fixed vector is \((G,K_ 0)\)-admissible at each point (a somewhat technical condition).

The main theorem is proved by applying the following to the distribution character \(\Theta_{\pi}\). Let \(\Theta\) be any admissible distribution on an open G-invariant subset U. Let \(\gamma \in G^ i\) be a semi-simple element of U and assume that \(\Theta\) is admissible at \(\gamma\). Then \(\Theta\) coincides with a locally integrable function in a neighborhood of \(\gamma\). Further, \(\Theta\) has a germ expansion described in terms of the Fourier transform of nilpotent orbits in a Lie algebra.

Let M be the connected component of the centralizer of \(\gamma\) in G. Then there exist unique complex coefficients \(c_{{\mathcal O}}\) such that \[ \Theta (\gamma \exp Y)=\sum c_{{\mathcal O}} \hat v_{{\mathcal O}}(Y) \] for Y close to 0 in Lie(M), where \({\mathcal O}\) ranges over the set of nilpotent M-orbits in Lie(M), \(v_{{\mathcal O}}\) is the invariant measure on the orbit, and \(\hat v{}_{{\mathcal O}}\) is its Fourier transform.

This extension is needed in work of Arthur and Clozel on base change. In particular, certain applications of the trace formula to the study of automorphic forms come from the consideration of outer automorphisms. For example, the “twisted orthogonality relations” arising in base change may be considered as ordinary orthogonality relations between elliptic characters of a non-connected group given as a semi-direct product of a connected group by a Galois automorphism.

The generalization to the non-connected case is an easy extension of Harish-Chandra’s methods. However, since the original article of Harish-Chandra [Lie Theor. Appl., Proc. Ann. Semin. Can. Math. Congr., Kingston 1977, 281-347 (1978; Zbl 0433.22012)] does not contain proofs of crucial lemmas (which do appear in unpublished notes), it is not clear from existing literature why the theorem extends.

Let Ad denote the action of G on its Lie algebra. Define the rank \(r_ i\) on each (Zariski) connected component \(G^ i\) of G to be the first non-zero power of T in the polynomial \(P_ i(T)=\det (T-Ad(g)+1),\) for g in \(G^ i\). Define the discriminant \(D_ G\) on each component \(G^ i\) to be the coefficient of \(T^{r_ i}\). An element \(\gamma\) of G is called regular if \(D_ G(\gamma)\neq 0\). First, one uses elementary representation theory of G, as in Harish-Chandra, to show that the distribution character of an irreducible admissible representation \(\pi\) with a non-zero \(K_ 0\)-fixed vector is \((G,K_ 0)\)-admissible at each point (a somewhat technical condition).

The main theorem is proved by applying the following to the distribution character \(\Theta_{\pi}\). Let \(\Theta\) be any admissible distribution on an open G-invariant subset U. Let \(\gamma \in G^ i\) be a semi-simple element of U and assume that \(\Theta\) is admissible at \(\gamma\). Then \(\Theta\) coincides with a locally integrable function in a neighborhood of \(\gamma\). Further, \(\Theta\) has a germ expansion described in terms of the Fourier transform of nilpotent orbits in a Lie algebra.

Let M be the connected component of the centralizer of \(\gamma\) in G. Then there exist unique complex coefficients \(c_{{\mathcal O}}\) such that \[ \Theta (\gamma \exp Y)=\sum c_{{\mathcal O}} \hat v_{{\mathcal O}}(Y) \] for Y close to 0 in Lie(M), where \({\mathcal O}\) ranges over the set of nilpotent M-orbits in Lie(M), \(v_{{\mathcal O}}\) is the invariant measure on the orbit, and \(\hat v{}_{{\mathcal O}}\) is its Fourier transform.

Reviewer: C.D.Keys

### MSC:

22E35 | Analysis on \(p\)-adic Lie groups |

22E50 | Representations of Lie and linear algebraic groups over local fields |

11F70 | Representation-theoretic methods; automorphic representations over local and global fields |